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Chapter II Spectrum (Local Theory)
47
Spectrum
(Local T h e o r y )
CHAPTER I 1
1. Spectrum of an element. Spectral radius Given any algebra with an identity element (no topology is considered), one sets the following.
Definition 1.1. Let E be an algebra with an identity element and x any element of E. Then, the spectmcm of x (denoted by SpE1xl) is the set of those complex numbers A , for which the element A-x in E does not have an inverse. Thus, one has, by definition, (1.1)
SpElxl = { X e C : A-x
is not invertible}.
Concerning the above notation, A-x stands, of course, for the element A.iE-x in t ' , where lE denotes the identity element of E . NOW, by ( 1 . 1 )
,
t h e zero element of @ belongs t o t h e spectrum of x i f , and
only i f , x i s a singular (i.e., non-invertible) element o f E. On the other hai,d, on the basis of the "circle operation" consi-
dered above (cf.Chapt. I; ( 6 . 7 ) ) ,
one extends the previous definition
to the case of an algebra which does not necessarily possess an identity element. Thus, we have the next.
Definition 1.2. Let E be an algebra and x any element of E . Then, the spectrum of x is the set of those non-zero complex numbers h (not. f o r which the element +z e E does not have a quasi-inverse in E , plus the zero element of @, unless x is regular (i.e., invertible).So
C,),
one has, by definition, (1.2)
SpE(xi = { AeC, : Ix x E E is not quasi-invertible]
{ o f @ , if x is singular}. In this respect, i f t h e algebra E does not have an i d e n t i t y element, we agree t h a t every xe E i s s i n g u l a r , so t h a t Oe SpE(x).
Thus, t h e above two d e f i n i t i o n s a r e , i n f a c t , equiualent i n ease t h e given algebra E does possess an i d e n t i t g element: This is easily seen, since a com-
48
11 SPECTRUM (LOCAL THEORY) 1
p l e x X EC, belongs t o SpE(x) i f , and only i f , - x
i s quasi-singular (i.e., does not have a quasi-inverse); now, this is readily proved by the relation
x
x - x = A ( l - - 1x )
(1.3)
A
(here the distinction between l e IR and 1 for l E € E is clear) and the fact that x o y = 0 i f , andonly i f ,( l - z ) ( l - y l = l ,
(1.4)
for any x , y in E. In connection with the previous Definition 1.2, we still note that it is also common to define the spectrum of an element x in an algebra E without identity, as the spectrum of x in the algebra E B C , obtained from E by adding an identity element (cf.Chapt.1;Section 5). Thus, if the algebra E does not have an identity element, the twosets SpE(xi and SpE8@ (x), with x e E , are naturally related (in fact, coincide). That is, we have.
Lemma 1 . I . Let E be an algebra (which does not necessarily have an identity element) and l e t E B C be t h e algebra obtained from E by "adding an i-
.
d e n t i t y " (cf Chapt. I; Section 5 I .
Then, one has
(1.5) f o r every element x i n E , where t h e two members 01 t h e l a s t r e l a t i o n are defined b y ( 1 . 2 ) and ( 1 . 1 ) , r e s p e c t i v e l y .
Proof. By the comments after Definition 1.2, one concludes that SpEeC (x) (which is given by (1.I) ) coincides with SpEec ( ix,O)) ( cf. (1.2)), for every element x in E l identified wi.th ( x , O l € E B C . Therefore, it suffices to prove that t h e l a s t s p e c s r m above coincides w i t h s p ( x l E
cJhich i s given by ( 1 . 2 ) . Thus, we first remark that a v e q element x s E i s a singular element o f t h e algebra E 8 C . ( Indeed, otherwise the relation (x,O). ( y , X I = ( x y +Ax,Ol =(O,l),
for some (y,X)€ E B C , would imply 0 = 1 i n C ,
.
which is acontradiction) Hence, 0 6 SpEBC(xl = SpEBC ( (x,O) I and similarly, 0 E SpE(xI (since we regard t h e algebra E a s lacking an i d e n t i t y e l e rnent; cf. also the comment after the en2 of this proof). Furthermore,
one verifies that an element x EE i s quasi-singular i n E i f , and only i f , (x,O) is quasi-singular i n E 8 C . (Otherwise, the relation ( x , O ) o ( y , A ) = (y,X)o(x,Ol =(O,O/, far some ( y , X I E E @ C , would yield x o y = y o x = 0,which is a contradiction to the hypothesis for x ; and conversely). NOW, the argument, as that after Definition 1.2, applied to the element with X#0, provides the desired conclusion. I
same
1 --€El
X
In connection with the preceding lemma, we note that if the gi-
SPECTRUM. SPECTRAL RADIUS
1.
49
ven a l g e b r a E has a n i d e n t i t y e l e m e n t , t h e n , s i n c e e v s r y e l e m e n t x € B
i s s i n g u l a r when c o n s i d e r e d a s a n e l e m e n t o f E B C (see a l s o t h e a b o v e , o n e c o n c l u d e s t h a t U ESpE 8 C ( x ) ; h o w e v e r , t h i s may n o t be t h e case f o r SpE(x), i f x h a p p e n s t o b e a r e g u l a r e l e m e n t o f E . I n t h a t ca-
proof)
se ( 1 . 5 ) i s n o l o n g e r t r u e , h e n c e , i f E has an i d e n t i t y element one g e t s , i n
general, t h e r e l a t i o n
=
s pE (2) 3pE@@ix)3
(1.6)
for every xEE. On t h e o t h e r h a n d , t h e f o l l o w i n g r e s u l t p r o v i d e s a c o n n e c t i o n b e t w e e n t h e s p e c t r a o f e l e m e n t s o f a g i v e n a l g e b r a a n d t h o s e of t h e s e e l e m e n t s i n a b i g g e r a l g e b r a . Namely, w e h a v e . P r o p o s i t i o n 1 . 1 . Let E , F be tuo algebras and b : E + F
an (algebra) mor-
ph.ism. Then, one has t h e r e l a t i o n
SPFl$iZi)E SPE(XCl,
(1.7)
f o r every element x i n E.
(The members o f t h e l a s t r e l a t i o n a r e d e f i n e d
.
b y t h e r e s p e c t i v e r e l a t i o n s t o ( 1 . 2 ) a b o v e ) In p a r t i c u l a r , i f E i s a sub-
algebra o f F, then one 7 b t a i n s (1
.a)
SPF(XI c SPEiX),
f o r every element x i n E. (The r e s p e c t i v e s p e c t r a a r e a l w a y s u n d e r s t o o d i n t h e sense of Definition 1 . 2 ) .
Proof. By Lemma 1 . 1
(see also i t s p r o o f ) , one h a s t h e r e l a t i o n
SPF(@(Xl)= SPF9@i!@(X/,0))
(1.9)
f o r every element
So a c o m p l e x number A # 0 b e l o n g s t o S p F ( @ ( d )
E.
J: f
i f , a n d o n l y i f , t h e e l e m e n t A-Qixi ( i . e . , A (O,l)-(@(x),O)) i n F BC
,
which i s t h e case o n l y i f A - 2 1
EBC:In fact,
i f -xox'=
t h e s i s f o r @,
one h a s
A
x @ix) 1
1 x'o-~:=O,
x
0
is singular ( i . e . , A(O,l)-(x,U)) i s s i n g u l a r i n
f o r some x ' ~ E , t h e n by h y p o 1
$ ( x ' ) = @ ( x ' l o -@(x)
A
=0 1
( i . e . , an algebra morphism "preserves t h e c i r c l e o p e r a t i o n " ) . Hence, -$(x) A
is
i n F o r , e q u i v a l e n t l y , A - $(x) i s
regular i n F @ C , thus a 1 c o n t r a d i c t i o n t o t h e h y p o t h e s i s f o r @(I) E F . T h e r e f o r e , - x is q u a s i -
quasi-regular
singular i n E ,
A
so t h a t A ~ S p ~ i M d .o r e o v e r , i f 0 6 S p F ( $ f x ) ) , t h e n $ ( x ) i s
s i n g u l a r i n F; h e n c e , by t h e f o r e g o i n g , x i s s i n g u l a r i n E as that is,
OeSpE(x), a n d
Now,
well,
t h i s finishes t h e proof. I
a n i m p o r t a n t n o t i o n which i s c o n n e c t e d w i t h t h e s p e c t r u m
of a n e l e m e n t i n a g i v e n a l g e b r a i s t h a t p r o v i d e d by t h e n e x t .
50
I1 SPECTRUM (LOCAL THEORY) D e f i n i t i o n 1.3. Given a n a l g e b r a E and an e l e m e n t x e E l one d e f i -
n e s t h e spectral radius o f x
( d e n o t e d by r E ( x ) ) , by t h e r e l a t i o n
(1.10)
w e g e t , of course, t h e r e l a t i o n
Thus, by ( 1 . 1 0 ) and Lemma 1 . 1 ,
r (xl =
(1.11)
f o r every element x e E NOW,
l e m e n t of
E
.
P
Ef3cl:(x)
it i s c l e a r by ( 1 . 1 0 ) t h a t r (xJ may b e , i n g e n e r a l , a n eE Et ( : = IR, U {tm}), s o t h a t it i s i m p o r t a n t t o know t h a t i n a
g i v e n a l g e b r a E l r (zl i s , i n E
f a c t , a p o s i t i v e r e a l number. O f c o u r s e ,
t h i s i s t r u e i n e v e r y Banach a l g e b r a a n d , a s w e s h a l l s e e , t h i s i s a l -
so t h e case i n more g e n e r a l t o p o l o g i c a l a l g e b r a s ( Q - a l g e b r a s , f o r i n stance; c f . Corollary 4 . 5 ) . F i n a l l y , as we s h a l l see i n t h e e n s u i n g d i s c u s s i o n , t h e " a l g e
-
b r a i c " d e f i n i t i o n of s p e c t r a l r a d i u s , g i v e n by ( l . l O ) l h a s f o r s u i t a b l e t o p o l o g i c a l a l g e b r a s a l s o a " t o p o l o g i c a l " c o u n t e r p a r t (see, f o r i n s t a n c e , Theorem 7.2
and a l s o t h e n e x t c h a p t e r ) .
2 . The r e s o l v e n t s e t I n c o n n e c t i o n w i t h t h e n o t i o n of t h e s p e c t r u m of a n e l e m e n t
3:
i n a g i v e n a l g e b r a E , it i s s t i l l of i m p o r t a n c e t h e complement o f t h e s p e c t r u m i n 02, a n o t i o n which w e s h a l l a l s o a p p l y i n t h e s e q u e l . T h u s ,
w e f i r s t give the respective formal d e f i n i t i o n . T h u s , g i v e n a n a l g e b r a E w i t h an i d e n t i t y e l e m e n t , one d e f i n e s t h e resolvent s e t o f an e l e m e n t x e E
( d e n o t e d by p i x ) )
t o b e t h e com-
p l e m e n t i n C o f t h e s p e c t r u m of LC.T h a t i s , o n e h a s , by d e f i n i t i o n pix) =
(2.1)
c SPE(Xl .
I n o t h e r w o r d s , t h e s e t p i x ) c o n s i s t s o f t h o s e complex numbers A,
for
which t h e r e s p e c t i v e e l e m e n t A - x E E i s i n v e r t i b l e . T h e r e f o r e ,
one d e f i n e s t h e f o l l o w i n g E-vaZued map R(x;XI =
(2.2)
with
h
E pix);
(
A-xI-'
,
w e c a l l i t t h e resolvent f u n c t i o n
of t h e e l e m e n t x e 0". S o
i f G d e n o t e s t h e g r o u p of i n v e r t i b l e e l e m e n t s of E l one h a s
~ 1 x 1= R ( x ;
(2.3)
for
every element
ZE
E
'
)-I (Gl ,
.
The n e x t l e m m a ( f i r s t resolvent e q u a t i o n ) w i l l b e a p p l i e d below.
3.
Lemma 2.1. Let
ALGEBRAS WITH C O N T I N U O U S I N V E R S I O N
51
E be an algebra w i t h an i d e n t i t y element and x an element of
E. Then, t h e resolvent f u n c t i o n of x s a t i s f i e s t h e r e l a t i o n R(x;h)
(2.4)
-
R(x;p) =
- (A - p l
R ( x ; A l R ( ~ ; p l,
f o r every p a i r ( A , u i of elements of pix). Proof. At first a routine verification shows that R (x;X I .R(x; p) = R(x;p l .R(x; A/ ,
(2.5)
f o r any A , p in p i x ) . Therefore, one obtains R(x;Al-R(x;ul
- (h-21
= R l x ; h ) E l ' ~ ; p ) (( V - X )
)
= -IX-pIRlx;hlR(a;p),
which is the desired relation (2.4). I In this respect, we still note that,besides the relation (2.4) above,there is another formula referring to the resolvent function for different values of (the variable) x e E ; (we consider thus R ( x ; X I as a function of x too). This is valid f o r any algebra E with an i d e n t i t y element. Namely, it is easily proved first that (X-X)
(2.6)
( R l z ; A)
-
R(y;X)I
(A- y l =
3:
-y
,
for any x, y in E and A in both of the resolvent sets p(xl and ~ ( L J ) ; thus, multiplying the last relation on the left by R ( x ; X ) and on the right by R ( y ; X) , one obtains (2.7)
R(x;hi -R(y;X)
for every A
E
also E. HILLE
= R(z;X/(x- y)Rly;A),
p i x i n p(y1 and any 2 , in - R.S. PHILLIPS [I ; p. 126 ff.]).
(
second resolvent equation ; see
3. Topological algebras w i t h continuous inversion Our task in this and the following section is to examine towhat extent certain fundamental properties connected with the notions we have already defined above (cf. Section 1 ) and which are valid, of course, in every Banach algebra, are still true for appropriate topological algebras. Thus, we start with the following.
Definition 3.1. Given a topological algebra E with an identity element, we shall say that E has a continuous i n v e r s i o n , whenever the ( E valued) map z-x-'
is continuous on its domain of definition (i.e., the set of regular elements of E ) . On the other hand, a topological algebra E is said to have a continuous quasi-inversion , if the quasi-inversion in E defined by (Chapt.
52
11 SPECTRUM (LOCAL THEORY)
I;( 6 . 6 ) ) is a continuous (E-valued) map on the set of quasi-regular elements of E . In this respect, we first have the next.
Lemma 3.1. Let
E be a l o c a l l y rn-convex algebra with an i d e n t i t y element. T??en,
E has a continuous inversion. More generally, every l o c a l l y m-convex algebra has a
continuous quasi-inversion. Proof. This goes on verbatim as in the case of a normed algebra, by considering instead of a norm a family of submultiplicative seminorms defining the topology of E (cf. Chapt.1; Proposition 3.2):Thus.
see, for instance, C.E. RICKART [l :p. 13, Theorem 1.4.8, and p. 19, Theorem 1. 4.221. I The inversion in a given topological algebra with an identity element may be continuous without the algebra to be locally m-convex,
as this is shown by the following result. In fact, the same result may be considered as a rather concrete instance of a more general (nonetheless, more technical) situation (see the concluding Remark below). Thus, we have.
Theorem 3.1. Let
E be a metrizable topological algebra with an i d e n t i t y e l e -
ment, for which the group G of regular elements i s , i n the r e l a t i v e topology, a separable Baire space. Then, the inversion i n E i s a continuous (E-viluedl map on G,
so that G i s , i n p a r t i c u l a r , a topological group with respect t o the r e l a t i v e topology * Proof. By hypothesis the group G is a Baire space which is also
metrizable and separable, hence second countable; thus, the assertion is now a consequence of a known relevant (and more general) result (cf. T . HUSAIN [I : p. 3 6 , Theorem 8 and p. 37, Theorem 9 1 ) . I
As an application, one also gets the following.
Corollary 3.1. Let F be a Frgchet algebra (cf. Chapt.1;Definition 1 . 5 1 with an i d e n t i t y element for which the group G of regular elements i s a se,arable G -set.Then,
6
E has acontinuous i n v e r s i o n , so t h a t G i s , i n p a r t i c u l a r , a topologi-
caZ group i n t h e r e l a t i v e topology. Proof. The assertion follows by the previous theorem and the fact that G, being by hypothesis a G6-set in E , is "topologically complete"
(Alexandroff;cf. , for instance, J.G. HOCKING-G.S. YOUNG [ I : p. 85, Theorem 2-76]), hence a Baire space. I On the other hand, one also obtains.
3.
Corollary 3.2. Let
ALGEBRAS WITH CONTINUOUS INVERSION
53
E be a Baire metrizable Q-algebra w i t h an i d e n t i t y e l e -
ment such t h a t i t s group G o f regular elements i s a separable subspace. Then, G i s i n t h e r e l a t i v e topology a topological group, so t h a t E has, moreover, a continuous inversion. I n p a r t i c u l a r , a Frgchet separable 4-algebra w i t h an i d e n t i t y element i s a topological algebra w i t h a continuous i n v e r s i o n , having t h e group o f regular e l e ments a topological group i n t h e r e l a t i v e topology. Proof. By hypotesis G is an open subset of E
(cf. Chapt. I ; Definition 6 . 2 ) , hence a Baire space too, which is also separable, so that the first assertion follows by Theorem 3.1. On the other hand, if E is, moreover, a separable Q-algebra, then the group G of regular elements, being an open subset of E, is also a separable subspace; thus, one is led to the previous case, and this finishes the proof. I
Remark.- According to the result of T . HUSAIN [I] applied in the proof of Theorem 3 . 1 , a group G , endowed with a topology T making the (group-operation) "multiplication in G " separately continuous (i.e., a semi-topological group) becomes a topological group, if G i s a normal Baire second countable space in the topology T. This occurs, in particular, by the hypothesis of Theorem 3 . 1 and its corollaries above. Moreover, by a known result of D. MONTGOMERY [I :p. 881 , Theorem 21 , a semi-topological group whose underlying topological space is complete metrizable and separable, is a topological group. Now this condition is satisfied, of course, if one has
il
Frgehet algebra E w i t h an i d e n t i t y element
and group G o f i n v e r t i b l e elements a separable G - s e t .
6 ilore generally, the last conclusion is valid, if E is a metrizable (topological) algebra with an identity element and G locally complete and separable. In this respect, we still note that by T . HUSAIN [I: p. 3 6 , Theorem 71 (cf. also I".+. WU [I: p . 452, Theorem] ), one concludes that: Every topological algebra E w i t h an i d e n t i t y element and group G of regular elements a Baire second countable space, has a continuous i n v e r s i o n , G becoming t h u s a topological group i n t h e r e l a t i v e topology.
In particular, t h e l a s t conclusion i s v a t i d f o r a Baire Q-algebra E w i t h an i d e n t i t y element and G second countable o r , more p a r t i c u z a r l y , if E i s a Baire metrizable separable Q-aZgebra w i t h an i d e n t i t y element.
In conclusion, some extra conditions of the kind considered above seems to be necessary, as it concerns the group G of regular elements of a topological algebra E (not locally m-convex!) withan identity element, if one wants G to be a topological group, hence E to ha-
54
I1 SPECTRUM (LOCAL THEORY)
ve a continuous inversion. So this need appears even if E is a F r g c h e t locally convex algebra, as this is indicated by the "Arens algebra" Lw f [ O , l ] l (cf. Chapt. I; Subsection 2. ( 4 ) ) , which does not have a coztinuous inversion (cf. R. ARENS [4: p. 6301 ) . Now, topological algebras exhibiting the properties of the algebras appeared in the second part of Corollary 3 . 2 will play an important role in the sequel,therefore, they are singled out by the next section.
4. Waelbroeck algebras The algebras in title of this section seems to be the appropriate ones for an important branch of the whole theory of topological algebras, as it is the "Functional Calculus" (cf. Chapt.VI;Section 3 , as well as Chapt.VIII;Section 8). They have been applied in this context (in particular, as locally convex and/or locally m-convex ones) already in the early 5 0 ' s by L . WAELBROECK [ I ; 2 1 . On the other hand, the same algebras, at least the locally mconvex ones, seems to constitute that particular class of topological algebras in which one can find many of the fundamental results of the classical Banach algebras theory. This will become clear in several instances throughout the ensuing discussion. Thus, the same class of algebras plays a significant role too in other parts of the theory of topological algebras or of its applications, whose discussion, however, would be beyond our scope in this book. ( But see, for instance, A. MALLIOS [20-24; 2 7 1 ) . We start with giving the relevant formal definition. Namely, we have. Definition 4.1, By a Waelbroeck aZgebra , we mean a topological algebra E with an identity element, in such a way that the group G of the invertible elements is an open subset of E and the inversion a continuous (E-valued) map on G.
In other words, a Waelbroeck aZgebra is, by definition, a Q-aZgebrn w i t h n continuous inversion. Furthermore, by the foregoing (see the last Xemark above) , a separable Baire (hence, in particular, a separabZe Frgchet) &-algebra w i t h an i d e n t i t y element isa Waelbroeck algebra. Now, the following variant of Lemma I ; 6 . 4 is needed below, if one is going to apply Definition I ;6 . 2 . Thus , a topological aZgebra E with an i d e n t i t y element i s a 9-algebra (cf. Definition I;6.2) i f , and only i f , the
4. WAELBROECK ALGEBRAS
55
s e t of regular elements of E i s a neighborhood of t h e i d e n t i t y element of E:The assertion can easily be proved by suitably modifying the argument in the proof of Lemma I;6.4. We can now formulate another useful version of the above Defi-
nition 4 . 1 via the next.
Proposition 4.1. Let E be a topological aZgebra w i t h an i d e n t i t y element e . Then, t h e f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : 11 E i s a Waelbroeck algebra (Definition 4.1 ).
2) The group G of regular elements of E i s a neighborhood of e and t h e inver-
s i o n i n E i s continuous a t e . 31 The i d e n t i t y element e has a neighborhood c o n s i s t i n g of i n v e r t i b l e e l e -
rnen?s and the i n v e r s i o n i n E i s con-tinuous a t e . Proof. We first remark that 2 ) is equivalent to 3), while 1 ) implies 2) by Definition 4.1. Thus, based on the above comment and by assuming 2 ) , one concludes that E is a Q-algebra, so that it suffices to prove that the inversion in E is a continuous (E-valued) map on G ,
assuming its continuity at the point x = e . Indeed, if x e l ; , the map x b e + x - ' ( x - x O 1 , x E E , is continuous at x = x , so that :here exists O -7 a neighborhood U of zero in E , such that e +n: A i x - x 1 E G , €or every
x - x e U ; i.e., x e x O + U , where x + U is a neighborhood of xo in
E.
Thus,
due to the relation x = x0 i e + x0- ' i x - x O ) l
(4.1) valid for every
3:
e
E,
the map 3: + + i e + x - ' i x
(4.2)
,
-z0i)-',
defined for every x e x O +U , is by hypothesis continuous at refore, by (4.1), one obtains
3:
= :co
with r e x O +U , thus the desired continuity of the map x + - + x - ' , the point x = x and this finishes the proof. I
. The-
x e E , at
0'
On the basis of the previous proof, one concludes that cond.3) of Proposition 4 . 1 can be s t a t e d f o r every regular element 3: i n E , which is, in fact, an equivalent form of cond. 1 ) of the same proposition, i.e., of Definition 4.1. Now, as an application of the foregoing, we are in a positionto state the following. Theorem 4.1.
Let E be a Waelbroeck aZgebra and
3:
any element of E . Then, the
56
I1 SPECTRUM (LOCAL THEORY)
resolvent s e t
p1x) of the clement x i s an open subset of C and the respective re-
- J an E-valued
solvent f u n c t i o n R ( x ; the r e l a t i o n
holomorphic map on
p ( ~ ) ,which also s a t i s f i e s
lim RIx;AI = 0
(4.4)
A+
OJ
("bounded a t t h e point a t i n f i n i t y " o f C), so t h a t it i s , i n p a r t i c u l a r , a bounded (holomorphic) map on p l x l .
Proof. W e f i r s t remark t h a t R ( x ; X
i s continuous as a f u n c t i o n of A,
with X E p ( x I , b y ( 2 . 2 ) and t h e c o n t i n u i t y o f t h e i n v e r s i o n i n E. Hence, ( 2 . 3 ) , one c o n c l u d e s t h a t is an open s u b s e t o f C . On t h e o t h e r h a n d , t h e " f i r s t r e s o l v e n t e q u a t i o n " ( c f . Lemma 2 . 1 ) i m p l i e s t h a t
by h y p o t h e s i s and a p p l y i n g t h e n o t a t i o n of
pix)
(4.5)
1
A-
IR1x;AI- R(x;Xoll = - R ( x ; X I R ( ~ : ; A , ) .
