We now develop an analytic functional calculus for a commuting n-tuple of operators on a Banach space. The form of the Cauchy-Weil integral t h a t we have developed makes this task almost as easy for n-operators as it is for a single operator.
Notation 4.1. I n this section, X will be a Banach space and (al, ..., a~) will be a com- muting tuple of operators on X. We set a = a l 8 1 -~ ... +ansnEAl[a, L(X)].
I n [15] we defined Sp (a, X) to be the set of all z E C n such t h a t z - a = (z 1 - a l ) s 1 + ... + ( z ~ - an)s n is singular on X. The set Sp (a, X) is always a compact n o n - e m p t y subset of the closed polydisc n~ of multiradius v = (Vl ... v~), where v, = l i m I]a~ll 1In (el. [15], w 3).
De/inition 4.2. I f / EO~(U) for some open set U ~ Sp (a, X), then for each x E X we define /(a) x = (2zd)~ ( R z _ = / ( z ) x) A d z I A . . . A dzn.
F o r the convenience of the reader, we recall briefly the steps involved in computing the above expression. Since z - a is non-singular except on Sp (a, X ) = U, it follows t h a t (~(U, X), ~o(U, X), z - a , ~ z ) is a Cauchy-Weil system (cf. Def. 1.8 and L e m m a 3.4).
Here, we set m = 0 in L e m m a 3.4 so t h a t the variables w 1 ... wm do not appear and we have
~0(U, X) = ~ I ( U , X). I t follows t h a t the m a p R~_~: H ' ( ~ ( U , X), ~)~H~+'(~o(U, X), ~ ) of Definition 1.9 is defined. With p = 0 this yields a m a p Rz_a: 9~(U, X)---> H~(~o( U, X), ~z).
Hence, if /E 9.I(U) then (z--->](z)x)E 9~( U, X) and R~_j(z)x has a representative g which is a differential form of degree n in d51 ... dSn with coefficients in ~0(U, X). Thus, g A dz 1 A ... A dz~ is a differential form of degree 2n with coefficients in ~0(U, X). We integrate this over U to obtain
(27~i)n/( a) x
= ji Rz_=/(z) x)
Adz 1 A ... A dz~ e X.Note t h a t if a = ( a ) (aEL(X)) is a singleton, then Sp (a, X ) = { z E C : z - a is not in- vertible} since the complex _F(X, z - a ) is simply the sequence 0 ~ X z - 5 X-->0. F u r t h e r more, b y L e m m a 3.14 we have
f Rz j ( z ) x A dz = ~r (z -- a)-l /(z) xdz
for a J o r d a n curve F = U which encloses Sp (~, X). Hence, in the case of a single operator, the expression/(~)x of Definition 4.2 agrees with t h a t determined b y the classical opera- tional calculus.
THEOREM 4.3. I / U is an open set containing S p ( a , X ) , then ]--->](a) is a continuous homomorphism o/the algebra 9~(U) into the Banach algebra (~)" o/ all operators on X which
THE ANALYTIC-FUnCTIONAL CALCULUS ]~O1~ SEV~I~AL COM2~UTI~NG OPERATORS 31
commute with all operators commuting with each a ~. Furthermore, 1 ( a ) = i d and z ,( a ) = a ~ ]or i = 1 , ..., n.
Proof. I t follows f r o m P r o p o s i t i o n 3.7 a n d C o r o l l a r y 3.10 t h a t ([, x)~f(zt)x is a con- t i n u o u s l i n e a r m a p of ~I(U) • X i n t o X a n d t h a t / ( ~ ) c o m m u t e s w i t h e a c h o p e r a t o r t h a t c o m m u t e s w i t h e a c h ai. H e n c e , ]~/(~) is a c o n t i n u o u s m a p i n t o (zr w i t h t h e n o r m t o p o l o g y . To p r o v e t h a t 1 (zr a n d z~(g)=a~ for e a c h i, n o t e t h a t for a n y p o l y n o m i a l P a n d x E X , t h e f u n t i o n z ~ P ( z ) x is in ~(C n, X). Since, z ~ - a i E ~ I ( C , X) for e a c h i, i t follows f r o m P r o p o s i t i o n 3.11 t h a t we m a y r e p l a c e U w i t h C n w i t h o u t loss of g e n e r a l i t y . I t t h e n follows f r o m C o r o l l a r y 3.15 t h a t
[.
P(~) x = | (R~_~P(z) x) A dzi A . . . A dzn (27d) n J c'~
1 n . X
-- (2~i) f Fl"" f p,~ (zi - a l ) - l ''' (zn -an)-l P(z) xdzl A ''' A dzn = P(ai
....an)
for F i , . - . , Fn s u f f i c i e n t l y large circles.
T o p r o v e t h a t / ~ ] ( o r is m u l t i p l i c a t i v e , n o t e t h a t
' {L }
/ (or) g(a)x = (2:~i)2n R~_~,/(z) (Rw