A,
T h e r e f o r e , by p a s s i n g t o t h e l i m i t i n t h e l a s t r e l a t i o n f o r A+h w i t h i n p ( x ) , and
0'
t a k i n g t h e c o n t i n u i t y of t h e i n v e r s i o n i n E i n t o ac-
c o u n t , one o b t a i n s
t h u s , RIx;AI i s an E-valued d i f f e r e n t i a b l e map on pixi
,
h e n c e by d e f i n i t i o n a
h o l o m o r p h i c map on p ( x ) . NOW, s i n c e , f o r e v e r y x e E , t h e map a c r e - a x :
i s c o n t i n u o u s , t h e r e e x i s t s a n e i g h b o r h o o d NE ( U l o f z e r o i n C , with e-ax e G , f o r every a e N E ( 0 ) (see a l s o P r o p o s i t i o n 4 . 1 ) Conse -
C+E
.
quently, the relation
makes s e n s e , f o r
1
(x(
with a =-
1
X , i.e.,
l i m i t i n t h e l a s t r e l a t i o n f o r h + m i n @,
( X ( > E ; thus, by t a k i n g t h e
one g e t s by t h e c o n t i n u i t y
of t h e i n v e r s i o n i n E t h e d e s i r e d r e l a t i o n ( 4 . 4 ) . i o n i s now a c o n s e q u e n c e o f
(4.4)
S o t h e f i n a l assert-
and t h e c o n t i n u i t y of t h e map R ( x ; X ) ,
with X E p(xl. I
Scholium 4.1.-
I n t h e p r e v i o u s p r o o f an E-valued map d e f i n e d on an
open s u b s e t of C was t a k e n a s " h o l o m o r p h i c " , b e i n g d i f f e r e n t i a b l e on
i t s domain of d e f i n i t i o n ( a n d h e n c e c o n t i n u o u s too: t h i s w a s t h e case f o r t h e map R(x;XI b y ( 4 . 6 ) and ( 4 . 7 ) ) . NOW, t h i s r e m i n d s , o f c o u r s e , t h e c a s e E = C , namely, t h a t of complex-valued h o l o m o r p h i c maps, w h i l e t h i s i s s t i l l v a l i d i f one c o n s i d e r s t h e c o m p o s i t i o n s of a n E-valued h o l o m o r p h i c map a s above w i t h ( c o m p l e x - v a l u e d ) c o n t i n u o u s l i n e a r forms on t h e g i v e n t o p o l o g i c a l a l g e b r a E ,
i . e . , w i t h e l e m e n t s o f t h e topolo-
4. WAELBROECK
57
AJAGEBRAS
g i c a l dual E' of E . I n t h i s r e s p e c t , i t would b e t h u s o f i m p o r t a n c e t o h a v e i n E a "good s u p p l y " o f s u c h f o r m s , a s t h i s i s , f o r example, t h e
case i f E i s a l o c a l l y convex a l g e b r a (Hahn-Banach). B u t , a s w e s h a l l see i n s u b s e q u e n t s e c t i o n s , t h i s may s t i l l happ e n i n t o p o l o g i c a l a l g e b r a s which a r e n o t n e c e s s a r i l y l o c a l l y convex. Thus, what one r e a l l y n e e d s i n t h i s c o n t e x t i s
c(
topological algebra E
admitting a t o t a l s e t of continuous l i n e a r forms, i n t h e s e n s e t h a t , t h e r e e-
x i s t s a s u b s e t A of E ' , s u c h t h a t , f o r e v e r y e l e m e n t x E E , w i t h x # O , t h e r e e x i s t s a n e l e m e n t f e A , w i t h f(x)# 0 . T h u s , as a l r e a d y n o t e d , i f E i s , i n p a r t i c u l a r ,
a l o c a l l y con-
i s s u c h a t o t a l s e t f o r E (Hahn-Banach); i n f a c t , t h i s i s v a l i d f o r every subset A of E' whose linear hull i s "weakly dense" i n
vex a l g e b r a , t h e n E ' E', i.e.
,
i f A i s a total set
Finally,
i n E',
,
t h e "weak t o p o l o g i c a l d u a l " of E .
i n c o n n e c t i o n w i t h Theorem 4 . 1 ,
w e s t i l l remark t h a t
by t h e p r o o f o f t h e same t h e o r e m , t h e r e e x i s t s a n e i g h b o r h o o d of t h e "point a t i n f i n i t y " of C contained i n p ( x ) ,
so t h a t
p i x ) i s a neighbor-
hood of t h e p o i n t a t i n f i n i t y o f C . F u r t h e r m o r e , one c a n p r o v e t h a t R ( x ; X I i s holomorphic a t t h i s p o i n t t o o (see, f o r i n s t a n c e ,
M.A.
NAYMARK [I : p . 172,
111, and p. 66, 5 1 2 1 ) .
As a n a p p l i c a t i o n of t h e a b o v e d i s c u s s i o n , w e come now t o t h e f o l l o w i n g b a s i c r e s u l t which i s v a l i d , i n f a c t , f o r a n a p p r o p r i a t e
class of topological algebras ( e . g . ,
l o c a l l y convex o n e s , c f . C o r o l -
having a continuous i n v e r s i o n . That i s , w e g e t .
lary 4.1)
Theorem 4.2. Let
E be a topological algebra w i t h an i d e n t i t y element and con-
tinuous inversion admitting, moreover, a t o t a l s e t o f continuous l i n e a r forms ( c f . S c h o l i u m 4.1). Then, one has
SPE(XI # 0 ,
(4. 8)
f o r every element x i n E. Proof. S u p p o s e t h a t SpE(xLII = @ , f o r some x e E . T h u s ,
p(x)= a : ,
by ( 2 . 1 1 ,
so t h a t t h e r e s o l v e n t f u n c t i o n R ( x ; X I i s defined for every X E C
being, moreover, a holomorphic map ( c f . t h e Remark b e l o w ) . T h e r e f o r e , if A
cE'
i s a t o t a l s e t o f c o n t i n u o u s l i n e a r f o r m s on E l t h e n t h e
X++f(R(x;h))
,
map
w i t h f e A , i s a complex-valued h o l o m o r p h i c map on C (i.e.,
a n e n t i r e f u n c t i o n ) which i s a l s o bounded by ( 4 . 4 )
,
s i n c e f i s continu-
o u s . Hence ( L i o u v i l l e ' s T h e o r e m ) , t h e l a t t e r map i s c o n s t a n t , so t h a t by ( 4 . 4 ) (4.9)
one o b t a i n s
f(R1x;XII = f ( ( A - x ) - ' )
= 0 ,
58
with A
I1 SPECTRUM (LOCAL THEORY)
€@
,
and f o r e v e r y f E A . T h u s , by h y p o t h e s i s f o r A and ( 4 . 9 ) , one
o b t a i n s fA-x)-'=
0
,i.e. ,
a c o n t r a d i c t i o n t o o u r h y p o t h e s i s , and t h i s
terminates t h e proof. I
Remark.fact, a
I n t h e p r o o f o f t h e p r e v i o u s t h e o r e m w e have u s e d ,
in
strengthened form of Theorem 4 . 1 ; t h a t i s , concerning t h e r e s o l v e n t
f u n c t i o n R l x ; A ) , x € E , t h e analogous concZusion t o t h a t of Theorem 4.1 i s s t i l l val i d , f o r every element x , t h e r e s o l v e n t s e t of which i s an open subset of @. The a s s e r t i o n i s c l e a r by t h e a r g u m e n t i n t h e p r o o f of t h e same t h e o r e m , and i t w a s t h a t e x a c t l y , which h a s been a p p l i e d i n t h e p r o o f of t h e above Theorem 4 . 2 .
Of c o u r s e , it i s t r u e f o r e v e r y Waelbroeck a l g e b r a
(Theorem 4 . 1 ) Now, t h e n e x t f u n d a m e n t a l c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e above and Lemma 3.1.
Corollary 4.1. L e t
E be a l o c a l l y convex algebra w i t h an i d e n t i t y eZement
and continuous i n v e r s i o n . Then, t h e spectrum
SpEfx) of every element x i n E i s a
non-empty subset of C. In p a r t i c u z a r , f o r every l o c a l l y m-convex algebra E w i t h an i d e n t i t y element, every element x i n E has a non-empty s p e c t m SpEfxl.
Scholium 4.2.- I f a t o p o l o g i c a l a l g e b r a E d o e s n o t n e c e s s a r i l y hav e a n i d e n t i t y e l e m e n t , o n e c a n c o n s i d e r , o f c o u r s e , a k i n d of a gene-
raZized WaeZbroeck aZgebra,
i n t h e sense t h a t
E i s again a &-algebra
( c f . De-
f i n i t i o n I : 6 . 3 ) and t h e quasi-inversion i n E is continuous. I n t h i s r e s p e c t ,
by a p p l y i n g Lemma I ; 6 . 4 , tion 4.1
.
one o b t a i n s a n o b v i o u s a n a l o g u e o f P r o p o s i -
On t h e o t h e r hand , i f a generalized Waelbroeck aZgebra has already
an i d e n t i t y eZement, then i t i s a WaeZbroeck algebra ( D e f i n i t i o n 4.1); t h i s f o l -
l o w s by Lemma I ; 6 . 4
and P r o p o s i t i o n 4.1
(see a l s o ( 1 . 4 ) ) . Moreover,
t h e ( t o p o l o g i c a l ) adjunction of an i d e n t i t y ( c f . C h a p t . I ; 5. ( 1 ) ) t o a gene raZized Waelbroeck algebra makes it a WaeZbroeck algebra. Now by t h e f o r e g o i n g ( s e e a l s o D e f i n i t i o n 1 . 2 ) , one g e t s t h e f o l l o w i n g s t r e n g t h e n e d form of t h e above C o r o l l a r y 4 . 1 .
Corollary 4.2. Let E be a ZocalZy convex algebra w i t h a continuous quasi-inv e r s i o n . In p a r t i c u l a r , suppose t h a t E i s any locaZly m-convex algebra (cf.Lemm a 3.1 ). Then, i n e i t h e r c a s e , one has
0 # spE(x)G c
(4.10)
f o r every element x i n E . I I n t h i s r e s p e c t , one f u r t h e r o b t a i n s t h e f o l l o w i n g r e s u l t which
4. WAELBROECK
59
ALGEBRAS
p r o v i d e s supplementary i n f o r m a t i o n c o n c e r n i n g ( 1 . 8 ) . Thus, w e have.
Lemma 4.1.
Let E be a closed subazgebra o f a topoZogica1 algebra F , where F has an i d e n t i t y element and continuous i n v e r s i o n . Moreover, suppose t h a t , f o r ever y x i n E , SpE(x) i s a closed subset o f C ( t a k e , f o r i n s t a n c e , a Waelbroeck a l g e b r a E). Then, concerning t h e boundaries o f t h e spectra of a given element x
i n E and F, r e s p e c t i v e l y , one has t h e reZation bd ( SpE(xl I 5 bd (SpF(xi )
(4.11)
.
Proof. By h y p o t h e s i s , one h a s t h e r e l a t i o n bd ISpElx) I = Sprixcl f o r every x E E. G C,
n el”,
n
,
T h e r e f o r e , i f A e b d ( S p E ( x ) ), t h e n A = l i m A n , w i t h A n € p ( x )
so t h a t one o b t a i n s
y = l h ( A - x)= A-x E E . n n NOW, i f y i s i n v e r t i b l e i n F , t h e n y - I = lirn(An-xl-l , by h y p o t h e s i s ; t h u s , y = A-x i s a l s o i n v e r t i b l e i n E , a c o n t r a d i c t i o n , s i n c e X€SpE(x). Therefore, y = A-x
i s n o t i n v e r t i b l e i n F , s o t h a t A E + ~ ( x ), which p r o by ( 1 . 8 ) and ( 2 . 1 ) . I
ves the desired relation (4.111,
Remark.-
I f t h e a l g e b r a s E a n d F i n t h e above lemma do n o t h a v e
n e c e s s a r i l y a n i d e n t i t y e l e m e n t ( n a m e l y , t h e same o n e ) , t h e n under t h e
r e s t o f t h e c o n d i t i o n s o f Lemma 4 . 1 , one o b t a i n s (4.12)
b d ( SpE(xJIC b d ( SpF(x)),
for every x i n E. ( C o n s i d e r t h e a l g e b r a s E O C and
FOC and
apply t h e pre-
v i o u s a r g u m e n t ) . Now t h e l a s t r e l a t i o n i s f u r t h e r j u s t i f i e d , s i n c e t h e a l g e b r a E may h a v e a n i d e n t i t y e l e m e n t which w i l l be n o t a n i d e n t i t y f o r F. The n e x t lemma w i l l p r e s e n t l y b e needed and s u p p l e m e n t s Lemma I; 6.4,
p r o v i d i n g a n o t h e r u s e f u l c h a r a c t e r i z a t i o n of & - a l g e b r a s .
Lemma 4.2.
Let E be a topologicaZ algebra and consider t h e foZZowing subset
of E (E) = {x E E : r,(x)
(4.13)
5 1),
where rE(x) is g i v e n by ( 1 . 1 0 ) . Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a Q-algebra. 2)
SIE) i s a neighborhood o f zero i n E.
Proof. I f E i s a & - a l g e b r a , t h e n (Lemma I ; 6.41,
Gq
is a neigh-
borhood of z e r o i n E , so t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U
60
I1 SPECTRUM (LOCAL THEORY)
of 0 6 E c o n t a i n e d i n G q .
We s h a l l show t h a t U
t h e n -1x is h G q , because
quasi-singular,
S ( E I : I n d e e d , i f x E E and
1x1 > 1 . B u t , t h a t i s a c o n t r a d i c t i o n , s i n c e -1x f U C
r E ( x l > I , t h e n by ( 1 - 1 0 ) t h e r e would e x i s t
A e S p E ( x l , with
1 and I, i s b a l a n c e d . Thus,
l i e s 2 ) . On t h e o t h e r h a n d , i f 2 ) i s v a l i d , t h e n p o i n t of
s ( E l ; hence, t h e s e t
moreover i s c o n t a i n e d i n G q . E
1 --.S(EI
is quasi-singular,
2
1 .S(EI 2
x
U C S ( E ) ,s o t h a t 1 ) i m -
is an i n t e r i o r
O& E
h a s a non-empty
i n t e r i o r t o o , and
1 T r u e , s i n c e o t h e r w i s e , i f a n e l e m e n t -x
2
( D e f i n i t i o n 1 .2), s o t h a t
then 2 ESpE(x)
by ( 1 . l o ) rE(xl 2 2 , t h a t i s , a c o n t r a d i c t i o n t o ( 4 . 1 3 ) : t h u s 2 ) i m p l i e s 1 ) a s w e l l (Lemma I ; 6 . 4 ) ,
and t h i s f i n i s h e s t h e p r o o f . I
The argument a p p l i e d i n t h e f i r s t p a r t of t h e p r e c e d i n g p r o o f i s s t i l l u s e d i n t h e p r o o f of t h e n e x t r e s u l t . Thus, w e h a v e .
Proposition 4.2. Let E be a c-aZgebra. Then, f o r every element x i n E , the spectrum of x is a ( p o s s i b l y empty) compact subset of @. On the other hand, f o r every l o c a l l y convex generalized Waelbroeck algebra ( S c h o l i u m 4 . 2 1 , hence, in part i c y l a r (see Lemma 3 . 1
),
f o r every l o c a l l y m-convex Q-algebra E , the spectrum of
every element x i n E is a non-empty compact subset of @ .
Proof. By h y p o t h e s i s , t h e s e t G q o f q u a s i - r e g u l a r e l e m e n t s o f i s a n e i g h b o r h o o d of z e r o i n E (Lemma I ; 6 . 4 ) , s o t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U o f 0 e E , w i t h U G G q . Moreover, i f x E E , t h e r e e x i s t s a > 0 , w i t h a x e U . T h u s , w e show t h a t r i x ) 2 .L
(4.14)
I n d e e d , i f r E ( x )> 1
1
E
, there
a
would e x i s t A € S p E ( x ) , w i t h I h / > 1
1 7 ,
so t h a t
1
s i n c e 1--1<1 and il i s b a l a n c e d , one c o n c l u d e s t h a t - -xa - - a x = -xx € Us xa G q ; t h a t i s , a c o n t r a d i c t i o n , and t h i s p r o v e s ( 4 . 1 4 ) , t h u s t h e f i r s t a s s e r t i o n of t h e s t a t e m e n t , t h e r e s t b e i n g s t r a i g h t f o r w a r d by t h e f o regoing (see C o r o l l a r y 4 . 2 ) . I The f o l l o w i n g e x h i b i t s a n o t h e r u s e f u l p r o p e r t y o f Waelbroeck a l g e b r a s , i n f a c t , of & - a l g e b r a s i n g e n e r a l , and i s a n immediate c o n s e quence o f
( 4 . 1 4 ) . Namely, w e h a v e .
Corollary 4.3. Let E be a Q-algebra. Moreover, consider the following subset of E (4.15)
B(E)={x&E: rE(x)<
+a).
Then,E=B(E); narneZy,the spectraZ radius of every element of E d e f i n e s a ( f i n i t e non-negative) r e a l number. F u r t h e r p r o p e r t i e s o f t h e above c l a s s of t o p o l o g i c a l a l g e b r a s
5. TOPOLOGICAL D I V I S I O N ALGEBRAS
61
w i l l repeatedly be encountered throughout the sequel.
5. Topological division algebras. Gel'fand-Mazur
Theorem
Our main o b j e c t i v e i n t h i s s e c t i o n i s t o d i s c u s s a " c o n t i n u o u s analogon" of a classical r e s u l t of F.G.Frobenius
referring to f i n i t e
d i m e n s i o n a l ( a s s o c i a t i v e ) d i v i s i o n a l g e b r a s . Namely, t o p r o v e , w i t h i n t h e g e n e r a l c o n t e x t o f t o p o l o g i c a l a l g e b r a s t h e o r y , t h e Gel'fand-Maazur
Theorem ( s e e Theorem 5 . 2 ) ,
i n i t i a l l y g i v e n f o r normed a l g e b r a s .
I n t h i s r e s p e c t , by a d i v i s i o n aZgebra w e mean, o f c o u r s e , a n a l g e b r a E w i t h a n i d e n t i t y e l e m e n t which i s a l s o a d i v i s i o n r i n g ( e v e r y n o n - z e r o e l e m e n t of E i s r e g u l a r ) . W e s i m p l y s t a t e t h e F r o b e n i u s t h e o r e m ( w i t h o u t p r o o f ) f o r con-
v e n i e n c e of r e f e r e n c e ; ( s e e , f o r i n s t a n c e , 14. KOCHENDORFFER [I : p. 353, The-
orem 1 0 . 5 . 1 1 ) .
Theorem 5.1. Let E be a f i n i t e dimensional r e a l ( l i n e a r a s s o c i a t i v e ) d i v i s i o n algebra. Then t h e only p o s s i b l e dimensions f o r E are 1 , 2 , and 4 . I n p a r t i c u l a r , t h e only f i n i t e dimensional d i v i s i o n aZgebra over t h e complexes i s the complex number f i e Zd i t s e I f . 1
Schol ium.-
The h y p o t h e s i s of a s s o c i a t i v i t y of t h e a l g e b r a s i n
-
v o l v e d i n t h e above t h e o r e m i s q u i t e e s s e n t i a l . T h u s , it i s a r a t h e r r e c e n t r e s u l t t h e s e t t l e m e n t of t h e q u e s t i o n of e x i s t e n c e o f d i v i s i o n a l g e b r a s which a r e n o t n e c e s s a r i l y a s s o c i a t i v e . i n g methods of A l g e b r a i c Topology it w a s p r o v e d t h a t
mensional r e a l
S o by a p p l y -
t h e only f i n i t e d i -
( n o t n e c e s s a r i l y a s s o c i a t i v e ) d i a i s i o n aZgebras are those o f
aimension 1 , 2 , 4 , and 8 ( c f . , f o r i n s t a n c e , R. B O T T - J . lary I]).
(real)
MILNOR [I: p. 87, Corol-
On t h e o t h e r h a n d , i t seems t h a t t h e r e d o e s n o t e x i s t a s y e t
a n y " s i m p l e a l g e b r a i c " ( ! ) p r o o f o f t h e l a s t r e s u l t , namely one t o a v o i d t h e m a c h i n e r y of A l g e b r a i c Topology i n v o l v e d i n t h e p r o o f s e x i s t -
ed so f a r . Now, our fundamental r e s u l t i n t h i s s e c t i o n , i . e . , t h e Gel'fand-
Maziir Theorem ( t h e n e x t Theorem 5 . 2 a n d / o r i t s C o r o l l a r y 5 . 1 ) w i l l be c o n c l u d e d a s a d i r e c t a p p l i c a t i o n of t h e above Theorem 4 . 2 rollary 4.1).
and i t s C o -
Thus, w e h a v e .
Theorem 5.2. Every topological d i v i s i o n algebra E having a contintnous i n v e r s i o n and a totaZ s e t of continuous l i n e a r forms i s , w i t h i n a topological-algebraic isomorphism, t h e f i e l d o f complex numbers.
Proof. I t
s u f f i c e s t o p r o v e t h a t e v e r y e l e m e n t x i n E i s of t h e
11 SPECTRUM (LOCAL THEORY)
62
form
x =Ae,
(5.1)
f o r some X E C (where e d e n o t e s t h e i d e n t i t y e l e m e n t o f El. O t h e r w i s e , s u p p o s e t h a t 3: - X e # 0 , f o r e v e r y h e @ ; t h e n , s i n c e E i s by h y p o t h e s i s a d i v i s i o n a l g e b r a , o n e c o n c l u d e s t h a t S p E ( x l = Q , which i s a c o n t r a d i c t i o n by h y p o t h e s i s a n d Theorem 4 . 2 .
Thus, t h e r e l a t i o n ( 5 . 1 ) h o l d s
t r u e f o r e v e r y x i n E and s u i t a b l e X e C , i n o t h e r words, t h e r e l a t i o n (5.2)
X-Xe:C-+E,
d e f i n e s an onto map between t h e algebras C and E. Moreover, i t i s r e a d i l y s e e n ( E i s supposed t o be n o n - t r i v i a l )
t h a t (5.2) i s a continuous ( a l -
g e b r a ) isomorphism of t h e ( t o p o l o g i c a l ) a l g e b r a s C a n d E . Now, t h e i n v e r s e map o f
(5.2) is continuous t o o . Indeed,
f o r every a > 0 , s i n c e
a e # O i n E , t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U of z e r o i n E , w i t h a e e U ( t h e t o p o l o g i c a l a l g e b r a E i s H a u s d o r f f , by h y p o t h e s i s ) . T h e r e f o r e , f o r e v e r y x = h e e U , one o b t a i n s have
01
a
5 1 , h e n c e -x = - * Xe = a e 01
t h e i n v e r s e map o f
x
x
I X ( < a, s i n c e o t h e r w i s e o n e would
e U , t h a t i s a c o n t r a d i c t i o n . Thus ,
( 5 . 2 ) i s c o n t i n u o u s a t O e E , h e n c e c o n t i n u o u s , and
t h i s t e r m i n a t e s t h e proof o f t h e theorem. I
Schol ium.of
(5.1); then,
The p r e c e d i n g p r o o f i s e s s e n t i a l l y t h e v e r i f i c a t i o n ( 5 . 2 ) i s , i n f a c t , a n a l g e b r a isomorphism o f C o n t o E
( w e assume t h a t e f 0 ) . T h u s , E i s a I - d i m e n s i o n a l
(Hausdorff topolo-
g i c a l ) v e c t o r s p a c e o v e r C , so t h a t it i s i n f a c t i s o m o q h i c t o C , as a t o -
pological v e c t o r space; i n d e e d , t h e l a s t p a r t of t h e above p r o o f w a s ess e n t i a l l y t h e v e r i f i c a t i o n o f t h i s f a c t . I n t h i s r e s p e c t , see a l s o , f o r i n s t a n c e , J . HORVhh! [l:p. 107, P r o p o s i t i o n 6 1 . NOW,
t h e f o l l o w i n g c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e a-
bove Theorem 5.2 and C o r o l l a r y 4 . 1 .
Corollary 5.1. Every l o c a l l y convex d i v i s i o n algebra having a continuous i n -
.
version i s (algebra) isomorphic (hence, t o p o l o g i c a l l y isomorphic; c f t h e p r e c e d i n g Scholium) t o t h e f i e l d of complex numbers. In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e for every Locally m-convex d i v i s i o n algebra. I C o n c e r n i n g t h e c o n c l u s i o n o f t h e Gel’fand-Mazur
Theorem g i v e n by
t h e above Theorem 5 . 2 , w e remark t h a t t h i s r e m a i n s t r u e u n d e r t h e weak-
er h y p o t h e s i s t h e s p a c e E ’ ( t h e t o p o l o g i c a l d u a l o f E ) t o be n o n - t r i v i a l . I n f a c t , i n t h e p r e s e n c e of t h e rest c o n d i t i o n s f o r E , w e a c t u a l l y g e t a n e q u i v a l e n c e e x p r e s s e d by t h e f o l l o w i n g lemma. Moreover,
6. MAXIMAL IDEALS
63
it i s i n t h e l a t t e r form t h a t w e s h a l l e n c o u n t e r Theorem 5 . 2 i n s u b -
s e q u e n t s e c t i o n s . Thus, w e have. LemM 5.1.
L e t E b e a topological d i v i s i o n algebra. Then, t h e following as-
s e r t i o n s are equivalent 1 ) The topological dual E’ o f E i s n o n - t r i v i a l .
21 E admits a t o t a l s e t of continuous %inear forms ( c f . Scholium 4 . 1 ).
Proof. I t i s c l e a r t h a t 2 ) i m p l i e s 1 ) ( f o r e v e r y t o p o l o g i c a l v e c t o r s p a c e E ) . O n t h e o t h e r h a n d , it i s e a s i l y s e e n t h a t 1 ) i m p l i e s 2 ) b y t h e s e p a r a t e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E and t h e f a c t t h a t E i s a division algebra. I T h u s , a n e q u i v a l e n t f o r m u l a t i o n o f Theorem 5 . 2 i s t o s a y t h a t :
The only (complex) topologica2 d i v i s i o n algebra w i t h a continu
-
ous i n v e r s i o n and n o n - t r i v i a l topological dua% i s t h e complex nwnber f i e l d i t s e l f .
(5.3) O r yet,
assuming t h a t by a
d i v i s i o n topological algebra w e s h a l l mean a
t o p o l o g i c a l d i v i s i o n a l g e b r a w i t h c o n t i n u o u s i n v e r s i o n , one c o n c l u d e s by t h e p r e v i o u s d i s c u s s i o n t h a t : A (complex) d i v i s i o n topological aZgebra is isomorphic t o C ( h e n -
c e , t o p o l o g i c a l l y i s o m o r p h i c ) i f , and only if, it possesses
(5.4)
a non-trivial
topological dual.
T h u s , some e x t r a c o n d i t i o n s a r e n e c e s s a r y , i n d e e d , t h a t a g i v e n (complex) topological d i v i s i o n algebra t o be t h e f i e l d C . In f a c t , t h e r e e x i s t d i v i s i o n t o p o l o g i c a l a l g e b r a s , i n t h e above s e n s e , t h a t a r e d i f f e r e n t from t h e complex number f i e l d i t s e l f . T h i s h a p p e n s , € o r i n s t a n ce, i n t h e (complex) a l g e b r a C t t l o f a l l r a t i o n a l f u n c t i o n s of one complex v a r i a b l e ; t h u s , t h e l a t t e r a l g e b r a s h o u l d a d m i t no n o n - t r i v i a l c o n t i n u o u s l i n e a r f o r m s ( c f . J.H.
WILLIAMSON [I : p. 7311 a n d / o r L . WAELBROECI:
[4:p. 134 f f . ] ) . 6. Maximal i d e a l s Given a n a l g e b r a E , a s u b s e t I of E i s s a i d t o be a l c f t ( r e s p . right) i d e a l of E l i f I i s a v e c t o r s u b s p a c e of E s u c h t h a t a x ( r e s p . x a ) b e l o n g s t o I , f o r a n y a e E and x el. Thus, I i s , i n p a r t i c u l a r , a suba l g e b r a of E s a t i s f y i n g a l s o t h e l a s t c o n d i t i o n . A s u b s e t of E which i s b o t h a l e f t and r i g h t i d e a l i s s a i d t o b e
a
2-sided i d e a l
of E .
I n a l l t h a t f o l l o w s by an i d e a l of an algebra E, We s h a l l always mean a
64
11 SPECTRUM (LOCAL THEORY)
2-sided i d e a l unless i t i s stated otherwise. Moreover, w e s h a l l a l w a y s unders t a n d , a s p a r t of t h e d e f i n i t i o n , t h a t a g i v e n i d e a l i n E w i l l b e a
proper non-trivial i d e a l , i.e.,
an i d e a l d i f f e r e n t from b o t h t h e t r i v i a l
i d e a l {Ol ( o r e l s e zero-ideal) The set of
and t h e whole a l g e b r a E .
( 2 - s i d e d ) i d e a l s o f E c a n be " p a r t i a l l y o r d e r e d " by
i n c l u s i o n , sc) t h a t by a mmimal ( 2 - s i d e d ) i d e a l w e s h a l l mean a n y maximal e l e m e n t of t h e l a t t e r s e t , i . e . ,
an element n o t g e n u i n l y c o n t a i n e d i n
any o t h e r element o f t h e s e t . Thus, on t h e b a s i s o f Lemma 6.1.
Zorn's Lemma
one now o b t a i n s .
Let E be an algebra w i t h an i d e n t i t y element. Then, every i d e a l
of E i s contained i n a maximal i d e a l .
Proof.
The set o f i d e a l s of E o r d e r e d by i n c l u s i o n i s a p a r t i a l -
l y o r d e r e d s e t , i n s u c h a way t h a t , f o r e v e r y chain (Le., t o t a l l y order-
ed s e t ) of i d e a l s o f E , t h e i r u n i o n i s e a s i l y s e e n t o b e a g a i n an i d e -
a l of E ; t h u s , t h e a s s e r t i o n f o l l o w s now by a n a p p l i c a t i o n of Z o r n ' s Lemma (see, f o r i n s t a n c e , N . BOURBAKI [l :Chap. 3 ; p. 3 5 , C o r o l l a i r e 11) I
.
On t h e o t h e r h a n d , t h e f o l l o w i n g p r o v i d e s a u s e f u l c r i t e r i o n t h a t a g i v e n i d e a l i n a commutative a l g e b r a t o b e maximal. Lemma 6.2.
Let E be a c o m t a t i v e algebra w i t h an i d e n t i t y element. Then,
the folloljing a s s e r t i o n s are equivalent: 1 ) M i s a maximal i d e a l of
E.
2) The quotient algebra E/M i s a (conunutative) d i v i s i o n algebra (hence, a
f i e l d over 0 .
Proof. Suppose t h a t M i s a maximal i d e a l of E and l e t a € E l w i t h a $ M I s o t h a t a + M = [ a ] # 0 i n E / M . Moreover, i f I i s t h e ( 2 - s i d e d ) id e a l of E g e n e r a t e d by M U { a ) , one h a s M G I , p r o p e r l y , so t h a t I =E , hence e = m t a x , w i t h m E M and X E E . T h u s , by t a k i n g t h e l a s t r e l a t i o n "modulo M"
one o b t a i n s
[el = [ax! = [a]-[zJ ,
s o t h a t [ a ] i s an i n v e r t i b l e e l e m e n t i n E / M , t h a t i s 1) implies 2). On t h e i s a d i v i s i o n a l g e b r a and I i s a n i d e a l of E which
o t h e r h a n d , i f E/M
p r o p e r l y c o n t a i n s M I t h e n t h e r e e x i s t s a n element a E i , w i t h a#M.Thus, (6.1)
i s t r u e , f o r some x e E l so t h a t e - a x E M G I ; h e n c e , e e l s i n c e a E I ,
t h a t i s I = E which p r o v e s t h a t 2) implies 1 ) a s w e l l ,
and t h i s f i n i s h e s
the proof. NOW, given a c o m t a t i v e algebra E w i t h an i d e n t i t y element, an element x in E i s i n v e r t i b l e i f , and o n l y i f , it does not belong t o any i d e a l of E : The con-
6.
65
MAXIMAL IDEALS
f o r a n y a € E , t h e principal i d e a l (c)= { a x : x e B } i n E g e n e r a t e d by a c o n t a i n s a , f o r x = e l so t h a t
d i t i o n i s o b v i o u s l y n e c e s s a r y . Moreover,
by h y p o t h e s i s a must be i n v e r t i b l e , and t h i s p r o v e s t h e a s s e r t i o n . Thus, a s a c o n s e q u e n c e of t h e l a s t s t a t e m e n t and Lemma 5 . 1 ,
we
conclude t h a t :
An element x of a given c o m t a t i v e algebra E w i t h an i d e n t i t y element i s i n v e r t i b l e i f , and only i f , it i s not contained i n
(6.2)
any maximal i d e a l o f E . The l a s t p r o p o s i t i o n i s o f t e n a p p l i e d i n t h e s e q u e l . The p r e c e d i n g c a n b e e x t e n d e d t o a l g e b r a s which do Rot n e c e s s a r i l y have i d e n t i t y elements: Thus, by a regular i d e a l
o f a g i v e n a l g e b r a E w e s h a l l mean a
( 2 - s i d e d ) i d e a l I o f E i n s u c h a way t h a t t h e q u o t i e n t a l g e b r a E / I has an i d e n t i t y element. T h e r e f o r e , i f
[ a ] = a + I i s s u c h i n E/I, we s h a l l
u s u a l l y c a l l a E E an i d e n t i t y of E modulo I ; hence , one g e t s by d e f i n i t i o n
[.I
(6.3)
=
rxi.1.r
=
,
o r , equivalently,
ax-xeI
(6.41
and x a - x e I ,
f o r eve ry element x i n E . N o w it i s c l e a r t h a t
i f the given algebra E has already an i d e n t i t y e -
lement, t h e n t h i s i s a n i d e n t i t y o f E modulo any ( 2 - s i d e d ) i d e a l o f E , h e n c e i n t h a t c a s e every i d e a l i n E i s regular. Thus, t h e d e s i r e d e x t e n s i o n of t h e p r e c e d i n g d i s c u s s i o n i s now subsummed i n t o t h e f o l l o w i n g . Lemma 6 . 3 . Let E be an algebra. Then, every regular i d e a l of E i s contained
i n a maximal regular i d e a l . In p a r t i c u l a r , a regular i d e a l M o f a c o m t a t i v e algebra E i s maximal i f , and only i f , E/M i s a l c o m t a t i v e l d i v i s i o n algebra (hence, i n p a r t i c u l a r , a f i e l d over C). Proof. Lemma 6 . 1 ,
The p r o o f o f t h e f i r s t a s s e r t i o n g o e s on s i m i l a r l y a s i n s i n c e t h e s e t of r e g u l a r i d e a l s o f E o r d e r e d by i n c l u s i o n
i s s t i l l a n " i n d u c t i v e o r d e r e d s e t " ( i . e . , e v e r y c h a i n i n t h e s e t ad-
m i t s an u p p e r bound; c f ; N . BOURBAKI [l: Chap. 3 ; p. 34, § 4]),
so t h a t t h e
a s s e r t i o n f o l l o w s a g a i n by Z o r n ' s Lemma. On t h e o t h e r h a n d , t h e
se-
cond p a r t of t h e lemma f o l l o w s by a p p l y i n g a s i m i l a r argument a s i n t h e proof o f Lemma 6.2,
t a k i n g t h u s a s e i n ( 6 . 1 ) a n i d e n t i t y of
E
modulo M . I F u r t h e r m o r e , i n a n a l o g y w i t h t h e comment f o l l o w i n g Lemma 6 . 2 ,
66
I1 SPECTRUM (LOCAL THEORY)
one g e t s t h e f o l l o w i n g r e s u l t . An “ a n a l y t i c ” form of it ( i n p a r t i c u -
l a r , of t h e n e x t C o r o l l a r y 6 . 1 ) , and of t h o s e r e m a r k s t o o , w i l l be c o n s i d e r e d i n s u b s e q u e n t s e c t i o n s (see C h a p t . 111).
Lemma 6.4. Let E be a commutative algebra. Then an element x of E i s quasisingular i f , and only i f , t h e following subset of E , (6.5)
I = { ~ - x Y :y e E 1
i s an i d e a l . Moreover, I i s , in f a c t , a regular i d e a l o f E not containing x, t h e l a t t e r being an i d e n t i t y of E modulo I.
Proof. I f I = E , t h e n by
X‘E
E is quasi-singular,
(6.5) x o y = O ,
t h e n I #E . T r u e , b e c a u s e
,if
f o r some y f E which i s a c o n t r a d i c t i o n .
T h u s , i f I f E , one e a s i l y p r o v e s t h a t I i s , i n f a c t , a r e g u l a r i d e a l of E w h i l e t h e g i v e n e l e m e n t x i s a n i d e n t i t y of E modulo I . M o r e o v e r , x & I , because, i f x e I
I , f o r e-
t h e n by ( 6 . 5 ) o n e g e t s y = ( y - x y l + x y
v e r y y e E , so I = E which i s a g a i n a c o n t r a d i c t i o n t o o u r h y p o t h e s i s . On t h e o t h e r h a n d , i f x e E i s q u a s i - r e g u l a r ,
t h e n it i s e a s i l y proved
b y ( 6 . 5 ) t h a t I = E , which i s s t i l l a c o n t r a d i c t i o n i f w e assume t h a t I i s an i d e a l , and t h i s f i n i s h e s t h e p r o o f . I
Now, it i s c l e a r b y t h e p r e c e d i n g p r o o f t h a t Lemma 6 . 4
is still
v a l i d € o r a non-commutative a l g e b r a E , b y t a k i n g i n s t e a d t h e e l e m e n t
a e E t o be r i g h t ( r e s p . l e f t ) q u a s i - s i n g u l a r ;
then, I is a r i g h t o r
l e f t i d e a l i n E , r e s p e c t i v e l y . F o r s i m p l i c i t y ’ s s a k e , however, w e a s sumed F commutative. W e c o n c l u d e w i t h t h e f o l l o w i n g .
Corollary 6.1. Let E be a c o m t a t i v e algebra.Then, an element
3:
in E
is
quasi-regular i f , and only i f , f o r every niazimal i ~ p t ? iud e~a l M o f E , there e x i s t s an element y e E i n such a way t h a t
x + y-xy = x
(6.6)
0
Proof. I f x e E i s q u a s i - r e g u l a r ,
y
E M.
t h e n x o y = O E M , f o r some y € E ,
and f o r e v e r y (maximal r e g u l a r ) i d e a l M of E . C o n v e r s e l y , i f x e E i s quasi-singular,
t h e n t h e i d e a l I of E g i v e n by ( 6 . 4 )
i s r e g u l a r , so
that(Lemma 6 . 3 ) , it i s c o n t a i n e d i n a maximal r e g u l a r i d e a l , s a y M , o f E . T h e r e f o r e , t h e r e e x i s t s by h y p o t h e s i s y E E , w i t h x+ y - x y e M I so t h a t s i n c e by ( 6 . 4 ) y - x y e I ~ M , f o r e v e r y ~ E E one , concludes t h a t x M ; thus, for
every y e E ,
one h a s y = ( y
-
€
x y l + x y e M , t h a t i s M = E . Name-
l y , a c o n t r a d i c t i o n and t h i s f i n i s h e s t h e p r o o f .
I
W e c o n s i d e r now, and t h i s i s of c o u r s e o u r c o n c e r n , maximal ide-
a l s i n t o p o l o g i c a l a l g e b r a s (see a l s o t h e n e x t s e c t i o n ) . Thus,we know
7.
61
CHARACTERS
f o r i n s t a n c e t h a t e v e r y maximal i d e a l i n any Banach a l g e b r a i s c l o s e d . On t h e o t h e r h a n d , a s w e p r e s e n t l y see, t h i s f a c t i s s t i l l v a l i d i n t h e more g e n e r a l c l a s s of 4-(topologica1)algebras. T h a t i s , w e h a v e .
Theorem 6.1. Let E be a &-algebra. Then, every mazima2 regu2ar i d e a l of E i s closed. In particu2ar, every maximal i d e a l o f a Q-a2gebra with an i d e n t i t y element i s closed. -
M =E.
C o n s i d e r a maximal r e g u l a r i d e a l M of E and suppose t h a t
Proof.
Thus, i f a i s a n i d e n t i t y of E modulo M and Gq t h e s e t of q u a s i -
r e g u l a r e l e m e n t s of E , t h e s e t a + G q i s a n ( o p e n ) neighborhood of a i n E (Lemma I ; 6.4) ; h e n c e , one o b t a i n s ia+Gq)
that is, a
-m
n
M
i0
,
f o r some e l e m e n t m E M . T h e r e f o r e , i f y e E
E Gq,
quasi-inverse of a
is the
one g e t s
-in,
la-rnioy= (a-m)+y- ia-m)y=O that is, a =rn+(ay- y l hypothesis f o r M .
-
M , that
-
my,
t h u s a E M , which i s a c o n t r a d i c t i o n t o t h e
Thus, M f E , so t h a t ( C o r o l l a r y I ; 1.1)
we obtain M =
i s t h e a s s e r t i o n , w h i l e t h e r e s t of t h e s t a t e m e n t i s c e r t a i n -
l y clear. I On t h e o t h e r hand, t h e s i t u a t i o n d e s c r i b e d by t h e above theorem
may be d i f f e r e n t f o r an a r b i t r a r y t o p o l o g i c a l a l g e b r a ( e v e n a l o c a l l y ri-convex
one) which i s n o t a & - a l g e b r a ; namely, t h e r e may e x i s t maxi-
m a l i d e a l s which a r e dense s u b s e t s of t h e g i v e n t o p o l o g i c a l a l g e b r a . Whether t h i s may n o t happen in case of a commutative Frgchet l o c a l l y in-eonvex
algebra seems t o be s t i l l an open q u e s t i o n (Michae2’s conjeetttre; see T . HUSAIN[2]). In t h i s respect, c f . a l s o t h e next section, i n p a r t i c u l a r ,
Theorem 1 . 1 .
7. Characters. Closed maximal ideals The s i t u a t i o n d e s c r i b e d b y Lemma 6 . 2 of t h e p r e v i o u s s e c t i o n i n c o n n e c t i o n w i t h t h e Gel’fand-Mazur Theorem ( c f . Theorem 5 . 2 o r i t s l e s s t e c h n i c a l form g i v e n by C o r o l l a r y 5 . 1 ) ,
leads u s t o our next o b j e c t i -
v e ; namely, t h e r e l a t i o n between maximal i d e a l s of a g i v e n t o p o l o g i c a l a l g e b r a E and (complex) a l g e b r a morphisms o f E i n t o t h e complexes, i n p a r t i c u l a r , c o n t i n u o u s o n e s . Thus, we s t a r t w i t h t h e f o l l o w i n g .
Definition 7.1. mean
a non-zero
L e t E be an a l g e b r a . Then, by a
character of E , we
(complex) morphism of E i n t o t h e ( a l g e b r a o f ) comple-
x e s C . The s e t of c h a r a c t e r s of E i s c a l l e d t h e algebraic speetiwv
of E
11 SPECTRUM (LOCAL THEORY)
68
and denoted by M ( E I . ( I t i s o n l y i n Chapt. V where w e c o n s i d e r
M(EI, i n
f a c t , a subspace of it a s an a p p r o p r i a t e t o p o l o g i c a l s p a c e ) . F u r t h e r more, t h e s e t M(E)+ = M(E)U {
(7.1)
u I,
i s c a l l e d t h e extended algebraic spectruv of t h e a l g e b r a E .
In t h i s respect, w e note t h a t
M(EJ may be t h e empty
s e t , even
i f L" i s a Banach a l g e b r a . However, t h e s i t u a t i o n i s q u i t e d i f f e r e n t , of c o u r s e , f o r a commutative Banach a l g e b r a ( i n t h e p r e s e n c e of a n i -
d e n t i t y e l e m e n t ) ; a s w e s h a l l see ( c f . Chapt. V ) t h i s i s s t i l l v a l i d f o r much more g e n e r a l (commutative!) t o p o l o g i c a l a l g e b r a s . Now, i f f i s a c h a r a c t e r of a g i v e n a l g e b r a E one h a s by d e f i n i t i o n the r e l a t i o n flxyl = f(xi.f(y),
(7.2)
f c r any x , y i n E l f b e i n g a l s o a (complex) l i n e a r form on t h e r e s p e c t i v e v e c t o r s p a c e E ; moreover, t h e r e i s an e l e m e n t x i n E , w i t h f i x ) # 0 . In p a r t i c u l a r ,
i f t h e a l g e b r a E has an i d e n t i t y element, say e , s a y -
i n g t h a t f i s " n o n - i d e n t i c a l l y z e r o " i s , of c o u r s e , e q u i v a l e n t w i t h f ( e l = l i n @,
thus f i s then " i d e n t i t y preserving".
On t h e o t h e r hand, one c o n c l u d e s t h a t every f e M ( E ) i s e s s e n t i a l l y a map of E onto C . I n d e e d , i t s u f f i c e s t o c o n s i d e r , f o r e v e r y X E C,
x
element - x
E E , where
f(x) =
c1
the
# 0 i n C by h y p o t h e s i s f o r f . (So one ob-
t a i n s , i n f a c t , t h a t e v e r y non-zero
(complex) l i n e a r form o n E i s an
o n t o map). Thus, our f i r s t statement r u n s a s follows.
Lemma 7.1.
Let E be an algebra and f a character of E . Then, the kernet of 5
the set
i.e.,
k e r ( f ) = C x e E : f ( x l = 01,
(7.3) ?:B
a
2-sided regular ninceimat i d e a t of E.
Proof. By ( 7 . 3 ) and t h e h y p o t h e s i s f o r f one c o n c l u d e s t h a t ker If) i s a 2 - s i d e d i d e a l of E . Moreover,
S E
1
if
f ( a ) = X # O i n C,
f o r some a
E l t h e e l e m e n t ~a E E i s an i d e n t i t y of E modulo ker if), so t h a t t h e l a t t e r i s a r e g u l a r i d e a l of E. On t h e o t h e r hand, s i n c e f i s a (comE
p l e x ) l i n e a r form on E , i t s k e r n e l
E of ximal be
codimension 1 (i. e . ,
( 7 . 3 ) w i l l be a v e c t o r subspace of
dim (Elker(f)l= I ) , h e n c e , e q u i v a l e n t l y , a ma-
v e c t o r subspace of E ; s o s i n c e ker (f) i s an i d e a l of E l it w i l l
t h e n a maximal i d e a l of E l and t h i s p r o v e s t h e a s s e r t i 0 n . I
7.
Remark.-
69
CHARACTERS
C o n c e r n i n g t h e l a s t p a r t of t h e above p r o o f , w e c o u l d
a p p l y , o f c o u r s e , a more e l e m e n t a r y ( h o w e v e r , n o t s i m p l e r ! ) argument t o p r o v e t h a t t h e regular i d e a l of E defined by (7.31 i s maximal.
Thus, sup-
p o s e t h a t I i s a n i d e a l o f E which p r o p e r l y c o n t a i n s k e r ( f I ; t h e r e f o -
re, t h e r e e x i s t s x e I , w i t h f ( x l $ 0.
T1a e E
of
t h e preceding proof
s i n c e f l u l = l , one o b t a i n s s i n c e f ( x ) # 0, E
x-f(x)u
E
k e r ( f l C I . Thus, f l x l u e I so t h a t ,
one c o n c l u d e s t h a t u e I , t h a t i s , a c o n t r a d i c t i o n t o
the hypothesis f o r I .
lement u
Hence, by u s i n g t h e e l e m e n t u =
( i . e . , an i d e n t i t y of E modulo k e r i f l ) ,
( I n t h i s r e s p e c t , w e u s e d t h e f a c t t h a t i f a n e-
E: i s an i d e n t i t y of E modulo an i d e a l I of E , then u
E
I , unless I = E .
Moreover, u i s a l s o an i d e n t i t y o f E modulo any other idea2 J of E containing I ) . T h u s , d e n o t i n g by M l E l t.he s e t of 2 - s i d e d r e g u l a r maximal ideals o f an a l g e b r a E , one g e t s by Lemma 7 . 1 t h e f o l l o w i n g map
9:
(7.4) Now,
M(EI-MIEI:
f w @(f/:= ker(fI.
t h e n e x t s t a t e m e n t s a y s t h a t t h e map 9 i c , i n f m t , one-to-one.
Namely, w e h a v e . Lemma 7.2.
Let E be an a2gehra and f , g any
TWO
characters of E. Then, f = g
if, and o n l y i f . k e r ( f l = kerigl. I n p a r t i c u z a r , t h e map @ defined by ( 7 . 4 ) i s an
injection.
Proof. Of c o u r s e , it s u f f i c e s t o p r o v e o n l y t h e " i f " p a r t of t h e a s s e r t i o n . Thus, i f a e E then f ( a ) =g(al - 1
,
i s an i d e n t i t y of E modulo ker(fi = k e r , ' g I ,
b e c a u s e by h y p o t h e s i s f, g a r e n o n - i d e n t i c a l l y z e r o .
Hence, one h a s
x - f ( x ) aF k e r i f ) = kerig) ,
(7.5) f o r every x e E
,
s o t h a t gix) = f f x ) g ( a l , i.e. ,f (x)= g ( x ) , f o r e v e r y x
E
E
,
and t h i s p r o v e s t h e a s s e r t i o n . I W e c o n s i d e r n e x t continuous characters of a g i v e n t o p o l o g i c a l a l -
g e b r a , i n p a r t i c u l a r , a s it c o n c e r n s t h e i r r e l a t i o n t o t h e ( c l o s e d reg u l a r ) maximal i d e a l s of t h e a l g e b r a ; t h e i r i n t e r r e l a t i o n i s e s t a b l i s h e d by a r e s t r i c t i o n o f t h e map ( 7 , 4 ) ( s e e Theorem 7 . 1 b e l o w ) . Thus, w e f i r s t have t h e f o l l o w i n g . Definition 7.2.
L e t E b e a g i v e n t o p o l o g i c a l a l g e b r a . Then by t h e
( o r s i m p l y spectrum) o f E l w e s h a l l mean t h e s e t of c o n t i n u o u s c h a r a c t e r s o f E , d e n o t e d by m l E i . (Thus an e l e m e n t f e mfE) topo2ogicaZ s p e c t m
w i l l b e a non-zero c o n t i n u o u s ( c o m p l e x ) morphism of E i n t o t h e ( a l g e b r a C of t h e ) complexes).
I1 SPECTRUM (LOCAL THEORY)
70
I t i s c l e a r t h a t a s i m i l a r comment t o t h a t f o l l o w i n g D e f i n i t i o n
7 . 1 i s h e r e i n o r d e r . We a l s o d e f e r f o r C h a p t e r V t o c o n s i d e r m ( E )
a s a s u i t a b l e t o p o l o g i c a l s p a c e , r e t a i n i n g however t h e t e r m i n o l o g y of Moreover, s i m i l a r l y t o ( 7 . 1 ) , w e s t i l l con-
t h e above D e f i n i t i o n 7 . 2 . sider the set
V"Y(El+ =
(7.6)
m(E.,u( 0 3 , ( o r s i m p l y extended spec -
which w e c a l l t h e extended topoZogica1 spectrum
tmun) of t h e t o p o l o g i c a l a l g e b r a E . Thus, w e c o n c l u d e t h a t ; t h e r e s t r i c t i o n o f t h e map 1 7 . 4 ) t o m ( E ) C M ( E ) d e f i n e s a oneto-one correspondence between continuous characters of E ( i . e . ,
(7.7)
e l e m e n t s of 97UE)) and closed regular maximal
( 2 - s i d e d ) i-
deals o f E . W e d e n o t e t h e l a s t s e t i n ( 7 . 7 ) by h4iE). Thus, it i s c l e a r t h a t , f o r e v e r y f e m ( E ) , one h a s kerifi e M ( E ) G M ( E ) , a s f o l l o w s from Lemma 7.1
and
t h e c o n t i n u i t y of f. Now,
it i s o u r main o b j e c t i v e i n t h e e n s u i n g d i s c u s s i o n t o en-
s u r e , under s u i t a b l e r e s t r i c t i o n s
on t h e p a r t i c u l a r t o p o l o g i c a l a l g e -
b r a E c o n s i d e r e d , t h a t t h e above c o r r e s p o n d e n c e ( 7 . 7 ) i s , i n f a c t , a b i j e c t i o n (see Theorem 7 . 1 and i t s C o r o l l a r y 7 . 1 .
On t h e o t h e r h a n d ,
t h e n e x t c h a p t e r p r o v i d e s a more g e n e r a l framework f o r t h e t o p o l o g i c a l algebras considered h e r e ) . W e n e e d f i r s t some more p r e l i m i n a r y m a t e r i a l . Th.us, s u p p o s e t h a t E i s a t o p o l o g i c a l a l g e b r a and I
a ( 2 - s i d e d ) closed i d e a l o f E . S o t h e
q u o t i e n t a l g e b r a E/I, b e i n g a H a u s d o r f f t o p o l o g i c a l v e c t o r s p a c e , i s
also a t o p o l o g i c a l a l g e b r a ; i . e . , t h e m u l t i p l i c a t i o n i n E / I i s separately continuous. I n d e e d , s i n c e t h e t o p o l o g y i n E / I
is a " f i n a l ( v e c t o r space)
t o p o l o g y " w i t h r e s p e c t t o t h e c a n o n i c a l ( q u o t i e n t ) map @ : E - + E / I , one h a s t o p r o v e t h e c o n t i n u i t y o f t h e map l a o @ : E+
E/I
,
f o r every a E E . Therefore, s i n c e
l.(@(x)) = &-@(xc)= @(a).@(xc) = @(ax) = @ ( l a ( x ) ) , with
x e E , one o b t a i n s la o @ = @ o La : E-+
(7.8) where
@ o l a , with a e E ,
E/I ,
i s c e r t a i n l y c o n t i n u o u s , and t h i s p r o v e s t h e
a s s e r t i o n . ( I n t h i s r e s p e c t , see a l s o C h a p t . I V ; S e c t i o n 2 ) . I n p a r t i c u l a r , i f E i s a l o c a l l y convex a l g e b r a and I a s b e f o r e ,
7.
CHARACTERS
71
t h e n one p r o v e s t h a t E/I i s a l o c a l l y c o n v e x a l g e b r a t o o , by a p p l y i n g a s i m i l a r argument to t h e above c o n c e r n i n g t h e ( q u o t i e n t ) l o c a l l y convex s p a c e E I I . F u r t h e r m o r e ,
i f E i s a l o c a l l y m-convex algebvaa, t h e n
this i s
s t i l l t r u e f o r a quotient algebra E / I : T h i s i s e a s i l y s e e n , s i n c e (7.9) with k = [x] e E / I ,
y i e l d s a s u b m u l t i p l i c a t i v e semi-norm on E / I , f o r any
s u c h p on E ( c f . a l s o Theorem 1;3.1). T h u s , w e now h a v e t h e f o l l o w i n g .
Lemma 7.3. Let E be a topological algebra with a continuous quasi-inversion and I a closed (2-sided) ideal of E. Then, the quotient algebra E / I i s a topologic a l algebra w i t h a continuous quasi-inversion t o o .
Proof. By t h e p r e c e d i n g E/I i s a t o p o l o g i c a l a l g e b r a . Moreover, i f x++x'denotes
t h e quasi-inversion
i n E , t h e n t h e c o n t i n u i t y of t h e
r e s p e c t i v e map i n E / I i s e q u i v a l e n t w i t h t h a t o f t h e map x -
[x']:E+E/I.
T h i s i s e a s i l y s e e n b y h y p o t h e s i s f o r E and t h e c o n t i n u i t y of t h e quot i e n t map @ : B + E / I ,
s i n c e @ i s a n a l g e b r a morphism p r e s e r v i n g
the
circle operation. I We a r e now i n t h e p o s i t i o n t o s t a t e t h e f i n a l b a s i c r e s u l t o f t h i s s e c t i o n . Namely, w e h a v e .
Theorem 7.1. Let E be a commutative l o c a l l y convex algebra w i t h a continuous quasi-inversion and M a closed regular maximal i d e a l of E . Then, the quotient algebra E/M i s the algebra C o f the complexes, w i t h i n a topological algebra isoniorphism. I n p a r t i c u l a r , the conclusion holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra E .
Proof. By Lemma 7 . 3 t h e q u o t i e n t a l g e b r a E/M i s a l o c a l l y convex algebra with a continuous quasi-inversion,
r e s p . , a l o c a l l y in-convex
a l g e b r a ( c f . a l s o ( 7 . 9 ) ) . Moreover, by h y p o t h e s i s f o r M a n d L e m m a 6 . 3 ,
E/M i s a d i v i s i o n a l g e b r a , s o t h a t B/M i s , i n f a c t , a (commutative) l o c a l l y convex d i v i s i o n algebra w i t h a continuous inversion
(the l a s t assertion is
e a s i l y d e r i v e d by Lemma 7 . 3 and ( 1 . 4 ) 1 , r e s p . , a ZocaZZy m-emvex d i v i s i o n
algebra.
S o t h e a s s e r t i o n f o l l o w s now f r o m an a p p l i c a t i o n of t h e G e l ' -
fand-Mazur Theorem ( s e e C o r o l l a r y 5 . 1 ) . I
Corollary 7.1. Let
E be a commutative l o c a l l y convex algebra w i t h a continu-
ous quasi-inversion, i n p a r t i c u l a r ( c f . Lemma 3 . 1 ) , any c o m t a t i v e l o c a l l y mconvex algebra. Then, every closed regular maximal i d e a l M of E gives r i s e t o a continuous character f of E ; namely, t h a t one defined by t h e following commutative d i -
72
I1 SPECTRUM (LOCAL THEORY)
agramm :
(7.10)
iiere j denotes t h e topological algebra isomorphism given by t h e above Theorem 7 . 1 . 1
F i n a l l y , one g e t s t h e f o l l o w i n g u s e f u l r e s u l t .
I t is mostly i n
t h i s form t h a t w e a r e g o i n g t o a p p l y Theorem 7 . 1 i n t h e s e q u e l . T h a t i s , we have.
Corollary 7.2. Let E be a c o m u t a t i v e localZy convex aZgebra w i t h a continuous quasi-inversion.
Then, t h e r e e x i s t s a one-to-one
and onto correspondence be -
tween t h e s e t o f continuous characters o f E ( i . e . , t h e spectrwn m(E)o f t h e algebra El and t h e s e t M(EI of cZosed regular maximal ideaZs o f E, given by t h e r e l a t i o n ( 7 . 4 1 . I n p a r t i c u l a r , t h i s i s t m e f o r every l o c a l l y m-convex algebra, so t h a t i n t h i s case one has t h e r e l a t i o n m(E)= i/d(El ,
(7.11)
within a bijection. Proof.
I t i s c l e a r ( c f . Lemma 7 . 2 )
i n d e e d one-to-one. f
&
t h a t t h e map i n q u e s t i o n i s
F u r t h e r m o r e , i f MeM:i’)
m(Eiw i t h k e r ( f ) = M , a s follows f r o m t h i s finishes the proof.1
t h e n by ( 7 . 1 0 ) o n e g e t s t h e r e l a t i o n f = jo@ i n ( 7 .
l o ) , and
On t h e o t h e r hahd, a u s e f u l c o n s e q u e n c e o f t h e p r e c e d i n g , i n c o n j u n c t i o n w i t h Theorem 6 . 1 ,
is t h e following r e s u l t extending t h e
s i t u a t i o n one h a s i n ( c o m m u t a t i v e ) Banach a l g e b r a s .
Corollary 7.3. L e t E be a &-algebra. Then, every character of
E is continu-
ous, i.e., one has
M(E) = m ( E l C E’,
(7.12)
where E’ denotes t h e topoZogicaZ dual o f E. In p a r t i c u l a r , if E i s a c o m t a t i v e Zocally convex Q-aZgebra w i t h a c o n t i nuous quasi-inversion, o r y e t any commutative Locally m-convex Q-algebra ( h e n c e , a
l o c a l l y convex i r e s p . l o c a l l y m-convex.) generazized Waelbroeck aZgebra ( S c h o l i u m then one o b t a i n s
4.2)),
M(EJ = MIE!
(7.13)
where
‘1
I
”
Pi-ocf.
‘m(El= M ( E 1 ,
stands f o r a b i j e c t i o n . I f f i s a c h a r a c t e r of E , t h e n i t s k e r n e l , b e i n g a r e g u l a r
7.
maximal ( 2 - s i d e d ) i d e a l of E: (Lemma 7 . 1 ) c l o s e d s u b s p a c e of E .
73
CHARACTERS
In particular,
,
i s a l s o by Theorem 6 . 1 a
ker(fI
i s a c l o s e d ( v e c t o r ) sub-
s p a c e of E of c o d i m e n s i o n 1 , so t h a t f i s a c o n t i n u o u s l i n e a r form on E
( c f . N . HORVhH [I: p. 107, P r o p o s i t i o n 71 )
,
h e n c e , i n p a r t i c u l a r , a con-
t i n u o u s c h a r a c t e r of E which p r o v e s ( 7 . 1 2 ) . Now, t h e r e s t o f t h e a s s e r t i o n i s a g a i n a c o n s e q u e n c e of Theorem 6 . 1 ,
Corollary 7 . 2 ,
and of
( 7 . 1 2 1 , and t h i s t e r m i n a t e s t h e p r o o f . I Scholium 7.1.
The h y p o t h e s i s i n t h e f i r s t p a r t of t h e p r e v i o u s co-
r o l l a r y t h a t E i s a 4 - a l g e b r a was made o n l y i n c o n n e c t i o n w i t h Theorem
6.1.
Thus, more g e n e r a l l y , t h e f o l l o w i n g i s t r u e : For every t o p o l o g i c a l
algebra E w i t h t h e p r o p e r t y t h a t every r e g u l a r maximal ( 2 - s i d e d ) i d e a l i n E is c l o s e d , one concludes t h a t every character o f E i s continuous. I n o t h e r words, i.he r e l a t i o n M ( E ) = M I E ) i m p l i e s MIEI = m ( E I . F o r c o n v e n i e n c e w e have u s e d i n s t e a d t h e s t r o n g e r , however l e s s t e c h n i c a l , h y p o t h e s i s t h a t t h e a l g e b r a E i s a &-algebra; t h i s w i l l be a l s o t h e case i n several instances below, w h i l e t h e more g e n e r a l s i t u a t i o n d i s c u s s e d h e r e w i l l be a l w a y s c l e a r from t h e c o n t e x t .
W e c o n c l u d e t h e s e c t i o n and t h i s c h a p t e r t o o w i t h c e r t a i n comments and some ( p a r t i a l ) r e s u l t s r e f e r r i n g t o Q - a l g e b r a s ; more p r e c i s e l y , t h i s h a s t o do w i t h a c e r t a i n p a r t i c u l a r n o t i o n , t h e Gel’fand
transform,
which a l s o w e a r e g o i n g t o a p p l y s e v e r a l t i m e s i n s u b s e q u e n t
c h a p t e r s of t h i s book. Thus, w e f i r s t have t h e n e x t .
Definition 7.3. L e t E b e a n a l g e b r a a n d M I E ) i t s a l g e b r a i c s p e c t r u m (i.e., t h e s e t of c h a r a c t e r s of E ; c f . D e f i n i t i o n 7 . 1 ) .
l e t x b e a n e l e m e n t of E. Then, t h e
Moreover,
Gel’fand transform o f x , d e n o t e d by
3 , i s d e f i n e d t o b e t h e map
2 : M(EI * C : f ct 2 i f I : = f 1x1 .
(7.14)
Thus, i f
F ( M I E 1 , CI d e n o t e s t h e s e t o f a l l complex-valued maps on
M I E ) , t h e r e s u l t i n g map
(7.15)
:E
+
F ( M ( E I , C I : x -%(xi
( g i v e n by ( 7 . 1 4 ) ) i s c a l l e d t h e
:=
2
Gel’fand map of t h e g i v e n a l g e b r a E.
I f E i s a topological algebra, the preceding terminology i s , i n particular,
r e s e r v e d f o r t h e r e s t r i c t i o n of
(7.15) to
t h e ( t o p o l o g i c a l ) s p e c t r u m of E ( s e e D e f i n i t i o n 7 . 2 ,
7 7 U E ) G M(E)
,
and C h a p t . V ) .
On t h e o t h e r h a n d , i f w e c o n s i d e r F ( M ( E I , C 1 a s a (complex) a l g e -
11 SPECTRUM ( L O C ~THEORY)
74
bra, with pointwise defined operations, we readily see by the preceding relations that t h e Gel'fand map o f an algebra E i s an algebra rnorphism of E into FfM(EI, C).
Thus, the relation
I~~S)CFIM(EI,CI
E* =
(7.16)
defines E A as a subalgebra of F I M f E I , C I , called the Gel'fand transform algebra of E .
NOW, if we are given a topological algebra E l the respective Gel'fand transform algebra E* consists, in effect, of (complex-valued) continuous functions on mIEI, the latter set being suitably topologized; as we shall see (cf. Chapt. V), one at least natural topology, we usually consider on T I T l E I , is the weakest one making the elements of E ^ continuous functions ( G e l ' f a n d t o p o l o g y ; but cf. also Chapt. IX) . Conclusively, we just apply (7.15) to derive some useful relations concerning the spectrum SpE (x) of an element x in a given alge-
bra E. But first we need the next lemma.(A form of a converse is also valid but deferred for Chapt. 111; see there Theorem 5.1). So we have.
Lemma 7.4. Let E be an aZgebra and x a quasi-regular element o f E. Then, one has t h e reZation f(xl # 1 ,
(7.17) f o r every character f of E.
Proof. If x o y = 0, for some y e E l
(7.18)
one gets
f ( x o y I = f ( x +y - x y I = f f x I + f f y I - f f x I f ( y I = O ,
for every f e M f E I . Thus, if f ( x l = l , for some f e M f E J , one gets by (7.18) I=O,i.e., a contradiction, and this proves the assertion. 1 Thus, a useful consequence of the previous lemma is now the following result, a "continuous analogon" of which will be given lateron (see Chapt. 111; Theorem 6.2, or even Corollary 7.5 below). That is, we have.
Corollary 7.4. L e t form
2
E b e an aZgebra and
R(O/ t h e range o f t h e Gel'fand trans-
o f an element x in E (cf. (7.15)). Then, one has
(7.19)
R(2)L S p E ( x c ) .
I n p a r t i c u l a r , one has t h e r e l a t i o n
(7.20)
~ ( M ( E I += I spE(x)
u {OI,
f o r every x i n E. Furthermore, one o b t a i n s
(7.21)
r E fxl =
sup J j Y x I J ,
S EM I E I
7. CHARACTERS
75
f o r every x i n E. Proof. If fixl=O, for some f EM~EI, then x can not be a regular element of E, so that (Definition l .2) 0 e SpElxl. On the other hand, if ?if) = ffxl= X f ; O in C , where f E MlEl, then one has f ( 1T z l = 1 ; hence (Lem1 should be a quasi-singular element of E, so A f SpE(xl and ma 7.4) , h x this proves (7.19). NOW if A E SpElxl , with A # 0, then (cf. Definition 1
1.2) - x
x
must be quasi-singular, so that (Lemma 7.4) there exists f 1
e MfEI, with f ( - xx ) = l , thus f l x l = ~ i f i = X ;hence, X E R i ; ) , finally obtains the relation (7.22)
spE(x)
so that one
u I O I = R( 2 i u {OI = ;IMIEI +i ,
which together with (7.19) proves (7.20). Finally (7.21) is now immediate by (1.10) and (7.20). I
Now the following proposition is still another application of the above Lemma 7.4 and will repeatedly be used in subsequent chapters.
Proposition 7.1. Let E be a Q-algebra. Then, i t s ( t o p o l o g i c a l ) spectmm mfEl i s an equicontinuous subset of E' ( t h e topologicaZ dual of E l and hence r e l a t i v e l y compact i n E,' (the weak topological dual of E ; see Chapt. V )
.
If U is a balanced neighborhood of O E E consisting of quasi-regular elements of E (Lemma I ; 6.4), then one has t h e r e l a t i o n Proof.
UE. f?7YfEllo,
(7.23)
where the second member of the last relation denotes the polar s e t of mrElr E'in
E: Indeed, if -3 E U and Iffx)I 2 1 , f o r some f €?;rZ(E), then 1 1 1 ffx) = A # 0 and 1-ix < 1. Thus -AX E U and f ( T z l = 1 , a contradiction ::i.nce --31 x is quasi-regular (Lemma 7.4). Therefore, one has ( f f d 51
(7.24)
,
for every f f 772(E/ and x E U which proves (7.23). Hence, (rrZ(E)IO is a neighborhood of 0 e E , therefore 772: f E l C ( m(El! O 0 an equicontinuous subset of E'. I Finally, we do have the next "continuous analogon" of Corollary 7.4, a stronger version of which will be derived in Chapter 111, just when the necessary preliminary material will be available. Thus, we have. Theorem 7 . 2 . Let E be a c o m t a t i v e localZy convex &-algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n ( i . e . , a commutative l o c a l l y convex WaeZbroeck
.
algebra; cf Definition 4.1). Then, for every element x i n E, one g e t s t h e r e l a -
76
11
SPECTRUM (LOCAL THEORY)
t i o n (cf. also the notation of Corollary 7.4)
R ( G I = 3 ( r n t E / ) = SPE(Xi.
(7.25)
In p a r t i c u l a r , one g e t s (7.26) Hence, the a s s e r t i o n holds t r u e f o r every commutative l o c a l l y m-convex 4-algebra with an i d e n t i t y element ( i . e . , a commutative l o c a l l y m-convex Waelbroeck algebra). I n t h i s r e s p e c t , if t h e given algebra E does not have an i d e n t i t y , one g e t s
S(7^n(EIf) = SpE(xl,
(7.27)
for every element x in E, while the r e l a t i o n ( 7 . 2 6 ) i s s t i l l t r u e . Boof.
By hypothesis and (7.12) one has M(EI = ??Z ( E l ,
so that by (7.19) R(;)rSpE(x), for every X E E. Moreover, for every h e SpElx), with h # 0, one concludes, based on the proof of (7.20), that h e R ( G ) (see also the comment after Definition 1.2). Thus, suppose that 0 e SpE(x:); then (Definition l.l), x can not be aregular element of E , so that x E M , for some maximal ideal M in E (cf.(6.2)). Hence, by (7.4) and (7.13) one obtains f(x1 =G(f) = O , for some f € MIE) = m(El, and this proves (7.25). Now (7.26) is immediate by (7.13) and (7.21) which actually finishes the proof, the rest of the assertion being now straightf0rward.m Now, given an algebra E and an element x of E, we denote by Z ( s ) the zero-set of the function 2 , the Gel'fand transform of x ; that is, one has by definition (7.28)
Z(?)=
{ f E M I E ) : G ( f ) = flx)=O}.
The following is now a direct consequence of the argument plied in the preceding proof. Namely, we have.
ap-
Corollary 7 . 5 . Let E be a c o m t a t i v e l o c a l l y convex Waelbroeck algebra (cf.
.
the above Theorem 7.2) Then, an element x i n E i s regular ( i . e . , i n v e r t i b l e ) if, and only i f , one has
Z ( 2 ) = G1;
(7.29) t h a t i s , equivalentzy, the r e l a t i o n
(7.30)
.;'.tf,
=f(x) f 0,
for every (continuous) character f of E. 1 In this respect, we finally note that a stronger version of the last result is still true, in case one has locally m-convex algebras. In fact, it suffices then to consider just advertibly complete alge-
3.
77
SCHUR'S LEMMA
b r a s i n s t e a d o f & - a l g e b r a s ( c f . T h e o r e m I ; 6.4. S e e a l s o C h a p t . 1 I I ; C o rollary 5.2). 8. Appendix: Schur's Lemma W e p r e s e n t i n t h i s a p p e n d i x a form of t h e c l a s s i c a l lemma
in
t i t l e which o f t e n a p p e a r s m a i n l y i n q u e s t i o n s c o n c e r n i n g t h e r e p r e s e n t a t i o n t h e o r y o f t o p o l o g i c a l a l g e b r a s . Though w e a r e n o t i n p a r t i c u l a r c o n c e r n e d w i t h t h a t p a r t of t h e t h e o r y i n t h i s book ( s e e , h o w e v e r , t h e f i n a l C h a p t e r X V ) , w e do d i s c u s s t h e r e s u l t i n q u e s t i o n m a i n l y as an a p p l i c a t i o n i n o u r c a s e o f t h e Gel'fand-Mazur
Theorem ( S e c -
tion 5). W e s t a r t w i t h a p r e l i m i n a r y comment on t h e r e l e v a n t t e r m i n o l o g y .
T h u s , g i v e n a (complex) a l g e b r a E and an ( a l g e b r a ) moryhism
of E i n t o t h e a l g e b r a o f l i n e a r endomorphisms ( o p e r a t o r s ) o f a (comp l e x ) v e c t o r s p a c e X , one d e f i n e s t h e map 41 a s a representation of t h e algebra
E
in
X
( t h e representation space). A l t e r n a t i v e l y , t h e s p a c e X
becomes a n E-mo&h*e v i a t h e r e l a t i o n
f o r a n y a 6 E and x e X .
( T h a t i s , o n e t a k e s a s a d e f i n i t i o n o f t h e re-
levant notion t h e r e l a t i o n (8.2) i n e i t h e r d i r e c t i o n , r e s p e c t i v e l y ) . Thus w e s p e a k o f an irreducible representation X I
of
(the algebra) E i n
i f t h e E-module X d e f i n e d by ( 8 . 2 ) i s i r r e d u c i b l e . T h a t i s , t h e r e
do n o t e x i s t o t h e r sub-E-modules
(03rX and t h e s p a c e X i t s e l f .
o f X e x c e p t t h e t r i v i a l o n e s , namely, (Thus i n t h i s case $ d o e s n o t decompose
( "reduce") t o any o t h e r " s i m p l e r "
,
non-trivial
,
(sub-) r e p r e s e n t a t i o n s
of E ) . Now a n i m p o r t a n t " s t r u c t u r a l i n g r e d i e n t " of t h e " g e o m e t r y " of t h e r e p r e s e n t a t i o n $ ( h e n c e , of t h e g i v e n a l g e b r a t o o ) i s t h o s e e l e ments T
€
L(X) which "commute w i t h t h e e l e m e n t s of E " ; i n f a c t t h e y f u l -
f il t h e r e l a t i o n
f o r every a e E . W e
d e n o t e by c($) t h e s e t of t h o s e T f L i X ) which s a t i s -
f y ( 8 . 3 ) and c a l l them intertwining operators o f t h e r e p r e s e n t a t i o n $ . Thus o n e d i s t i n g u i s h e s now t h a t p a r t of LIX) which b e h a v e s r e l a t i v e t o t h e "domain o f c o e f f i c i e n t s " E ( X b e i n g c o n s i d e r e d a s a n Emodule), e x a c t l y a s i f
E was a " n u m e r i c a l domain". Now, t h a n k s t o I .
Schur, t h i s d e s i r e i s , i n f a c t , v e r y c l o s e t o t h e r e a l i t y concerning
11 SPECTRUM (LOCAL THEORY)
78
at least "simple" (read irreducible) representations. Thus, his classical lemma says that: For every i r r e d u c i b l e representation 6 , as in (8.1), c ( @I
(8.4)
i s a d i v i s i o n subalgebra of L I X I .
(Here the identity operator on X , idX= 1 , is the identity element of ~ ( 4 ) . For a proof of (8.1) see, f o r instance, F.F. BONSAL- J . DUNKAN [l:p. 121, Proposition 6 1 ) . Now in case of a c o m t a t i v e topological algebra, the previous picture is further amended by appealing to Gel'fand-Mazur. In this respect, by looking at continuous representations one considers on L ( X ) any topology Gmaking the latter space a topological algebra (Definition I; 1.1. See also Example I;2.(1)). Thus we now have. Lemma 8.1. Let E be a c o m t a t i v e topo2ogical algebra w i t h an i d e n t i t y e l e -
ment and continuous i n v e r s i o n . Then, every continuous representation of E , say
@:
f o r which E/ker(@) (in t h e q u o t i e n t topology) has a non-trivial topological dual, i s one-dimensiona2.
E -Le(X),
Proof, By hypothesis ker(6) is a closed ideal of E, so E/ker($), endowed with the respective quotient topology, is a topological algera of the same type as E (see also Lemma 7.3). Moreover, one has
(8.5)
E/ker(O) = @(E) E L ( X ) ,
within an (algebra) isomorphism, while $ ( E l is a (complex)division algebra (Schur's Lemma; cf. (8.3)). Thus E/ker(@l is a division topological algebra satisfying, by hypothesis, (5.4), hence (topologically) isomorphic to C, and this proves the assertion. In addition, (8.5) becomes, in effect, a topological (algebraic) isomorphism.l
As an illustration of the above, we still give the following less technical result. Namely, we have.
Corollary 8.1. L e t E be a coirnnutatiue Zocally convex algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n . Then, t h e only continuous i r r e d u c i b l e r e p r e s e n t a t i o n s of E are i t s continuous characters ( i . e . , t h e elements of ??Z(Ei, t h e ( t o p o l o g i c a l ) spectrum of E; see Definition 7.2).
In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra w i t h an i d e n t i t y element ( see also Lemma 3.1 ) . I
Spectrum
(Local T h e o r y )
CHAPTER I 1
1. Spectrum of an element. Spectral radius Given any algebra with an identity element (no topology is considered), one sets the following.
Definition 1.1. Let E be an algebra with an identity element and x any element of E. Then, the spectmcm of x (denoted by SpE1xl) is the set of those complex numbers A , for which the element A-x in E does not have an inverse. Thus, one has, by definition, (1.1)
SpElxl = { X e C : A-x
is not invertible}.
Concerning the above notation, A-x stands, of course, for the element A.iE-x in t ' , where lE denotes the identity element of E . NOW, by ( 1 . 1 )
,
t h e zero element of @ belongs t o t h e spectrum of x i f , and
only i f , x i s a singular (i.e., non-invertible) element o f E. On the other hai,d, on the basis of the "circle operation" consi-
dered above (cf.Chapt. I; ( 6 . 7 ) ) ,
one extends the previous definition
to the case of an algebra which does not necessarily possess an identity element. Thus, we have the next.
Definition 1.2. Let E be an algebra and x any element of E . Then, the spectrum of x is the set of those non-zero complex numbers h (not. f o r which the element +z e E does not have a quasi-inverse in E , plus the zero element of @, unless x is regular (i.e., invertible).So
C,),
one has, by definition, (1.2)
SpE(xi = { AeC, : Ix x E E is not quasi-invertible]
{ o f @ , if x is singular}. In this respect, i f t h e algebra E does not have an i d e n t i t y element, we agree t h a t every xe E i s s i n g u l a r , so t h a t Oe SpE(x).
Thus, t h e above two d e f i n i t i o n s a r e , i n f a c t , equiualent i n ease t h e given algebra E does possess an i d e n t i t g element: This is easily seen, since a com-
48
11 SPECTRUM (LOCAL THEORY) 1
p l e x X EC, belongs t o SpE(x) i f , and only i f , - x
i s quasi-singular (i.e., does not have a quasi-inverse); now, this is readily proved by the relation
x
x - x = A ( l - - 1x )
(1.3)
A
(here the distinction between l e IR and 1 for l E € E is clear) and the fact that x o y = 0 i f , andonly i f ,( l - z ) ( l - y l = l ,
(1.4)
for any x , y in E. In connection with the previous Definition 1.2, we still note that it is also common to define the spectrum of an element x in an algebra E without identity, as the spectrum of x in the algebra E B C , obtained from E by adding an identity element (cf.Chapt.1;Section 5). Thus, if the algebra E does not have an identity element, the twosets SpE(xi and SpE8@ (x), with x e E , are naturally related (in fact, coincide). That is, we have.
Lemma 1 . I . Let E be an algebra (which does not necessarily have an identity element) and l e t E B C be t h e algebra obtained from E by "adding an i-
.
d e n t i t y " (cf Chapt. I; Section 5 I .
Then, one has
(1.5) f o r every element x i n E , where t h e two members 01 t h e l a s t r e l a t i o n are defined b y ( 1 . 2 ) and ( 1 . 1 ) , r e s p e c t i v e l y .
Proof. By the comments after Definition 1.2, one concludes that SpEeC (x) (which is given by (1.I) ) coincides with SpEec ( ix,O)) ( cf. (1.2)), for every element x in E l identified wi.th ( x , O l € E B C . Therefore, it suffices to prove that t h e l a s t s p e c s r m above coincides w i t h s p ( x l E
cJhich i s given by ( 1 . 2 ) . Thus, we first remark that a v e q element x s E i s a singular element o f t h e algebra E 8 C . ( Indeed, otherwise the relation (x,O). ( y , X I = ( x y +Ax,Ol =(O,l),
for some (y,X)€ E B C , would imply 0 = 1 i n C ,
.
which is acontradiction) Hence, 0 6 SpEBC(xl = SpEBC ( (x,O) I and similarly, 0 E SpE(xI (since we regard t h e algebra E a s lacking an i d e n t i t y e l e rnent; cf. also the comment after the en2 of this proof). Furthermore,
one verifies that an element x EE i s quasi-singular i n E i f , and only i f , (x,O) is quasi-singular i n E 8 C . (Otherwise, the relation ( x , O ) o ( y , A ) = (y,X)o(x,Ol =(O,O/, far some ( y , X I E E @ C , would yield x o y = y o x = 0,which is a contradiction to the hypothesis for x ; and conversely). NOW, the argument, as that after Definition 1.2, applied to the element with X#0, provides the desired conclusion. I
same
1 --€El
X
In connection with the preceding lemma, we note that if the gi-
SPECTRUM. SPECTRAL RADIUS
1.
49
ven a l g e b r a E has a n i d e n t i t y e l e m e n t , t h e n , s i n c e e v s r y e l e m e n t x € B
i s s i n g u l a r when c o n s i d e r e d a s a n e l e m e n t o f E B C (see a l s o t h e a b o v e , o n e c o n c l u d e s t h a t U ESpE 8 C ( x ) ; h o w e v e r , t h i s may n o t be t h e case f o r SpE(x), i f x h a p p e n s t o b e a r e g u l a r e l e m e n t o f E . I n t h a t ca-
proof)
se ( 1 . 5 ) i s n o l o n g e r t r u e , h e n c e , i f E has an i d e n t i t y element one g e t s , i n
general, t h e r e l a t i o n
=
s pE (2) 3pE@@ix)3
(1.6)
for every xEE. On t h e o t h e r h a n d , t h e f o l l o w i n g r e s u l t p r o v i d e s a c o n n e c t i o n b e t w e e n t h e s p e c t r a o f e l e m e n t s o f a g i v e n a l g e b r a a n d t h o s e of t h e s e e l e m e n t s i n a b i g g e r a l g e b r a . Namely, w e h a v e . P r o p o s i t i o n 1 . 1 . Let E , F be tuo algebras and b : E + F
an (algebra) mor-
ph.ism. Then, one has t h e r e l a t i o n
SPFl$iZi)E SPE(XCl,
(1.7)
f o r every element x i n E.
(The members o f t h e l a s t r e l a t i o n a r e d e f i n e d
.
b y t h e r e s p e c t i v e r e l a t i o n s t o ( 1 . 2 ) a b o v e ) In p a r t i c u l a r , i f E i s a sub-
algebra o f F, then one 7 b t a i n s (1
.a)
SPF(XI c SPEiX),
f o r every element x i n E. (The r e s p e c t i v e s p e c t r a a r e a l w a y s u n d e r s t o o d i n t h e sense of Definition 1 . 2 ) .
Proof. By Lemma 1 . 1
(see also i t s p r o o f ) , one h a s t h e r e l a t i o n
SPF(@(Xl)= SPF9@i!@(X/,0))
(1.9)
f o r every element
So a c o m p l e x number A # 0 b e l o n g s t o S p F ( @ ( d )
E.
J: f
i f , a n d o n l y i f , t h e e l e m e n t A-Qixi ( i . e . , A (O,l)-(@(x),O)) i n F BC
,
which i s t h e case o n l y i f A - 2 1
EBC:In fact,
i f -xox'=
t h e s i s f o r @,
one h a s
A
x @ix) 1
1 x'o-~:=O,
x
0
is singular ( i . e . , A(O,l)-(x,U)) i s s i n g u l a r i n
f o r some x ' ~ E , t h e n by h y p o 1
$ ( x ' ) = @ ( x ' l o -@(x)
A
=0 1
( i . e . , an algebra morphism "preserves t h e c i r c l e o p e r a t i o n " ) . Hence, -$(x) A
is
i n F o r , e q u i v a l e n t l y , A - $(x) i s
regular i n F @ C , thus a 1 c o n t r a d i c t i o n t o t h e h y p o t h e s i s f o r @(I) E F . T h e r e f o r e , - x is q u a s i -
quasi-regular
singular i n E ,
A
so t h a t A ~ S p ~ i M d .o r e o v e r , i f 0 6 S p F ( $ f x ) ) , t h e n $ ( x ) i s
s i n g u l a r i n F; h e n c e , by t h e f o r e g o i n g , x i s s i n g u l a r i n E as that is,
OeSpE(x), a n d
Now,
well,
t h i s finishes t h e proof. I
a n i m p o r t a n t n o t i o n which i s c o n n e c t e d w i t h t h e s p e c t r u m
of a n e l e m e n t i n a g i v e n a l g e b r a i s t h a t p r o v i d e d by t h e n e x t .
50
I1 SPECTRUM (LOCAL THEORY) D e f i n i t i o n 1.3. Given a n a l g e b r a E and an e l e m e n t x e E l one d e f i -
n e s t h e spectral radius o f x
( d e n o t e d by r E ( x ) ) , by t h e r e l a t i o n
(1.10)
w e g e t , of course, t h e r e l a t i o n
Thus, by ( 1 . 1 0 ) and Lemma 1 . 1 ,
r (xl =
(1.11)
f o r every element x e E NOW,
l e m e n t of
E
.
P
Ef3cl:(x)
it i s c l e a r by ( 1 . 1 0 ) t h a t r (xJ may b e , i n g e n e r a l , a n eE Et ( : = IR, U {tm}), s o t h a t it i s i m p o r t a n t t o know t h a t i n a
g i v e n a l g e b r a E l r (zl i s , i n E
f a c t , a p o s i t i v e r e a l number. O f c o u r s e ,
t h i s i s t r u e i n e v e r y Banach a l g e b r a a n d , a s w e s h a l l s e e , t h i s i s a l -
so t h e case i n more g e n e r a l t o p o l o g i c a l a l g e b r a s ( Q - a l g e b r a s , f o r i n stance; c f . Corollary 4 . 5 ) . F i n a l l y , as we s h a l l see i n t h e e n s u i n g d i s c u s s i o n , t h e " a l g e
-
b r a i c " d e f i n i t i o n of s p e c t r a l r a d i u s , g i v e n by ( l . l O ) l h a s f o r s u i t a b l e t o p o l o g i c a l a l g e b r a s a l s o a " t o p o l o g i c a l " c o u n t e r p a r t (see, f o r i n s t a n c e , Theorem 7.2
and a l s o t h e n e x t c h a p t e r ) .
2 . The r e s o l v e n t s e t I n c o n n e c t i o n w i t h t h e n o t i o n of t h e s p e c t r u m of a n e l e m e n t
3:
i n a g i v e n a l g e b r a E , it i s s t i l l of i m p o r t a n c e t h e complement o f t h e s p e c t r u m i n 02, a n o t i o n which w e s h a l l a l s o a p p l y i n t h e s e q u e l . T h u s ,
w e f i r s t give the respective formal d e f i n i t i o n . T h u s , g i v e n a n a l g e b r a E w i t h an i d e n t i t y e l e m e n t , one d e f i n e s t h e resolvent s e t o f an e l e m e n t x e E
( d e n o t e d by p i x ) )
t o b e t h e com-
p l e m e n t i n C o f t h e s p e c t r u m of LC.T h a t i s , o n e h a s , by d e f i n i t i o n pix) =
(2.1)
c SPE(Xl .
I n o t h e r w o r d s , t h e s e t p i x ) c o n s i s t s o f t h o s e complex numbers A,
for
which t h e r e s p e c t i v e e l e m e n t A - x E E i s i n v e r t i b l e . T h e r e f o r e ,
one d e f i n e s t h e f o l l o w i n g E-vaZued map R(x;XI =
(2.2)
with
h
E pix);
(
A-xI-'
,
w e c a l l i t t h e resolvent f u n c t i o n
of t h e e l e m e n t x e 0". S o
i f G d e n o t e s t h e g r o u p of i n v e r t i b l e e l e m e n t s of E l one h a s
~ 1 x 1= R ( x ;
(2.3)
for
every element
ZE
E
'
)-I (Gl ,
.
The n e x t l e m m a ( f i r s t resolvent e q u a t i o n ) w i l l b e a p p l i e d below.
3.
Lemma 2.1. Let
ALGEBRAS WITH C O N T I N U O U S I N V E R S I O N
51
E be an algebra w i t h an i d e n t i t y element and x an element of
E. Then, t h e resolvent f u n c t i o n of x s a t i s f i e s t h e r e l a t i o n R(x;h)
(2.4)
-
R(x;p) =
- (A - p l
R ( x ; A l R ( ~ ; p l,
f o r every p a i r ( A , u i of elements of pix). Proof. At first a routine verification shows that R (x;X I .R(x; p) = R(x;p l .R(x; A/ ,
(2.5)
f o r any A , p in p i x ) . Therefore, one obtains R(x;Al-R(x;ul
- (h-21
= R l x ; h ) E l ' ~ ; p ) (( V - X )
)
= -IX-pIRlx;hlR(a;p),
which is the desired relation (2.4). I In this respect, we still note that,besides the relation (2.4) above,there is another formula referring to the resolvent function for different values of (the variable) x e E ; (we consider thus R ( x ; X I as a function of x too). This is valid f o r any algebra E with an i d e n t i t y element. Namely, it is easily proved first that (X-X)
(2.6)
( R l z ; A)
-
R(y;X)I
(A- y l =
3:
-y
,
for any x, y in E and A in both of the resolvent sets p(xl and ~ ( L J ) ; thus, multiplying the last relation on the left by R ( x ; X ) and on the right by R ( y ; X) , one obtains (2.7)
R(x;hi -R(y;X)
for every A
E
also E. HILLE
= R(z;X/(x- y)Rly;A),
p i x i n p(y1 and any 2 , in - R.S. PHILLIPS [I ; p. 126 ff.]).
(
second resolvent equation ; see
3. Topological algebras w i t h continuous inversion Our task in this and the following section is to examine towhat extent certain fundamental properties connected with the notions we have already defined above (cf. Section 1 ) and which are valid, of course, in every Banach algebra, are still true for appropriate topological algebras. Thus, we start with the following.
Definition 3.1. Given a topological algebra E with an identity element, we shall say that E has a continuous i n v e r s i o n , whenever the ( E valued) map z-x-'
is continuous on its domain of definition (i.e., the set of regular elements of E ) . On the other hand, a topological algebra E is said to have a continuous quasi-inversion , if the quasi-inversion in E defined by (Chapt.
52
11 SPECTRUM (LOCAL THEORY)
I;( 6 . 6 ) ) is a continuous (E-valued) map on the set of quasi-regular elements of E . In this respect, we first have the next.
Lemma 3.1. Let
E be a l o c a l l y rn-convex algebra with an i d e n t i t y element. T??en,
E has a continuous inversion. More generally, every l o c a l l y m-convex algebra has a
continuous quasi-inversion. Proof. This goes on verbatim as in the case of a normed algebra, by considering instead of a norm a family of submultiplicative seminorms defining the topology of E (cf. Chapt.1; Proposition 3.2):Thus.
see, for instance, C.E. RICKART [l :p. 13, Theorem 1.4.8, and p. 19, Theorem 1. 4.221. I The inversion in a given topological algebra with an identity element may be continuous without the algebra to be locally m-convex,
as this is shown by the following result. In fact, the same result may be considered as a rather concrete instance of a more general (nonetheless, more technical) situation (see the concluding Remark below). Thus, we have.
Theorem 3.1. Let
E be a metrizable topological algebra with an i d e n t i t y e l e -
ment, for which the group G of regular elements i s , i n the r e l a t i v e topology, a separable Baire space. Then, the inversion i n E i s a continuous (E-viluedl map on G,
so that G i s , i n p a r t i c u l a r , a topological group with respect t o the r e l a t i v e topology * Proof. By hypothesis the group G is a Baire space which is also
metrizable and separable, hence second countable; thus, the assertion is now a consequence of a known relevant (and more general) result (cf. T . HUSAIN [I : p. 3 6 , Theorem 8 and p. 37, Theorem 9 1 ) . I
As an application, one also gets the following.
Corollary 3.1. Let F be a Frgchet algebra (cf. Chapt.1;Definition 1 . 5 1 with an i d e n t i t y element for which the group G of regular elements i s a se,arable G -set.Then,
6
E has acontinuous i n v e r s i o n , so t h a t G i s , i n p a r t i c u l a r , a topologi-
caZ group i n t h e r e l a t i v e topology. Proof. The assertion follows by the previous theorem and the fact that G, being by hypothesis a G6-set in E , is "topologically complete"
(Alexandroff;cf. , for instance, J.G. HOCKING-G.S. YOUNG [ I : p. 85, Theorem 2-76]), hence a Baire space. I On the other hand, one also obtains.
3.
Corollary 3.2. Let
ALGEBRAS WITH CONTINUOUS INVERSION
53
E be a Baire metrizable Q-algebra w i t h an i d e n t i t y e l e -
ment such t h a t i t s group G o f regular elements i s a separable subspace. Then, G i s i n t h e r e l a t i v e topology a topological group, so t h a t E has, moreover, a continuous inversion. I n p a r t i c u l a r , a Frgchet separable 4-algebra w i t h an i d e n t i t y element i s a topological algebra w i t h a continuous i n v e r s i o n , having t h e group o f regular e l e ments a topological group i n t h e r e l a t i v e topology. Proof. By hypotesis G is an open subset of E
(cf. Chapt. I ; Definition 6 . 2 ) , hence a Baire space too, which is also separable, so that the first assertion follows by Theorem 3.1. On the other hand, if E is, moreover, a separable Q-algebra, then the group G of regular elements, being an open subset of E, is also a separable subspace; thus, one is led to the previous case, and this finishes the proof. I
Remark.- According to the result of T . HUSAIN [I] applied in the proof of Theorem 3 . 1 , a group G , endowed with a topology T making the (group-operation) "multiplication in G " separately continuous (i.e., a semi-topological group) becomes a topological group, if G i s a normal Baire second countable space in the topology T. This occurs, in particular, by the hypothesis of Theorem 3 . 1 and its corollaries above. Moreover, by a known result of D. MONTGOMERY [I :p. 881 , Theorem 21 , a semi-topological group whose underlying topological space is complete metrizable and separable, is a topological group. Now this condition is satisfied, of course, if one has
il
Frgehet algebra E w i t h an i d e n t i t y element
and group G o f i n v e r t i b l e elements a separable G - s e t .
6 ilore generally, the last conclusion is valid, if E is a metrizable (topological) algebra with an identity element and G locally complete and separable. In this respect, we still note that by T . HUSAIN [I: p. 3 6 , Theorem 71 (cf. also I".+. WU [I: p . 452, Theorem] ), one concludes that: Every topological algebra E w i t h an i d e n t i t y element and group G of regular elements a Baire second countable space, has a continuous i n v e r s i o n , G becoming t h u s a topological group i n t h e r e l a t i v e topology.
In particular, t h e l a s t conclusion i s v a t i d f o r a Baire Q-algebra E w i t h an i d e n t i t y element and G second countable o r , more p a r t i c u z a r l y , if E i s a Baire metrizable separable Q-aZgebra w i t h an i d e n t i t y element.
In conclusion, some extra conditions of the kind considered above seems to be necessary, as it concerns the group G of regular elements of a topological algebra E (not locally m-convex!) withan identity element, if one wants G to be a topological group, hence E to ha-
54
I1 SPECTRUM (LOCAL THEORY)
ve a continuous inversion. So this need appears even if E is a F r g c h e t locally convex algebra, as this is indicated by the "Arens algebra" Lw f [ O , l ] l (cf. Chapt. I; Subsection 2. ( 4 ) ) , which does not have a coztinuous inversion (cf. R. ARENS [4: p. 6301 ) . Now, topological algebras exhibiting the properties of the algebras appeared in the second part of Corollary 3 . 2 will play an important role in the sequel,therefore, they are singled out by the next section.
4. Waelbroeck algebras The algebras in title of this section seems to be the appropriate ones for an important branch of the whole theory of topological algebras, as it is the "Functional Calculus" (cf. Chapt.VI;Section 3 , as well as Chapt.VIII;Section 8). They have been applied in this context (in particular, as locally convex and/or locally m-convex ones) already in the early 5 0 ' s by L . WAELBROECK [ I ; 2 1 . On the other hand, the same algebras, at least the locally mconvex ones, seems to constitute that particular class of topological algebras in which one can find many of the fundamental results of the classical Banach algebras theory. This will become clear in several instances throughout the ensuing discussion. Thus, the same class of algebras plays a significant role too in other parts of the theory of topological algebras or of its applications, whose discussion, however, would be beyond our scope in this book. ( But see, for instance, A. MALLIOS [20-24; 2 7 1 ) . We start with giving the relevant formal definition. Namely, we have. Definition 4.1, By a Waelbroeck aZgebra , we mean a topological algebra E with an identity element, in such a way that the group G of the invertible elements is an open subset of E and the inversion a continuous (E-valued) map on G.
In other words, a Waelbroeck aZgebra is, by definition, a Q-aZgebrn w i t h n continuous inversion. Furthermore, by the foregoing (see the last Xemark above) , a separable Baire (hence, in particular, a separabZe Frgchet) &-algebra w i t h an i d e n t i t y element isa Waelbroeck algebra. Now, the following variant of Lemma I ; 6 . 4 is needed below, if one is going to apply Definition I ;6 . 2 . Thus , a topological aZgebra E with an i d e n t i t y element i s a 9-algebra (cf. Definition I;6.2) i f , and only i f , the
4. WAELBROECK ALGEBRAS
55
s e t of regular elements of E i s a neighborhood of t h e i d e n t i t y element of E:The assertion can easily be proved by suitably modifying the argument in the proof of Lemma I;6.4. We can now formulate another useful version of the above Defi-
nition 4 . 1 via the next.
Proposition 4.1. Let E be a topological aZgebra w i t h an i d e n t i t y element e . Then, t h e f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : 11 E i s a Waelbroeck algebra (Definition 4.1 ).
2) The group G of regular elements of E i s a neighborhood of e and t h e inver-
s i o n i n E i s continuous a t e . 31 The i d e n t i t y element e has a neighborhood c o n s i s t i n g of i n v e r t i b l e e l e -
rnen?s and the i n v e r s i o n i n E i s con-tinuous a t e . Proof. We first remark that 2 ) is equivalent to 3), while 1 ) implies 2) by Definition 4.1. Thus, based on the above comment and by assuming 2 ) , one concludes that E is a Q-algebra, so that it suffices to prove that the inversion in E is a continuous (E-valued) map on G ,
assuming its continuity at the point x = e . Indeed, if x e l ; , the map x b e + x - ' ( x - x O 1 , x E E , is continuous at x = x , so that :here exists O -7 a neighborhood U of zero in E , such that e +n: A i x - x 1 E G , €or every
x - x e U ; i.e., x e x O + U , where x + U is a neighborhood of xo in
E.
Thus,
due to the relation x = x0 i e + x0- ' i x - x O ) l
(4.1) valid for every
3:
e
E,
the map 3: + + i e + x - ' i x
(4.2)
,
-z0i)-',
defined for every x e x O +U , is by hypothesis continuous at refore, by (4.1), one obtains
3:
= :co
with r e x O +U , thus the desired continuity of the map x + - + x - ' , the point x = x and this finishes the proof. I
. The-
x e E , at
0'
On the basis of the previous proof, one concludes that cond.3) of Proposition 4 . 1 can be s t a t e d f o r every regular element 3: i n E , which is, in fact, an equivalent form of cond. 1 ) of the same proposition, i.e., of Definition 4.1. Now, as an application of the foregoing, we are in a positionto state the following. Theorem 4.1.
Let E be a Waelbroeck aZgebra and
3:
any element of E . Then, the
56
I1 SPECTRUM (LOCAL THEORY)
resolvent s e t
p1x) of the clement x i s an open subset of C and the respective re-
- J an E-valued
solvent f u n c t i o n R ( x ; the r e l a t i o n
holomorphic map on
p ( ~ ) ,which also s a t i s f i e s
lim RIx;AI = 0
(4.4)
A+
OJ
("bounded a t t h e point a t i n f i n i t y " o f C), so t h a t it i s , i n p a r t i c u l a r , a bounded (holomorphic) map on p l x l .
Proof. W e f i r s t remark t h a t R ( x ; X
i s continuous as a f u n c t i o n of A,
with X E p ( x I , b y ( 2 . 2 ) and t h e c o n t i n u i t y o f t h e i n v e r s i o n i n E. Hence, ( 2 . 3 ) , one c o n c l u d e s t h a t is an open s u b s e t o f C . On t h e o t h e r h a n d , t h e " f i r s t r e s o l v e n t e q u a t i o n " ( c f . Lemma 2 . 1 ) i m p l i e s t h a t
by h y p o t h e s i s and a p p l y i n g t h e n o t a t i o n of
pix)
(4.5)
1
A-
IR1x;AI- R(x;Xoll = - R ( x ; X I R ( ~ : ; A , ) .
A,
T h e r e f o r e , by p a s s i n g t o t h e l i m i t i n t h e l a s t r e l a t i o n f o r A+h w i t h i n p ( x ) , and
0'
t a k i n g t h e c o n t i n u i t y of t h e i n v e r s i o n i n E i n t o ac-
c o u n t , one o b t a i n s
t h u s , RIx;AI i s an E-valued d i f f e r e n t i a b l e map on pixi
,
h e n c e by d e f i n i t i o n a
h o l o m o r p h i c map on p ( x ) . NOW, s i n c e , f o r e v e r y x e E , t h e map a c r e - a x :
i s c o n t i n u o u s , t h e r e e x i s t s a n e i g h b o r h o o d NE ( U l o f z e r o i n C , with e-ax e G , f o r every a e N E ( 0 ) (see a l s o P r o p o s i t i o n 4 . 1 ) Conse -
C+E
.
quently, the relation
makes s e n s e , f o r
1
(x(
with a =-
1
X , i.e.,
l i m i t i n t h e l a s t r e l a t i o n f o r h + m i n @,
( X ( > E ; thus, by t a k i n g t h e
one g e t s by t h e c o n t i n u i t y
of t h e i n v e r s i o n i n E t h e d e s i r e d r e l a t i o n ( 4 . 4 ) . i o n i s now a c o n s e q u e n c e o f
(4.4)
S o t h e f i n a l assert-
and t h e c o n t i n u i t y of t h e map R ( x ; X ) ,
with X E p(xl. I
Scholium 4.1.-
I n t h e p r e v i o u s p r o o f an E-valued map d e f i n e d on an
open s u b s e t of C was t a k e n a s " h o l o m o r p h i c " , b e i n g d i f f e r e n t i a b l e on
i t s domain of d e f i n i t i o n ( a n d h e n c e c o n t i n u o u s too: t h i s w a s t h e case f o r t h e map R(x;XI b y ( 4 . 6 ) and ( 4 . 7 ) ) . NOW, t h i s r e m i n d s , o f c o u r s e , t h e c a s e E = C , namely, t h a t of complex-valued h o l o m o r p h i c maps, w h i l e t h i s i s s t i l l v a l i d i f one c o n s i d e r s t h e c o m p o s i t i o n s of a n E-valued h o l o m o r p h i c map a s above w i t h ( c o m p l e x - v a l u e d ) c o n t i n u o u s l i n e a r forms on t h e g i v e n t o p o l o g i c a l a l g e b r a E ,
i . e . , w i t h e l e m e n t s o f t h e topolo-
4. WAELBROECK
57
AJAGEBRAS
g i c a l dual E' of E . I n t h i s r e s p e c t , i t would b e t h u s o f i m p o r t a n c e t o h a v e i n E a "good s u p p l y " o f s u c h f o r m s , a s t h i s i s , f o r example, t h e
case i f E i s a l o c a l l y convex a l g e b r a (Hahn-Banach). B u t , a s w e s h a l l see i n s u b s e q u e n t s e c t i o n s , t h i s may s t i l l happ e n i n t o p o l o g i c a l a l g e b r a s which a r e n o t n e c e s s a r i l y l o c a l l y convex. Thus, what one r e a l l y n e e d s i n t h i s c o n t e x t i s
c(
topological algebra E
admitting a t o t a l s e t of continuous l i n e a r forms, i n t h e s e n s e t h a t , t h e r e e-
x i s t s a s u b s e t A of E ' , s u c h t h a t , f o r e v e r y e l e m e n t x E E , w i t h x # O , t h e r e e x i s t s a n e l e m e n t f e A , w i t h f(x)# 0 . T h u s , as a l r e a d y n o t e d , i f E i s , i n p a r t i c u l a r ,
a l o c a l l y con-
i s s u c h a t o t a l s e t f o r E (Hahn-Banach); i n f a c t , t h i s i s v a l i d f o r every subset A of E' whose linear hull i s "weakly dense" i n
vex a l g e b r a , t h e n E ' E', i.e.
,
i f A i s a total set
Finally,
i n E',
,
t h e "weak t o p o l o g i c a l d u a l " of E .
i n c o n n e c t i o n w i t h Theorem 4 . 1 ,
w e s t i l l remark t h a t
by t h e p r o o f o f t h e same t h e o r e m , t h e r e e x i s t s a n e i g h b o r h o o d of t h e "point a t i n f i n i t y " of C contained i n p ( x ) ,
so t h a t
p i x ) i s a neighbor-
hood of t h e p o i n t a t i n f i n i t y o f C . F u r t h e r m o r e , one c a n p r o v e t h a t R ( x ; X I i s holomorphic a t t h i s p o i n t t o o (see, f o r i n s t a n c e ,
M.A.
NAYMARK [I : p . 172,
111, and p. 66, 5 1 2 1 ) .
As a n a p p l i c a t i o n of t h e a b o v e d i s c u s s i o n , w e come now t o t h e f o l l o w i n g b a s i c r e s u l t which i s v a l i d , i n f a c t , f o r a n a p p r o p r i a t e
class of topological algebras ( e . g . ,
l o c a l l y convex o n e s , c f . C o r o l -
having a continuous i n v e r s i o n . That i s , w e g e t .
lary 4.1)
Theorem 4.2. Let
E be a topological algebra w i t h an i d e n t i t y element and con-
tinuous inversion admitting, moreover, a t o t a l s e t o f continuous l i n e a r forms ( c f . S c h o l i u m 4.1). Then, one has
SPE(XI # 0 ,
(4. 8)
f o r every element x i n E. Proof. S u p p o s e t h a t SpE(xLII = @ , f o r some x e E . T h u s ,
p(x)= a : ,
by ( 2 . 1 1 ,
so t h a t t h e r e s o l v e n t f u n c t i o n R ( x ; X I i s defined for every X E C
being, moreover, a holomorphic map ( c f . t h e Remark b e l o w ) . T h e r e f o r e , if A
cE'
i s a t o t a l s e t o f c o n t i n u o u s l i n e a r f o r m s on E l t h e n t h e
X++f(R(x;h))
,
map
w i t h f e A , i s a complex-valued h o l o m o r p h i c map on C (i.e.,
a n e n t i r e f u n c t i o n ) which i s a l s o bounded by ( 4 . 4 )
,
s i n c e f i s continu-
o u s . Hence ( L i o u v i l l e ' s T h e o r e m ) , t h e l a t t e r map i s c o n s t a n t , so t h a t by ( 4 . 4 ) (4.9)
one o b t a i n s
f(R1x;XII = f ( ( A - x ) - ' )
= 0 ,
58
with A
I1 SPECTRUM (LOCAL THEORY)
€@
,
and f o r e v e r y f E A . T h u s , by h y p o t h e s i s f o r A and ( 4 . 9 ) , one
o b t a i n s fA-x)-'=
0
,i.e. ,
a c o n t r a d i c t i o n t o o u r h y p o t h e s i s , and t h i s
terminates t h e proof. I
Remark.fact, a
I n t h e p r o o f o f t h e p r e v i o u s t h e o r e m w e have u s e d ,
in
strengthened form of Theorem 4 . 1 ; t h a t i s , concerning t h e r e s o l v e n t
f u n c t i o n R l x ; A ) , x € E , t h e analogous concZusion t o t h a t of Theorem 4.1 i s s t i l l val i d , f o r every element x , t h e r e s o l v e n t s e t of which i s an open subset of @. The a s s e r t i o n i s c l e a r by t h e a r g u m e n t i n t h e p r o o f of t h e same t h e o r e m , and i t w a s t h a t e x a c t l y , which h a s been a p p l i e d i n t h e p r o o f of t h e above Theorem 4 . 2 .
Of c o u r s e , it i s t r u e f o r e v e r y Waelbroeck a l g e b r a
(Theorem 4 . 1 ) Now, t h e n e x t f u n d a m e n t a l c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e above and Lemma 3.1.
Corollary 4.1. L e t
E be a l o c a l l y convex algebra w i t h an i d e n t i t y eZement
and continuous i n v e r s i o n . Then, t h e spectrum
SpEfx) of every element x i n E i s a
non-empty subset of C. In p a r t i c u z a r , f o r every l o c a l l y m-convex algebra E w i t h an i d e n t i t y element, every element x i n E has a non-empty s p e c t m SpEfxl.
Scholium 4.2.- I f a t o p o l o g i c a l a l g e b r a E d o e s n o t n e c e s s a r i l y hav e a n i d e n t i t y e l e m e n t , o n e c a n c o n s i d e r , o f c o u r s e , a k i n d of a gene-
raZized WaeZbroeck aZgebra,
i n t h e sense t h a t
E i s again a &-algebra
( c f . De-
f i n i t i o n I : 6 . 3 ) and t h e quasi-inversion i n E is continuous. I n t h i s r e s p e c t ,
by a p p l y i n g Lemma I ; 6 . 4 , tion 4.1
.
one o b t a i n s a n o b v i o u s a n a l o g u e o f P r o p o s i -
On t h e o t h e r hand , i f a generalized Waelbroeck aZgebra has already
an i d e n t i t y eZement, then i t i s a WaeZbroeck algebra ( D e f i n i t i o n 4.1); t h i s f o l -
l o w s by Lemma I ; 6 . 4
and P r o p o s i t i o n 4.1
(see a l s o ( 1 . 4 ) ) . Moreover,
t h e ( t o p o l o g i c a l ) adjunction of an i d e n t i t y ( c f . C h a p t . I ; 5. ( 1 ) ) t o a gene raZized Waelbroeck algebra makes it a WaeZbroeck algebra. Now by t h e f o r e g o i n g ( s e e a l s o D e f i n i t i o n 1 . 2 ) , one g e t s t h e f o l l o w i n g s t r e n g t h e n e d form of t h e above C o r o l l a r y 4 . 1 .
Corollary 4.2. Let E be a ZocalZy convex algebra w i t h a continuous quasi-inv e r s i o n . In p a r t i c u l a r , suppose t h a t E i s any locaZly m-convex algebra (cf.Lemm a 3.1 ). Then, i n e i t h e r c a s e , one has
0 # spE(x)G c
(4.10)
f o r every element x i n E . I I n t h i s r e s p e c t , one f u r t h e r o b t a i n s t h e f o l l o w i n g r e s u l t which
4. WAELBROECK
59
ALGEBRAS
p r o v i d e s supplementary i n f o r m a t i o n c o n c e r n i n g ( 1 . 8 ) . Thus, w e have.
Lemma 4.1.
Let E be a closed subazgebra o f a topoZogica1 algebra F , where F has an i d e n t i t y element and continuous i n v e r s i o n . Moreover, suppose t h a t , f o r ever y x i n E , SpE(x) i s a closed subset o f C ( t a k e , f o r i n s t a n c e , a Waelbroeck a l g e b r a E). Then, concerning t h e boundaries o f t h e spectra of a given element x
i n E and F, r e s p e c t i v e l y , one has t h e reZation bd ( SpE(xl I 5 bd (SpF(xi )
(4.11)
.
Proof. By h y p o t h e s i s , one h a s t h e r e l a t i o n bd ISpElx) I = Sprixcl f o r every x E E. G C,
n el”,
n
,
T h e r e f o r e , i f A e b d ( S p E ( x ) ), t h e n A = l i m A n , w i t h A n € p ( x )
so t h a t one o b t a i n s
y = l h ( A - x)= A-x E E . n n NOW, i f y i s i n v e r t i b l e i n F , t h e n y - I = lirn(An-xl-l , by h y p o t h e s i s ; t h u s , y = A-x i s a l s o i n v e r t i b l e i n E , a c o n t r a d i c t i o n , s i n c e X€SpE(x). Therefore, y = A-x
i s n o t i n v e r t i b l e i n F , s o t h a t A E + ~ ( x ), which p r o by ( 1 . 8 ) and ( 2 . 1 ) . I
ves the desired relation (4.111,
Remark.-
I f t h e a l g e b r a s E a n d F i n t h e above lemma do n o t h a v e
n e c e s s a r i l y a n i d e n t i t y e l e m e n t ( n a m e l y , t h e same o n e ) , t h e n under t h e
r e s t o f t h e c o n d i t i o n s o f Lemma 4 . 1 , one o b t a i n s (4.12)
b d ( SpE(xJIC b d ( SpF(x)),
for every x i n E. ( C o n s i d e r t h e a l g e b r a s E O C and
FOC and
apply t h e pre-
v i o u s a r g u m e n t ) . Now t h e l a s t r e l a t i o n i s f u r t h e r j u s t i f i e d , s i n c e t h e a l g e b r a E may h a v e a n i d e n t i t y e l e m e n t which w i l l be n o t a n i d e n t i t y f o r F. The n e x t lemma w i l l p r e s e n t l y b e needed and s u p p l e m e n t s Lemma I; 6.4,
p r o v i d i n g a n o t h e r u s e f u l c h a r a c t e r i z a t i o n of & - a l g e b r a s .
Lemma 4.2.
Let E be a topologicaZ algebra and consider t h e foZZowing subset
of E (E) = {x E E : r,(x)
(4.13)
5 1),
where rE(x) is g i v e n by ( 1 . 1 0 ) . Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a Q-algebra. 2)
SIE) i s a neighborhood o f zero i n E.
Proof. I f E i s a & - a l g e b r a , t h e n (Lemma I ; 6.41,
Gq
is a neigh-
borhood of z e r o i n E , so t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U
60
I1 SPECTRUM (LOCAL THEORY)
of 0 6 E c o n t a i n e d i n G q .
We s h a l l show t h a t U
t h e n -1x is h G q , because
quasi-singular,
S ( E I : I n d e e d , i f x E E and
1x1 > 1 . B u t , t h a t i s a c o n t r a d i c t i o n , s i n c e -1x f U C
r E ( x l > I , t h e n by ( 1 - 1 0 ) t h e r e would e x i s t
A e S p E ( x l , with
1 and I, i s b a l a n c e d . Thus,
l i e s 2 ) . On t h e o t h e r h a n d , i f 2 ) i s v a l i d , t h e n p o i n t of
s ( E l ; hence, t h e s e t
moreover i s c o n t a i n e d i n G q . E
1 --.S(EI
is quasi-singular,
2
1 .S(EI 2
x
U C S ( E ) ,s o t h a t 1 ) i m -
is an i n t e r i o r
O& E
h a s a non-empty
i n t e r i o r t o o , and
1 T r u e , s i n c e o t h e r w i s e , i f a n e l e m e n t -x
2
( D e f i n i t i o n 1 .2), s o t h a t
then 2 ESpE(x)
by ( 1 . l o ) rE(xl 2 2 , t h a t i s , a c o n t r a d i c t i o n t o ( 4 . 1 3 ) : t h u s 2 ) i m p l i e s 1 ) a s w e l l (Lemma I ; 6 . 4 ) ,
and t h i s f i n i s h e s t h e p r o o f . I
The argument a p p l i e d i n t h e f i r s t p a r t of t h e p r e c e d i n g p r o o f i s s t i l l u s e d i n t h e p r o o f of t h e n e x t r e s u l t . Thus, w e h a v e .
Proposition 4.2. Let E be a c-aZgebra. Then, f o r every element x i n E , the spectrum of x is a ( p o s s i b l y empty) compact subset of @. On the other hand, f o r every l o c a l l y convex generalized Waelbroeck algebra ( S c h o l i u m 4 . 2 1 , hence, in part i c y l a r (see Lemma 3 . 1
),
f o r every l o c a l l y m-convex Q-algebra E , the spectrum of
every element x i n E is a non-empty compact subset of @ .
Proof. By h y p o t h e s i s , t h e s e t G q o f q u a s i - r e g u l a r e l e m e n t s o f i s a n e i g h b o r h o o d of z e r o i n E (Lemma I ; 6 . 4 ) , s o t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U o f 0 e E , w i t h U G G q . Moreover, i f x E E , t h e r e e x i s t s a > 0 , w i t h a x e U . T h u s , w e show t h a t r i x ) 2 .L
(4.14)
I n d e e d , i f r E ( x )> 1
1
E
, there
a
would e x i s t A € S p E ( x ) , w i t h I h / > 1
1 7 ,
so t h a t
1
s i n c e 1--1<1 and il i s b a l a n c e d , one c o n c l u d e s t h a t - -xa - - a x = -xx € Us xa G q ; t h a t i s , a c o n t r a d i c t i o n , and t h i s p r o v e s ( 4 . 1 4 ) , t h u s t h e f i r s t a s s e r t i o n of t h e s t a t e m e n t , t h e r e s t b e i n g s t r a i g h t f o r w a r d by t h e f o regoing (see C o r o l l a r y 4 . 2 ) . I The f o l l o w i n g e x h i b i t s a n o t h e r u s e f u l p r o p e r t y o f Waelbroeck a l g e b r a s , i n f a c t , of & - a l g e b r a s i n g e n e r a l , and i s a n immediate c o n s e quence o f
( 4 . 1 4 ) . Namely, w e h a v e .
Corollary 4.3. Let E be a Q-algebra. Moreover, consider the following subset of E (4.15)
B(E)={x&E: rE(x)<
+a).
Then,E=B(E); narneZy,the spectraZ radius of every element of E d e f i n e s a ( f i n i t e non-negative) r e a l number. F u r t h e r p r o p e r t i e s o f t h e above c l a s s of t o p o l o g i c a l a l g e b r a s
5. TOPOLOGICAL D I V I S I O N ALGEBRAS
61
w i l l repeatedly be encountered throughout the sequel.
5. Topological division algebras. Gel'fand-Mazur
Theorem
Our main o b j e c t i v e i n t h i s s e c t i o n i s t o d i s c u s s a " c o n t i n u o u s analogon" of a classical r e s u l t of F.G.Frobenius
referring to f i n i t e
d i m e n s i o n a l ( a s s o c i a t i v e ) d i v i s i o n a l g e b r a s . Namely, t o p r o v e , w i t h i n t h e g e n e r a l c o n t e x t o f t o p o l o g i c a l a l g e b r a s t h e o r y , t h e Gel'fand-Maazur
Theorem ( s e e Theorem 5 . 2 ) ,
i n i t i a l l y g i v e n f o r normed a l g e b r a s .
I n t h i s r e s p e c t , by a d i v i s i o n aZgebra w e mean, o f c o u r s e , a n a l g e b r a E w i t h a n i d e n t i t y e l e m e n t which i s a l s o a d i v i s i o n r i n g ( e v e r y n o n - z e r o e l e m e n t of E i s r e g u l a r ) . W e s i m p l y s t a t e t h e F r o b e n i u s t h e o r e m ( w i t h o u t p r o o f ) f o r con-
v e n i e n c e of r e f e r e n c e ; ( s e e , f o r i n s t a n c e , 14. KOCHENDORFFER [I : p. 353, The-
orem 1 0 . 5 . 1 1 ) .
Theorem 5.1. Let E be a f i n i t e dimensional r e a l ( l i n e a r a s s o c i a t i v e ) d i v i s i o n algebra. Then t h e only p o s s i b l e dimensions f o r E are 1 , 2 , and 4 . I n p a r t i c u l a r , t h e only f i n i t e dimensional d i v i s i o n aZgebra over t h e complexes i s the complex number f i e Zd i t s e I f . 1
Schol ium.-
The h y p o t h e s i s of a s s o c i a t i v i t y of t h e a l g e b r a s i n
-
v o l v e d i n t h e above t h e o r e m i s q u i t e e s s e n t i a l . T h u s , it i s a r a t h e r r e c e n t r e s u l t t h e s e t t l e m e n t of t h e q u e s t i o n of e x i s t e n c e o f d i v i s i o n a l g e b r a s which a r e n o t n e c e s s a r i l y a s s o c i a t i v e . i n g methods of A l g e b r a i c Topology it w a s p r o v e d t h a t
mensional r e a l
S o by a p p l y -
t h e only f i n i t e d i -
( n o t n e c e s s a r i l y a s s o c i a t i v e ) d i a i s i o n aZgebras are those o f
aimension 1 , 2 , 4 , and 8 ( c f . , f o r i n s t a n c e , R. B O T T - J . lary I]).
(real)
MILNOR [I: p. 87, Corol-
On t h e o t h e r h a n d , i t seems t h a t t h e r e d o e s n o t e x i s t a s y e t
a n y " s i m p l e a l g e b r a i c " ( ! ) p r o o f o f t h e l a s t r e s u l t , namely one t o a v o i d t h e m a c h i n e r y of A l g e b r a i c Topology i n v o l v e d i n t h e p r o o f s e x i s t -
ed so f a r . Now, our fundamental r e s u l t i n t h i s s e c t i o n , i . e . , t h e Gel'fand-
Maziir Theorem ( t h e n e x t Theorem 5 . 2 a n d / o r i t s C o r o l l a r y 5 . 1 ) w i l l be c o n c l u d e d a s a d i r e c t a p p l i c a t i o n of t h e above Theorem 4 . 2 rollary 4.1).
and i t s C o -
Thus, w e h a v e .
Theorem 5.2. Every topological d i v i s i o n algebra E having a contintnous i n v e r s i o n and a totaZ s e t of continuous l i n e a r forms i s , w i t h i n a topological-algebraic isomorphism, t h e f i e l d o f complex numbers.
Proof. I t
s u f f i c e s t o p r o v e t h a t e v e r y e l e m e n t x i n E i s of t h e
11 SPECTRUM (LOCAL THEORY)
62
form
x =Ae,
(5.1)
f o r some X E C (where e d e n o t e s t h e i d e n t i t y e l e m e n t o f El. O t h e r w i s e , s u p p o s e t h a t 3: - X e # 0 , f o r e v e r y h e @ ; t h e n , s i n c e E i s by h y p o t h e s i s a d i v i s i o n a l g e b r a , o n e c o n c l u d e s t h a t S p E ( x l = Q , which i s a c o n t r a d i c t i o n by h y p o t h e s i s a n d Theorem 4 . 2 .
Thus, t h e r e l a t i o n ( 5 . 1 ) h o l d s
t r u e f o r e v e r y x i n E and s u i t a b l e X e C , i n o t h e r words, t h e r e l a t i o n (5.2)
X-Xe:C-+E,
d e f i n e s an onto map between t h e algebras C and E. Moreover, i t i s r e a d i l y s e e n ( E i s supposed t o be n o n - t r i v i a l )
t h a t (5.2) i s a continuous ( a l -
g e b r a ) isomorphism of t h e ( t o p o l o g i c a l ) a l g e b r a s C a n d E . Now, t h e i n v e r s e map o f
(5.2) is continuous t o o . Indeed,
f o r every a > 0 , s i n c e
a e # O i n E , t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U of z e r o i n E , w i t h a e e U ( t h e t o p o l o g i c a l a l g e b r a E i s H a u s d o r f f , by h y p o t h e s i s ) . T h e r e f o r e , f o r e v e r y x = h e e U , one o b t a i n s have
01
a
5 1 , h e n c e -x = - * Xe = a e 01
t h e i n v e r s e map o f
x
x
I X ( < a, s i n c e o t h e r w i s e o n e would
e U , t h a t i s a c o n t r a d i c t i o n . Thus ,
( 5 . 2 ) i s c o n t i n u o u s a t O e E , h e n c e c o n t i n u o u s , and
t h i s t e r m i n a t e s t h e proof o f t h e theorem. I
Schol ium.of
(5.1); then,
The p r e c e d i n g p r o o f i s e s s e n t i a l l y t h e v e r i f i c a t i o n ( 5 . 2 ) i s , i n f a c t , a n a l g e b r a isomorphism o f C o n t o E
( w e assume t h a t e f 0 ) . T h u s , E i s a I - d i m e n s i o n a l
(Hausdorff topolo-
g i c a l ) v e c t o r s p a c e o v e r C , so t h a t it i s i n f a c t i s o m o q h i c t o C , as a t o -
pological v e c t o r space; i n d e e d , t h e l a s t p a r t of t h e above p r o o f w a s ess e n t i a l l y t h e v e r i f i c a t i o n o f t h i s f a c t . I n t h i s r e s p e c t , see a l s o , f o r i n s t a n c e , J . HORVhh! [l:p. 107, P r o p o s i t i o n 6 1 . NOW,
t h e f o l l o w i n g c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e a-
bove Theorem 5.2 and C o r o l l a r y 4 . 1 .
Corollary 5.1. Every l o c a l l y convex d i v i s i o n algebra having a continuous i n -
.
version i s (algebra) isomorphic (hence, t o p o l o g i c a l l y isomorphic; c f t h e p r e c e d i n g Scholium) t o t h e f i e l d of complex numbers. In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e for every Locally m-convex d i v i s i o n algebra. I C o n c e r n i n g t h e c o n c l u s i o n o f t h e Gel’fand-Mazur
Theorem g i v e n by
t h e above Theorem 5 . 2 , w e remark t h a t t h i s r e m a i n s t r u e u n d e r t h e weak-
er h y p o t h e s i s t h e s p a c e E ’ ( t h e t o p o l o g i c a l d u a l o f E ) t o be n o n - t r i v i a l . I n f a c t , i n t h e p r e s e n c e of t h e rest c o n d i t i o n s f o r E , w e a c t u a l l y g e t a n e q u i v a l e n c e e x p r e s s e d by t h e f o l l o w i n g lemma. Moreover,
6. MAXIMAL IDEALS
63
it i s i n t h e l a t t e r form t h a t w e s h a l l e n c o u n t e r Theorem 5 . 2 i n s u b -
s e q u e n t s e c t i o n s . Thus, w e have. LemM 5.1.
L e t E b e a topological d i v i s i o n algebra. Then, t h e following as-
s e r t i o n s are equivalent 1 ) The topological dual E’ o f E i s n o n - t r i v i a l .
21 E admits a t o t a l s e t of continuous %inear forms ( c f . Scholium 4 . 1 ).
Proof. I t i s c l e a r t h a t 2 ) i m p l i e s 1 ) ( f o r e v e r y t o p o l o g i c a l v e c t o r s p a c e E ) . O n t h e o t h e r h a n d , it i s e a s i l y s e e n t h a t 1 ) i m p l i e s 2 ) b y t h e s e p a r a t e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E and t h e f a c t t h a t E i s a division algebra. I T h u s , a n e q u i v a l e n t f o r m u l a t i o n o f Theorem 5 . 2 i s t o s a y t h a t :
The only (complex) topologica2 d i v i s i o n algebra w i t h a continu
-
ous i n v e r s i o n and n o n - t r i v i a l topological dua% i s t h e complex nwnber f i e l d i t s e l f .
(5.3) O r yet,
assuming t h a t by a
d i v i s i o n topological algebra w e s h a l l mean a
t o p o l o g i c a l d i v i s i o n a l g e b r a w i t h c o n t i n u o u s i n v e r s i o n , one c o n c l u d e s by t h e p r e v i o u s d i s c u s s i o n t h a t : A (complex) d i v i s i o n topological aZgebra is isomorphic t o C ( h e n -
c e , t o p o l o g i c a l l y i s o m o r p h i c ) i f , and only if, it possesses
(5.4)
a non-trivial
topological dual.
T h u s , some e x t r a c o n d i t i o n s a r e n e c e s s a r y , i n d e e d , t h a t a g i v e n (complex) topological d i v i s i o n algebra t o be t h e f i e l d C . In f a c t , t h e r e e x i s t d i v i s i o n t o p o l o g i c a l a l g e b r a s , i n t h e above s e n s e , t h a t a r e d i f f e r e n t from t h e complex number f i e l d i t s e l f . T h i s h a p p e n s , € o r i n s t a n ce, i n t h e (complex) a l g e b r a C t t l o f a l l r a t i o n a l f u n c t i o n s of one complex v a r i a b l e ; t h u s , t h e l a t t e r a l g e b r a s h o u l d a d m i t no n o n - t r i v i a l c o n t i n u o u s l i n e a r f o r m s ( c f . J.H.
WILLIAMSON [I : p. 7311 a n d / o r L . WAELBROECI:
[4:p. 134 f f . ] ) . 6. Maximal i d e a l s Given a n a l g e b r a E , a s u b s e t I of E i s s a i d t o be a l c f t ( r e s p . right) i d e a l of E l i f I i s a v e c t o r s u b s p a c e of E s u c h t h a t a x ( r e s p . x a ) b e l o n g s t o I , f o r a n y a e E and x el. Thus, I i s , i n p a r t i c u l a r , a suba l g e b r a of E s a t i s f y i n g a l s o t h e l a s t c o n d i t i o n . A s u b s e t of E which i s b o t h a l e f t and r i g h t i d e a l i s s a i d t o b e
a
2-sided i d e a l
of E .
I n a l l t h a t f o l l o w s by an i d e a l of an algebra E, We s h a l l always mean a
64
11 SPECTRUM (LOCAL THEORY)
2-sided i d e a l unless i t i s stated otherwise. Moreover, w e s h a l l a l w a y s unders t a n d , a s p a r t of t h e d e f i n i t i o n , t h a t a g i v e n i d e a l i n E w i l l b e a
proper non-trivial i d e a l , i.e.,
an i d e a l d i f f e r e n t from b o t h t h e t r i v i a l
i d e a l {Ol ( o r e l s e zero-ideal) The set of
and t h e whole a l g e b r a E .
( 2 - s i d e d ) i d e a l s o f E c a n be " p a r t i a l l y o r d e r e d " by
i n c l u s i o n , sc) t h a t by a mmimal ( 2 - s i d e d ) i d e a l w e s h a l l mean a n y maximal e l e m e n t of t h e l a t t e r s e t , i . e . ,
an element n o t g e n u i n l y c o n t a i n e d i n
any o t h e r element o f t h e s e t . Thus, on t h e b a s i s o f Lemma 6.1.
Zorn's Lemma
one now o b t a i n s .
Let E be an algebra w i t h an i d e n t i t y element. Then, every i d e a l
of E i s contained i n a maximal i d e a l .
Proof.
The set o f i d e a l s of E o r d e r e d by i n c l u s i o n i s a p a r t i a l -
l y o r d e r e d s e t , i n s u c h a way t h a t , f o r e v e r y chain (Le., t o t a l l y order-
ed s e t ) of i d e a l s o f E , t h e i r u n i o n i s e a s i l y s e e n t o b e a g a i n an i d e -
a l of E ; t h u s , t h e a s s e r t i o n f o l l o w s now by a n a p p l i c a t i o n of Z o r n ' s Lemma (see, f o r i n s t a n c e , N . BOURBAKI [l :Chap. 3 ; p. 3 5 , C o r o l l a i r e 11) I
.
On t h e o t h e r h a n d , t h e f o l l o w i n g p r o v i d e s a u s e f u l c r i t e r i o n t h a t a g i v e n i d e a l i n a commutative a l g e b r a t o b e maximal. Lemma 6.2.
Let E be a c o m t a t i v e algebra w i t h an i d e n t i t y element. Then,
the folloljing a s s e r t i o n s are equivalent: 1 ) M i s a maximal i d e a l of
E.
2) The quotient algebra E/M i s a (conunutative) d i v i s i o n algebra (hence, a
f i e l d over 0 .
Proof. Suppose t h a t M i s a maximal i d e a l of E and l e t a € E l w i t h a $ M I s o t h a t a + M = [ a ] # 0 i n E / M . Moreover, i f I i s t h e ( 2 - s i d e d ) id e a l of E g e n e r a t e d by M U { a ) , one h a s M G I , p r o p e r l y , so t h a t I =E , hence e = m t a x , w i t h m E M and X E E . T h u s , by t a k i n g t h e l a s t r e l a t i o n "modulo M"
one o b t a i n s
[el = [ax! = [a]-[zJ ,
s o t h a t [ a ] i s an i n v e r t i b l e e l e m e n t i n E / M , t h a t i s 1) implies 2). On t h e i s a d i v i s i o n a l g e b r a and I i s a n i d e a l of E which
o t h e r h a n d , i f E/M
p r o p e r l y c o n t a i n s M I t h e n t h e r e e x i s t s a n element a E i , w i t h a#M.Thus, (6.1)
i s t r u e , f o r some x e E l so t h a t e - a x E M G I ; h e n c e , e e l s i n c e a E I ,
t h a t i s I = E which p r o v e s t h a t 2) implies 1 ) a s w e l l ,
and t h i s f i n i s h e s
the proof. NOW, given a c o m t a t i v e algebra E w i t h an i d e n t i t y element, an element x in E i s i n v e r t i b l e i f , and o n l y i f , it does not belong t o any i d e a l of E : The con-
6.
65
MAXIMAL IDEALS
f o r a n y a € E , t h e principal i d e a l (c)= { a x : x e B } i n E g e n e r a t e d by a c o n t a i n s a , f o r x = e l so t h a t
d i t i o n i s o b v i o u s l y n e c e s s a r y . Moreover,
by h y p o t h e s i s a must be i n v e r t i b l e , and t h i s p r o v e s t h e a s s e r t i o n . Thus, a s a c o n s e q u e n c e of t h e l a s t s t a t e m e n t and Lemma 5 . 1 ,
we
conclude t h a t :
An element x of a given c o m t a t i v e algebra E w i t h an i d e n t i t y element i s i n v e r t i b l e i f , and only i f , it i s not contained i n
(6.2)
any maximal i d e a l o f E . The l a s t p r o p o s i t i o n i s o f t e n a p p l i e d i n t h e s e q u e l . The p r e c e d i n g c a n b e e x t e n d e d t o a l g e b r a s which do Rot n e c e s s a r i l y have i d e n t i t y elements: Thus, by a regular i d e a l
o f a g i v e n a l g e b r a E w e s h a l l mean a
( 2 - s i d e d ) i d e a l I o f E i n s u c h a way t h a t t h e q u o t i e n t a l g e b r a E / I has an i d e n t i t y element. T h e r e f o r e , i f
[ a ] = a + I i s s u c h i n E/I, we s h a l l
u s u a l l y c a l l a E E an i d e n t i t y of E modulo I ; hence , one g e t s by d e f i n i t i o n
[.I
(6.3)
=
rxi.1.r
=
,
o r , equivalently,
ax-xeI
(6.41
and x a - x e I ,
f o r eve ry element x i n E . N o w it i s c l e a r t h a t
i f the given algebra E has already an i d e n t i t y e -
lement, t h e n t h i s i s a n i d e n t i t y o f E modulo any ( 2 - s i d e d ) i d e a l o f E , h e n c e i n t h a t c a s e every i d e a l i n E i s regular. Thus, t h e d e s i r e d e x t e n s i o n of t h e p r e c e d i n g d i s c u s s i o n i s now subsummed i n t o t h e f o l l o w i n g . Lemma 6 . 3 . Let E be an algebra. Then, every regular i d e a l of E i s contained
i n a maximal regular i d e a l . In p a r t i c u l a r , a regular i d e a l M o f a c o m t a t i v e algebra E i s maximal i f , and only i f , E/M i s a l c o m t a t i v e l d i v i s i o n algebra (hence, i n p a r t i c u l a r , a f i e l d over C). Proof. Lemma 6 . 1 ,
The p r o o f o f t h e f i r s t a s s e r t i o n g o e s on s i m i l a r l y a s i n s i n c e t h e s e t of r e g u l a r i d e a l s o f E o r d e r e d by i n c l u s i o n
i s s t i l l a n " i n d u c t i v e o r d e r e d s e t " ( i . e . , e v e r y c h a i n i n t h e s e t ad-
m i t s an u p p e r bound; c f ; N . BOURBAKI [l: Chap. 3 ; p. 34, § 4]),
so t h a t t h e
a s s e r t i o n f o l l o w s a g a i n by Z o r n ' s Lemma. On t h e o t h e r h a n d , t h e
se-
cond p a r t of t h e lemma f o l l o w s by a p p l y i n g a s i m i l a r argument a s i n t h e proof o f Lemma 6.2,
t a k i n g t h u s a s e i n ( 6 . 1 ) a n i d e n t i t y of
E
modulo M . I F u r t h e r m o r e , i n a n a l o g y w i t h t h e comment f o l l o w i n g Lemma 6 . 2 ,
66
I1 SPECTRUM (LOCAL THEORY)
one g e t s t h e f o l l o w i n g r e s u l t . An “ a n a l y t i c ” form of it ( i n p a r t i c u -
l a r , of t h e n e x t C o r o l l a r y 6 . 1 ) , and of t h o s e r e m a r k s t o o , w i l l be c o n s i d e r e d i n s u b s e q u e n t s e c t i o n s (see C h a p t . 111).
Lemma 6.4. Let E be a commutative algebra. Then an element x of E i s quasisingular i f , and only i f , t h e following subset of E , (6.5)
I = { ~ - x Y :y e E 1
i s an i d e a l . Moreover, I i s , in f a c t , a regular i d e a l o f E not containing x, t h e l a t t e r being an i d e n t i t y of E modulo I.
Proof. I f I = E , t h e n by
X‘E
E is quasi-singular,
(6.5) x o y = O ,
t h e n I #E . T r u e , b e c a u s e
,if
f o r some y f E which i s a c o n t r a d i c t i o n .
T h u s , i f I f E , one e a s i l y p r o v e s t h a t I i s , i n f a c t , a r e g u l a r i d e a l of E w h i l e t h e g i v e n e l e m e n t x i s a n i d e n t i t y of E modulo I . M o r e o v e r , x & I , because, i f x e I
I , f o r e-
t h e n by ( 6 . 5 ) o n e g e t s y = ( y - x y l + x y
v e r y y e E , so I = E which i s a g a i n a c o n t r a d i c t i o n t o o u r h y p o t h e s i s . On t h e o t h e r h a n d , i f x e E i s q u a s i - r e g u l a r ,
t h e n it i s e a s i l y proved
b y ( 6 . 5 ) t h a t I = E , which i s s t i l l a c o n t r a d i c t i o n i f w e assume t h a t I i s an i d e a l , and t h i s f i n i s h e s t h e p r o o f . I
Now, it i s c l e a r b y t h e p r e c e d i n g p r o o f t h a t Lemma 6 . 4
is still
v a l i d € o r a non-commutative a l g e b r a E , b y t a k i n g i n s t e a d t h e e l e m e n t
a e E t o be r i g h t ( r e s p . l e f t ) q u a s i - s i n g u l a r ;
then, I is a r i g h t o r
l e f t i d e a l i n E , r e s p e c t i v e l y . F o r s i m p l i c i t y ’ s s a k e , however, w e a s sumed F commutative. W e c o n c l u d e w i t h t h e f o l l o w i n g .
Corollary 6.1. Let E be a c o m t a t i v e algebra.Then, an element
3:
in E
is
quasi-regular i f , and only i f , f o r every niazimal i ~ p t ? iud e~a l M o f E , there e x i s t s an element y e E i n such a way t h a t
x + y-xy = x
(6.6)
0
Proof. I f x e E i s q u a s i - r e g u l a r ,
y
E M.
t h e n x o y = O E M , f o r some y € E ,
and f o r e v e r y (maximal r e g u l a r ) i d e a l M of E . C o n v e r s e l y , i f x e E i s quasi-singular,
t h e n t h e i d e a l I of E g i v e n by ( 6 . 4 )
i s r e g u l a r , so
that(Lemma 6 . 3 ) , it i s c o n t a i n e d i n a maximal r e g u l a r i d e a l , s a y M , o f E . T h e r e f o r e , t h e r e e x i s t s by h y p o t h e s i s y E E , w i t h x+ y - x y e M I so t h a t s i n c e by ( 6 . 4 ) y - x y e I ~ M , f o r e v e r y ~ E E one , concludes t h a t x M ; thus, for
every y e E ,
one h a s y = ( y
-
€
x y l + x y e M , t h a t i s M = E . Name-
l y , a c o n t r a d i c t i o n and t h i s f i n i s h e s t h e p r o o f .
I
W e c o n s i d e r now, and t h i s i s of c o u r s e o u r c o n c e r n , maximal ide-
a l s i n t o p o l o g i c a l a l g e b r a s (see a l s o t h e n e x t s e c t i o n ) . Thus,we know
7.
61
CHARACTERS
f o r i n s t a n c e t h a t e v e r y maximal i d e a l i n any Banach a l g e b r a i s c l o s e d . On t h e o t h e r h a n d , a s w e p r e s e n t l y see, t h i s f a c t i s s t i l l v a l i d i n t h e more g e n e r a l c l a s s of 4-(topologica1)algebras. T h a t i s , w e h a v e .
Theorem 6.1. Let E be a &-algebra. Then, every mazima2 regu2ar i d e a l of E i s closed. In particu2ar, every maximal i d e a l o f a Q-a2gebra with an i d e n t i t y element i s closed. -
M =E.
C o n s i d e r a maximal r e g u l a r i d e a l M of E and suppose t h a t
Proof.
Thus, i f a i s a n i d e n t i t y of E modulo M and Gq t h e s e t of q u a s i -
r e g u l a r e l e m e n t s of E , t h e s e t a + G q i s a n ( o p e n ) neighborhood of a i n E (Lemma I ; 6.4) ; h e n c e , one o b t a i n s ia+Gq)
that is, a
-m
n
M
i0
,
f o r some e l e m e n t m E M . T h e r e f o r e , i f y e E
E Gq,
quasi-inverse of a
is the
one g e t s
-in,
la-rnioy= (a-m)+y- ia-m)y=O that is, a =rn+(ay- y l hypothesis f o r M .
-
M , that
-
my,
t h u s a E M , which i s a c o n t r a d i c t i o n t o t h e
Thus, M f E , so t h a t ( C o r o l l a r y I ; 1.1)
we obtain M =
i s t h e a s s e r t i o n , w h i l e t h e r e s t of t h e s t a t e m e n t i s c e r t a i n -
l y clear. I On t h e o t h e r hand, t h e s i t u a t i o n d e s c r i b e d by t h e above theorem
may be d i f f e r e n t f o r an a r b i t r a r y t o p o l o g i c a l a l g e b r a ( e v e n a l o c a l l y ri-convex
one) which i s n o t a & - a l g e b r a ; namely, t h e r e may e x i s t maxi-
m a l i d e a l s which a r e dense s u b s e t s of t h e g i v e n t o p o l o g i c a l a l g e b r a . Whether t h i s may n o t happen in case of a commutative Frgchet l o c a l l y in-eonvex
algebra seems t o be s t i l l an open q u e s t i o n (Michae2’s conjeetttre; see T . HUSAIN[2]). In t h i s respect, c f . a l s o t h e next section, i n p a r t i c u l a r ,
Theorem 1 . 1 .
7. Characters. Closed maximal ideals The s i t u a t i o n d e s c r i b e d b y Lemma 6 . 2 of t h e p r e v i o u s s e c t i o n i n c o n n e c t i o n w i t h t h e Gel’fand-Mazur Theorem ( c f . Theorem 5 . 2 o r i t s l e s s t e c h n i c a l form g i v e n by C o r o l l a r y 5 . 1 ) ,
leads u s t o our next o b j e c t i -
v e ; namely, t h e r e l a t i o n between maximal i d e a l s of a g i v e n t o p o l o g i c a l a l g e b r a E and (complex) a l g e b r a morphisms o f E i n t o t h e complexes, i n p a r t i c u l a r , c o n t i n u o u s o n e s . Thus, we s t a r t w i t h t h e f o l l o w i n g .
Definition 7.1. mean
a non-zero
L e t E be an a l g e b r a . Then, by a
character of E , we
(complex) morphism of E i n t o t h e ( a l g e b r a o f ) comple-
x e s C . The s e t of c h a r a c t e r s of E i s c a l l e d t h e algebraic speetiwv
of E
11 SPECTRUM (LOCAL THEORY)
68
and denoted by M ( E I . ( I t i s o n l y i n Chapt. V where w e c o n s i d e r
M(EI, i n
f a c t , a subspace of it a s an a p p r o p r i a t e t o p o l o g i c a l s p a c e ) . F u r t h e r more, t h e s e t M(E)+ = M(E)U {
(7.1)
u I,
i s c a l l e d t h e extended algebraic spectruv of t h e a l g e b r a E .
In t h i s respect, w e note t h a t
M(EJ may be t h e empty
s e t , even
i f L" i s a Banach a l g e b r a . However, t h e s i t u a t i o n i s q u i t e d i f f e r e n t , of c o u r s e , f o r a commutative Banach a l g e b r a ( i n t h e p r e s e n c e of a n i -
d e n t i t y e l e m e n t ) ; a s w e s h a l l see ( c f . Chapt. V ) t h i s i s s t i l l v a l i d f o r much more g e n e r a l (commutative!) t o p o l o g i c a l a l g e b r a s . Now, i f f i s a c h a r a c t e r of a g i v e n a l g e b r a E one h a s by d e f i n i t i o n the r e l a t i o n flxyl = f(xi.f(y),
(7.2)
f c r any x , y i n E l f b e i n g a l s o a (complex) l i n e a r form on t h e r e s p e c t i v e v e c t o r s p a c e E ; moreover, t h e r e i s an e l e m e n t x i n E , w i t h f i x ) # 0 . In p a r t i c u l a r ,
i f t h e a l g e b r a E has an i d e n t i t y element, say e , s a y -
i n g t h a t f i s " n o n - i d e n t i c a l l y z e r o " i s , of c o u r s e , e q u i v a l e n t w i t h f ( e l = l i n @,
thus f i s then " i d e n t i t y preserving".
On t h e o t h e r hand, one c o n c l u d e s t h a t every f e M ( E ) i s e s s e n t i a l l y a map of E onto C . I n d e e d , i t s u f f i c e s t o c o n s i d e r , f o r e v e r y X E C,
x
element - x
E E , where
f(x) =
c1
the
# 0 i n C by h y p o t h e s i s f o r f . (So one ob-
t a i n s , i n f a c t , t h a t e v e r y non-zero
(complex) l i n e a r form o n E i s an
o n t o map). Thus, our f i r s t statement r u n s a s follows.
Lemma 7.1.
Let E be an algebra and f a character of E . Then, the kernet of 5
the set
i.e.,
k e r ( f ) = C x e E : f ( x l = 01,
(7.3) ?:B
a
2-sided regular ninceimat i d e a t of E.
Proof. By ( 7 . 3 ) and t h e h y p o t h e s i s f o r f one c o n c l u d e s t h a t ker If) i s a 2 - s i d e d i d e a l of E . Moreover,
S E
1
if
f ( a ) = X # O i n C,
f o r some a
E l t h e e l e m e n t ~a E E i s an i d e n t i t y of E modulo ker if), so t h a t t h e l a t t e r i s a r e g u l a r i d e a l of E. On t h e o t h e r hand, s i n c e f i s a (comE
p l e x ) l i n e a r form on E , i t s k e r n e l
E of ximal be
codimension 1 (i. e . ,
( 7 . 3 ) w i l l be a v e c t o r subspace of
dim (Elker(f)l= I ) , h e n c e , e q u i v a l e n t l y , a ma-
v e c t o r subspace of E ; s o s i n c e ker (f) i s an i d e a l of E l it w i l l
t h e n a maximal i d e a l of E l and t h i s p r o v e s t h e a s s e r t i 0 n . I
7.
Remark.-
69
CHARACTERS
C o n c e r n i n g t h e l a s t p a r t of t h e above p r o o f , w e c o u l d
a p p l y , o f c o u r s e , a more e l e m e n t a r y ( h o w e v e r , n o t s i m p l e r ! ) argument t o p r o v e t h a t t h e regular i d e a l of E defined by (7.31 i s maximal.
Thus, sup-
p o s e t h a t I i s a n i d e a l o f E which p r o p e r l y c o n t a i n s k e r ( f I ; t h e r e f o -
re, t h e r e e x i s t s x e I , w i t h f ( x l $ 0.
T1a e E
of
t h e preceding proof
s i n c e f l u l = l , one o b t a i n s s i n c e f ( x ) # 0, E
x-f(x)u
E
k e r ( f l C I . Thus, f l x l u e I so t h a t ,
one c o n c l u d e s t h a t u e I , t h a t i s , a c o n t r a d i c t i o n t o
the hypothesis f o r I .
lement u
Hence, by u s i n g t h e e l e m e n t u =
( i . e . , an i d e n t i t y of E modulo k e r i f l ) ,
( I n t h i s r e s p e c t , w e u s e d t h e f a c t t h a t i f a n e-
E: i s an i d e n t i t y of E modulo an i d e a l I of E , then u
E
I , unless I = E .
Moreover, u i s a l s o an i d e n t i t y o f E modulo any other idea2 J of E containing I ) . T h u s , d e n o t i n g by M l E l t.he s e t of 2 - s i d e d r e g u l a r maximal ideals o f an a l g e b r a E , one g e t s by Lemma 7 . 1 t h e f o l l o w i n g map
9:
(7.4) Now,
M(EI-MIEI:
f w @(f/:= ker(fI.
t h e n e x t s t a t e m e n t s a y s t h a t t h e map 9 i c , i n f m t , one-to-one.
Namely, w e h a v e . Lemma 7.2.
Let E be an a2gehra and f , g any
TWO
characters of E. Then, f = g
if, and o n l y i f . k e r ( f l = kerigl. I n p a r t i c u z a r , t h e map @ defined by ( 7 . 4 ) i s an
injection.
Proof. Of c o u r s e , it s u f f i c e s t o p r o v e o n l y t h e " i f " p a r t of t h e a s s e r t i o n . Thus, i f a e E then f ( a ) =g(al - 1
,
i s an i d e n t i t y of E modulo ker(fi = k e r , ' g I ,
b e c a u s e by h y p o t h e s i s f, g a r e n o n - i d e n t i c a l l y z e r o .
Hence, one h a s
x - f ( x ) aF k e r i f ) = kerig) ,
(7.5) f o r every x e E
,
s o t h a t gix) = f f x ) g ( a l , i.e. ,f (x)= g ( x ) , f o r e v e r y x
E
E
,
and t h i s p r o v e s t h e a s s e r t i o n . I W e c o n s i d e r n e x t continuous characters of a g i v e n t o p o l o g i c a l a l -
g e b r a , i n p a r t i c u l a r , a s it c o n c e r n s t h e i r r e l a t i o n t o t h e ( c l o s e d reg u l a r ) maximal i d e a l s of t h e a l g e b r a ; t h e i r i n t e r r e l a t i o n i s e s t a b l i s h e d by a r e s t r i c t i o n o f t h e map ( 7 , 4 ) ( s e e Theorem 7 . 1 b e l o w ) . Thus, w e f i r s t have t h e f o l l o w i n g . Definition 7.2.
L e t E b e a g i v e n t o p o l o g i c a l a l g e b r a . Then by t h e
( o r s i m p l y spectrum) o f E l w e s h a l l mean t h e s e t of c o n t i n u o u s c h a r a c t e r s o f E , d e n o t e d by m l E i . (Thus an e l e m e n t f e mfE) topo2ogicaZ s p e c t m
w i l l b e a non-zero c o n t i n u o u s ( c o m p l e x ) morphism of E i n t o t h e ( a l g e b r a C of t h e ) complexes).
I1 SPECTRUM (LOCAL THEORY)
70
I t i s c l e a r t h a t a s i m i l a r comment t o t h a t f o l l o w i n g D e f i n i t i o n
7 . 1 i s h e r e i n o r d e r . We a l s o d e f e r f o r C h a p t e r V t o c o n s i d e r m ( E )
a s a s u i t a b l e t o p o l o g i c a l s p a c e , r e t a i n i n g however t h e t e r m i n o l o g y of Moreover, s i m i l a r l y t o ( 7 . 1 ) , w e s t i l l con-
t h e above D e f i n i t i o n 7 . 2 . sider the set
V"Y(El+ =
(7.6)
m(E.,u( 0 3 , ( o r s i m p l y extended spec -
which w e c a l l t h e extended topoZogica1 spectrum
tmun) of t h e t o p o l o g i c a l a l g e b r a E . Thus, w e c o n c l u d e t h a t ; t h e r e s t r i c t i o n o f t h e map 1 7 . 4 ) t o m ( E ) C M ( E ) d e f i n e s a oneto-one correspondence between continuous characters of E ( i . e . ,
(7.7)
e l e m e n t s of 97UE)) and closed regular maximal
( 2 - s i d e d ) i-
deals o f E . W e d e n o t e t h e l a s t s e t i n ( 7 . 7 ) by h4iE). Thus, it i s c l e a r t h a t , f o r e v e r y f e m ( E ) , one h a s kerifi e M ( E ) G M ( E ) , a s f o l l o w s from Lemma 7.1
and
t h e c o n t i n u i t y of f. Now,
it i s o u r main o b j e c t i v e i n t h e e n s u i n g d i s c u s s i o n t o en-
s u r e , under s u i t a b l e r e s t r i c t i o n s
on t h e p a r t i c u l a r t o p o l o g i c a l a l g e -
b r a E c o n s i d e r e d , t h a t t h e above c o r r e s p o n d e n c e ( 7 . 7 ) i s , i n f a c t , a b i j e c t i o n (see Theorem 7 . 1 and i t s C o r o l l a r y 7 . 1 .
On t h e o t h e r h a n d ,
t h e n e x t c h a p t e r p r o v i d e s a more g e n e r a l framework f o r t h e t o p o l o g i c a l algebras considered h e r e ) . W e n e e d f i r s t some more p r e l i m i n a r y m a t e r i a l . Th.us, s u p p o s e t h a t E i s a t o p o l o g i c a l a l g e b r a and I
a ( 2 - s i d e d ) closed i d e a l o f E . S o t h e
q u o t i e n t a l g e b r a E/I, b e i n g a H a u s d o r f f t o p o l o g i c a l v e c t o r s p a c e , i s
also a t o p o l o g i c a l a l g e b r a ; i . e . , t h e m u l t i p l i c a t i o n i n E / I i s separately continuous. I n d e e d , s i n c e t h e t o p o l o g y i n E / I
is a " f i n a l ( v e c t o r space)
t o p o l o g y " w i t h r e s p e c t t o t h e c a n o n i c a l ( q u o t i e n t ) map @ : E - + E / I , one h a s t o p r o v e t h e c o n t i n u i t y o f t h e map l a o @ : E+
E/I
,
f o r every a E E . Therefore, s i n c e
l.(@(x)) = &-@(xc)= @(a).@(xc) = @(ax) = @ ( l a ( x ) ) , with
x e E , one o b t a i n s la o @ = @ o La : E-+
(7.8) where
@ o l a , with a e E ,
E/I ,
i s c e r t a i n l y c o n t i n u o u s , and t h i s p r o v e s t h e
a s s e r t i o n . ( I n t h i s r e s p e c t , see a l s o C h a p t . I V ; S e c t i o n 2 ) . I n p a r t i c u l a r , i f E i s a l o c a l l y convex a l g e b r a and I a s b e f o r e ,
7.
CHARACTERS
71
t h e n one p r o v e s t h a t E/I i s a l o c a l l y c o n v e x a l g e b r a t o o , by a p p l y i n g a s i m i l a r argument to t h e above c o n c e r n i n g t h e ( q u o t i e n t ) l o c a l l y convex s p a c e E I I . F u r t h e r m o r e ,
i f E i s a l o c a l l y m-convex algebvaa, t h e n
this i s
s t i l l t r u e f o r a quotient algebra E / I : T h i s i s e a s i l y s e e n , s i n c e (7.9) with k = [x] e E / I ,
y i e l d s a s u b m u l t i p l i c a t i v e semi-norm on E / I , f o r any
s u c h p on E ( c f . a l s o Theorem 1;3.1). T h u s , w e now h a v e t h e f o l l o w i n g .
Lemma 7.3. Let E be a topological algebra with a continuous quasi-inversion and I a closed (2-sided) ideal of E. Then, the quotient algebra E / I i s a topologic a l algebra w i t h a continuous quasi-inversion t o o .
Proof. By t h e p r e c e d i n g E/I i s a t o p o l o g i c a l a l g e b r a . Moreover, i f x++x'denotes
t h e quasi-inversion
i n E , t h e n t h e c o n t i n u i t y of t h e
r e s p e c t i v e map i n E / I i s e q u i v a l e n t w i t h t h a t o f t h e map x -
[x']:E+E/I.
T h i s i s e a s i l y s e e n b y h y p o t h e s i s f o r E and t h e c o n t i n u i t y of t h e quot i e n t map @ : B + E / I ,
s i n c e @ i s a n a l g e b r a morphism p r e s e r v i n g
the
circle operation. I We a r e now i n t h e p o s i t i o n t o s t a t e t h e f i n a l b a s i c r e s u l t o f t h i s s e c t i o n . Namely, w e h a v e .
Theorem 7.1. Let E be a commutative l o c a l l y convex algebra w i t h a continuous quasi-inversion and M a closed regular maximal i d e a l of E . Then, the quotient algebra E/M i s the algebra C o f the complexes, w i t h i n a topological algebra isoniorphism. I n p a r t i c u l a r , the conclusion holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra E .
Proof. By Lemma 7 . 3 t h e q u o t i e n t a l g e b r a E/M i s a l o c a l l y convex algebra with a continuous quasi-inversion,
r e s p . , a l o c a l l y in-convex
a l g e b r a ( c f . a l s o ( 7 . 9 ) ) . Moreover, by h y p o t h e s i s f o r M a n d L e m m a 6 . 3 ,
E/M i s a d i v i s i o n a l g e b r a , s o t h a t B/M i s , i n f a c t , a (commutative) l o c a l l y convex d i v i s i o n algebra w i t h a continuous inversion
(the l a s t assertion is
e a s i l y d e r i v e d by Lemma 7 . 3 and ( 1 . 4 ) 1 , r e s p . , a ZocaZZy m-emvex d i v i s i o n
algebra.
S o t h e a s s e r t i o n f o l l o w s now f r o m an a p p l i c a t i o n of t h e G e l ' -
fand-Mazur Theorem ( s e e C o r o l l a r y 5 . 1 ) . I
Corollary 7.1. Let
E be a commutative l o c a l l y convex algebra w i t h a continu-
ous quasi-inversion, i n p a r t i c u l a r ( c f . Lemma 3 . 1 ) , any c o m t a t i v e l o c a l l y mconvex algebra. Then, every closed regular maximal i d e a l M of E gives r i s e t o a continuous character f of E ; namely, t h a t one defined by t h e following commutative d i -
72
I1 SPECTRUM (LOCAL THEORY)
agramm :
(7.10)
iiere j denotes t h e topological algebra isomorphism given by t h e above Theorem 7 . 1 . 1
F i n a l l y , one g e t s t h e f o l l o w i n g u s e f u l r e s u l t .
I t is mostly i n
t h i s form t h a t w e a r e g o i n g t o a p p l y Theorem 7 . 1 i n t h e s e q u e l . T h a t i s , we have.
Corollary 7.2. Let E be a c o m u t a t i v e localZy convex aZgebra w i t h a continuous quasi-inversion.
Then, t h e r e e x i s t s a one-to-one
and onto correspondence be -
tween t h e s e t o f continuous characters o f E ( i . e . , t h e spectrwn m(E)o f t h e algebra El and t h e s e t M(EI of cZosed regular maximal ideaZs o f E, given by t h e r e l a t i o n ( 7 . 4 1 . I n p a r t i c u l a r , t h i s i s t m e f o r every l o c a l l y m-convex algebra, so t h a t i n t h i s case one has t h e r e l a t i o n m(E)= i/d(El ,
(7.11)
within a bijection. Proof.
I t i s c l e a r ( c f . Lemma 7 . 2 )
i n d e e d one-to-one. f
&
t h a t t h e map i n q u e s t i o n i s
F u r t h e r m o r e , i f MeM:i’)
m(Eiw i t h k e r ( f ) = M , a s follows f r o m t h i s finishes the proof.1
t h e n by ( 7 . 1 0 ) o n e g e t s t h e r e l a t i o n f = jo@ i n ( 7 .
l o ) , and
On t h e o t h e r hahd, a u s e f u l c o n s e q u e n c e o f t h e p r e c e d i n g , i n c o n j u n c t i o n w i t h Theorem 6 . 1 ,
is t h e following r e s u l t extending t h e
s i t u a t i o n one h a s i n ( c o m m u t a t i v e ) Banach a l g e b r a s .
Corollary 7.3. L e t E be a &-algebra. Then, every character of
E is continu-
ous, i.e., one has
M(E) = m ( E l C E’,
(7.12)
where E’ denotes t h e topoZogicaZ dual o f E. In p a r t i c u l a r , if E i s a c o m t a t i v e Zocally convex Q-aZgebra w i t h a c o n t i nuous quasi-inversion, o r y e t any commutative Locally m-convex Q-algebra ( h e n c e , a
l o c a l l y convex i r e s p . l o c a l l y m-convex.) generazized Waelbroeck aZgebra ( S c h o l i u m then one o b t a i n s
4.2)),
M(EJ = MIE!
(7.13)
where
‘1
I
”
Pi-ocf.
‘m(El= M ( E 1 ,
stands f o r a b i j e c t i o n . I f f i s a c h a r a c t e r of E , t h e n i t s k e r n e l , b e i n g a r e g u l a r
7.
maximal ( 2 - s i d e d ) i d e a l of E: (Lemma 7 . 1 ) c l o s e d s u b s p a c e of E .
73
CHARACTERS
In particular,
,
i s a l s o by Theorem 6 . 1 a
ker(fI
i s a c l o s e d ( v e c t o r ) sub-
s p a c e of E of c o d i m e n s i o n 1 , so t h a t f i s a c o n t i n u o u s l i n e a r form on E
( c f . N . HORVhH [I: p. 107, P r o p o s i t i o n 71 )
,
h e n c e , i n p a r t i c u l a r , a con-
t i n u o u s c h a r a c t e r of E which p r o v e s ( 7 . 1 2 ) . Now, t h e r e s t o f t h e a s s e r t i o n i s a g a i n a c o n s e q u e n c e of Theorem 6 . 1 ,
Corollary 7 . 2 ,
and of
( 7 . 1 2 1 , and t h i s t e r m i n a t e s t h e p r o o f . I Scholium 7.1.
The h y p o t h e s i s i n t h e f i r s t p a r t of t h e p r e v i o u s co-
r o l l a r y t h a t E i s a 4 - a l g e b r a was made o n l y i n c o n n e c t i o n w i t h Theorem
6.1.
Thus, more g e n e r a l l y , t h e f o l l o w i n g i s t r u e : For every t o p o l o g i c a l
algebra E w i t h t h e p r o p e r t y t h a t every r e g u l a r maximal ( 2 - s i d e d ) i d e a l i n E is c l o s e d , one concludes t h a t every character o f E i s continuous. I n o t h e r words, i.he r e l a t i o n M ( E ) = M I E ) i m p l i e s MIEI = m ( E I . F o r c o n v e n i e n c e w e have u s e d i n s t e a d t h e s t r o n g e r , however l e s s t e c h n i c a l , h y p o t h e s i s t h a t t h e a l g e b r a E i s a &-algebra; t h i s w i l l be a l s o t h e case i n several instances below, w h i l e t h e more g e n e r a l s i t u a t i o n d i s c u s s e d h e r e w i l l be a l w a y s c l e a r from t h e c o n t e x t .
W e c o n c l u d e t h e s e c t i o n and t h i s c h a p t e r t o o w i t h c e r t a i n comments and some ( p a r t i a l ) r e s u l t s r e f e r r i n g t o Q - a l g e b r a s ; more p r e c i s e l y , t h i s h a s t o do w i t h a c e r t a i n p a r t i c u l a r n o t i o n , t h e Gel’fand
transform,
which a l s o w e a r e g o i n g t o a p p l y s e v e r a l t i m e s i n s u b s e q u e n t
c h a p t e r s of t h i s book. Thus, w e f i r s t have t h e n e x t .
Definition 7.3. L e t E b e a n a l g e b r a a n d M I E ) i t s a l g e b r a i c s p e c t r u m (i.e., t h e s e t of c h a r a c t e r s of E ; c f . D e f i n i t i o n 7 . 1 ) .
l e t x b e a n e l e m e n t of E. Then, t h e
Moreover,
Gel’fand transform o f x , d e n o t e d by
3 , i s d e f i n e d t o b e t h e map
2 : M(EI * C : f ct 2 i f I : = f 1x1 .
(7.14)
Thus, i f
F ( M I E 1 , CI d e n o t e s t h e s e t o f a l l complex-valued maps on
M I E ) , t h e r e s u l t i n g map
(7.15)
:E
+
F ( M ( E I , C I : x -%(xi
( g i v e n by ( 7 . 1 4 ) ) i s c a l l e d t h e
:=
2
Gel’fand map of t h e g i v e n a l g e b r a E.
I f E i s a topological algebra, the preceding terminology i s , i n particular,
r e s e r v e d f o r t h e r e s t r i c t i o n of
(7.15) to
t h e ( t o p o l o g i c a l ) s p e c t r u m of E ( s e e D e f i n i t i o n 7 . 2 ,
7 7 U E ) G M(E)
,
and C h a p t . V ) .
On t h e o t h e r h a n d , i f w e c o n s i d e r F ( M ( E I , C 1 a s a (complex) a l g e -
11 SPECTRUM ( L O C ~THEORY)
74
bra, with pointwise defined operations, we readily see by the preceding relations that t h e Gel'fand map o f an algebra E i s an algebra rnorphism of E into FfM(EI, C).
Thus, the relation
I~~S)CFIM(EI,CI
E* =
(7.16)
defines E A as a subalgebra of F I M f E I , C I , called the Gel'fand transform algebra of E .
NOW, if we are given a topological algebra E l the respective Gel'fand transform algebra E* consists, in effect, of (complex-valued) continuous functions on mIEI, the latter set being suitably topologized; as we shall see (cf. Chapt. V), one at least natural topology, we usually consider on T I T l E I , is the weakest one making the elements of E ^ continuous functions ( G e l ' f a n d t o p o l o g y ; but cf. also Chapt. IX) . Conclusively, we just apply (7.15) to derive some useful relations concerning the spectrum SpE (x) of an element x in a given alge-
bra E. But first we need the next lemma.(A form of a converse is also valid but deferred for Chapt. 111; see there Theorem 5.1). So we have.
Lemma 7.4. Let E be an aZgebra and x a quasi-regular element o f E. Then, one has t h e reZation f(xl # 1 ,
(7.17) f o r every character f of E.
Proof. If x o y = 0, for some y e E l
(7.18)
one gets
f ( x o y I = f ( x +y - x y I = f f x I + f f y I - f f x I f ( y I = O ,
for every f e M f E I . Thus, if f ( x l = l , for some f e M f E J , one gets by (7.18) I=O,i.e., a contradiction, and this proves the assertion. 1 Thus, a useful consequence of the previous lemma is now the following result, a "continuous analogon" of which will be given lateron (see Chapt. 111; Theorem 6.2, or even Corollary 7.5 below). That is, we have.
Corollary 7.4. L e t form
2
E b e an aZgebra and
R(O/ t h e range o f t h e Gel'fand trans-
o f an element x in E (cf. (7.15)). Then, one has
(7.19)
R(2)L S p E ( x c ) .
I n p a r t i c u l a r , one has t h e r e l a t i o n
(7.20)
~ ( M ( E I += I spE(x)
u {OI,
f o r every x i n E. Furthermore, one o b t a i n s
(7.21)
r E fxl =
sup J j Y x I J ,
S EM I E I
7. CHARACTERS
75
f o r every x i n E. Proof. If fixl=O, for some f EM~EI, then x can not be a regular element of E, so that (Definition l .2) 0 e SpElxl. On the other hand, if ?if) = ffxl= X f ; O in C , where f E MlEl, then one has f ( 1T z l = 1 ; hence (Lem1 should be a quasi-singular element of E, so A f SpE(xl and ma 7.4) , h x this proves (7.19). NOW if A E SpElxl , with A # 0, then (cf. Definition 1
1.2) - x
x
must be quasi-singular, so that (Lemma 7.4) there exists f 1
e MfEI, with f ( - xx ) = l , thus f l x l = ~ i f i = X ;hence, X E R i ; ) , finally obtains the relation (7.22)
spE(x)
so that one
u I O I = R( 2 i u {OI = ;IMIEI +i ,
which together with (7.19) proves (7.20). Finally (7.21) is now immediate by (1.10) and (7.20). I
Now the following proposition is still another application of the above Lemma 7.4 and will repeatedly be used in subsequent chapters.
Proposition 7.1. Let E be a Q-algebra. Then, i t s ( t o p o l o g i c a l ) spectmm mfEl i s an equicontinuous subset of E' ( t h e topologicaZ dual of E l and hence r e l a t i v e l y compact i n E,' (the weak topological dual of E ; see Chapt. V )
.
If U is a balanced neighborhood of O E E consisting of quasi-regular elements of E (Lemma I ; 6.4), then one has t h e r e l a t i o n Proof.
UE. f?7YfEllo,
(7.23)
where the second member of the last relation denotes the polar s e t of mrElr E'in
E: Indeed, if -3 E U and Iffx)I 2 1 , f o r some f €?;rZ(E), then 1 1 1 ffx) = A # 0 and 1-ix < 1. Thus -AX E U and f ( T z l = 1 , a contradiction ::i.nce --31 x is quasi-regular (Lemma 7.4). Therefore, one has ( f f d 51
(7.24)
,
for every f f 772(E/ and x E U which proves (7.23). Hence, (rrZ(E)IO is a neighborhood of 0 e E , therefore 772: f E l C ( m(El! O 0 an equicontinuous subset of E'. I Finally, we do have the next "continuous analogon" of Corollary 7.4, a stronger version of which will be derived in Chapter 111, just when the necessary preliminary material will be available. Thus, we have. Theorem 7 . 2 . Let E be a c o m t a t i v e localZy convex &-algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n ( i . e . , a commutative l o c a l l y convex WaeZbroeck
.
algebra; cf Definition 4.1). Then, for every element x i n E, one g e t s t h e r e l a -
76
11
SPECTRUM (LOCAL THEORY)
t i o n (cf. also the notation of Corollary 7.4)
R ( G I = 3 ( r n t E / ) = SPE(Xi.
(7.25)
In p a r t i c u l a r , one g e t s (7.26) Hence, the a s s e r t i o n holds t r u e f o r every commutative l o c a l l y m-convex 4-algebra with an i d e n t i t y element ( i . e . , a commutative l o c a l l y m-convex Waelbroeck algebra). I n t h i s r e s p e c t , if t h e given algebra E does not have an i d e n t i t y , one g e t s
S(7^n(EIf) = SpE(xl,
(7.27)
for every element x in E, while the r e l a t i o n ( 7 . 2 6 ) i s s t i l l t r u e . Boof.
By hypothesis and (7.12) one has M(EI = ??Z ( E l ,
so that by (7.19) R(;)rSpE(x), for every X E E. Moreover, for every h e SpElx), with h # 0, one concludes, based on the proof of (7.20), that h e R ( G ) (see also the comment after Definition 1.2). Thus, suppose that 0 e SpE(x:); then (Definition l.l), x can not be aregular element of E , so that x E M , for some maximal ideal M in E (cf.(6.2)). Hence, by (7.4) and (7.13) one obtains f(x1 =G(f) = O , for some f € MIE) = m(El, and this proves (7.25). Now (7.26) is immediate by (7.13) and (7.21) which actually finishes the proof, the rest of the assertion being now straightf0rward.m Now, given an algebra E and an element x of E, we denote by Z ( s ) the zero-set of the function 2 , the Gel'fand transform of x ; that is, one has by definition (7.28)
Z(?)=
{ f E M I E ) : G ( f ) = flx)=O}.
The following is now a direct consequence of the argument plied in the preceding proof. Namely, we have.
ap-
Corollary 7 . 5 . Let E be a c o m t a t i v e l o c a l l y convex Waelbroeck algebra (cf.
.
the above Theorem 7.2) Then, an element x i n E i s regular ( i . e . , i n v e r t i b l e ) if, and only i f , one has
Z ( 2 ) = G1;
(7.29) t h a t i s , equivalentzy, the r e l a t i o n
(7.30)
.;'.tf,
=f(x) f 0,
for every (continuous) character f of E. 1 In this respect, we finally note that a stronger version of the last result is still true, in case one has locally m-convex algebras. In fact, it suffices then to consider just advertibly complete alge-
3.
77
SCHUR'S LEMMA
b r a s i n s t e a d o f & - a l g e b r a s ( c f . T h e o r e m I ; 6.4. S e e a l s o C h a p t . 1 I I ; C o rollary 5.2). 8. Appendix: Schur's Lemma W e p r e s e n t i n t h i s a p p e n d i x a form of t h e c l a s s i c a l lemma
in
t i t l e which o f t e n a p p e a r s m a i n l y i n q u e s t i o n s c o n c e r n i n g t h e r e p r e s e n t a t i o n t h e o r y o f t o p o l o g i c a l a l g e b r a s . Though w e a r e n o t i n p a r t i c u l a r c o n c e r n e d w i t h t h a t p a r t of t h e t h e o r y i n t h i s book ( s e e , h o w e v e r , t h e f i n a l C h a p t e r X V ) , w e do d i s c u s s t h e r e s u l t i n q u e s t i o n m a i n l y as an a p p l i c a t i o n i n o u r c a s e o f t h e Gel'fand-Mazur
Theorem ( S e c -
tion 5). W e s t a r t w i t h a p r e l i m i n a r y comment on t h e r e l e v a n t t e r m i n o l o g y .
T h u s , g i v e n a (complex) a l g e b r a E and an ( a l g e b r a ) moryhism
of E i n t o t h e a l g e b r a o f l i n e a r endomorphisms ( o p e r a t o r s ) o f a (comp l e x ) v e c t o r s p a c e X , one d e f i n e s t h e map 41 a s a representation of t h e algebra
E
in
X
( t h e representation space). A l t e r n a t i v e l y , t h e s p a c e X
becomes a n E-mo&h*e v i a t h e r e l a t i o n
f o r a n y a 6 E and x e X .
( T h a t i s , o n e t a k e s a s a d e f i n i t i o n o f t h e re-
levant notion t h e r e l a t i o n (8.2) i n e i t h e r d i r e c t i o n , r e s p e c t i v e l y ) . Thus w e s p e a k o f an irreducible representation X I
of
(the algebra) E i n
i f t h e E-module X d e f i n e d by ( 8 . 2 ) i s i r r e d u c i b l e . T h a t i s , t h e r e
do n o t e x i s t o t h e r sub-E-modules
(03rX and t h e s p a c e X i t s e l f .
o f X e x c e p t t h e t r i v i a l o n e s , namely, (Thus i n t h i s case $ d o e s n o t decompose
( "reduce") t o any o t h e r " s i m p l e r "
,
non-trivial
,
(sub-) r e p r e s e n t a t i o n s
of E ) . Now a n i m p o r t a n t " s t r u c t u r a l i n g r e d i e n t " of t h e " g e o m e t r y " of t h e r e p r e s e n t a t i o n $ ( h e n c e , of t h e g i v e n a l g e b r a t o o ) i s t h o s e e l e ments T
€
L(X) which "commute w i t h t h e e l e m e n t s of E " ; i n f a c t t h e y f u l -
f il t h e r e l a t i o n
f o r every a e E . W e
d e n o t e by c($) t h e s e t of t h o s e T f L i X ) which s a t i s -
f y ( 8 . 3 ) and c a l l them intertwining operators o f t h e r e p r e s e n t a t i o n $ . Thus o n e d i s t i n g u i s h e s now t h a t p a r t of LIX) which b e h a v e s r e l a t i v e t o t h e "domain o f c o e f f i c i e n t s " E ( X b e i n g c o n s i d e r e d a s a n Emodule), e x a c t l y a s i f
E was a " n u m e r i c a l domain". Now, t h a n k s t o I .
Schur, t h i s d e s i r e i s , i n f a c t , v e r y c l o s e t o t h e r e a l i t y concerning
11 SPECTRUM (LOCAL THEORY)
78
at least "simple" (read irreducible) representations. Thus, his classical lemma says that: For every i r r e d u c i b l e representation 6 , as in (8.1), c ( @I
(8.4)
i s a d i v i s i o n subalgebra of L I X I .
(Here the identity operator on X , idX= 1 , is the identity element of ~ ( 4 ) . For a proof of (8.1) see, f o r instance, F.F. BONSAL- J . DUNKAN [l:p. 121, Proposition 6 1 ) . Now in case of a c o m t a t i v e topological algebra, the previous picture is further amended by appealing to Gel'fand-Mazur. In this respect, by looking at continuous representations one considers on L ( X ) any topology Gmaking the latter space a topological algebra (Definition I; 1.1. See also Example I;2.(1)). Thus we now have. Lemma 8.1. Let E be a c o m t a t i v e topo2ogical algebra w i t h an i d e n t i t y e l e -
ment and continuous i n v e r s i o n . Then, every continuous representation of E , say
@:
f o r which E/ker(@) (in t h e q u o t i e n t topology) has a non-trivial topological dual, i s one-dimensiona2.
E -Le(X),
Proof, By hypothesis ker(6) is a closed ideal of E, so E/ker($), endowed with the respective quotient topology, is a topological algera of the same type as E (see also Lemma 7.3). Moreover, one has
(8.5)
E/ker(O) = @(E) E L ( X ) ,
within an (algebra) isomorphism, while $ ( E l is a (complex)division algebra (Schur's Lemma; cf. (8.3)). Thus E/ker(@l is a division topological algebra satisfying, by hypothesis, (5.4), hence (topologically) isomorphic to C, and this proves the assertion. In addition, (8.5) becomes, in effect, a topological (algebraic) isomorphism.l
As an illustration of the above, we still give the following less technical result. Namely, we have.
Corollary 8.1. L e t E be a coirnnutatiue Zocally convex algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n . Then, t h e only continuous i r r e d u c i b l e r e p r e s e n t a t i o n s of E are i t s continuous characters ( i . e . , t h e elements of ??Z(Ei, t h e ( t o p o l o g i c a l ) spectrum of E; see Definition 7.2).
In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra w i t h an i d e n t i t y element ( see also Lemma 3.1 ) . I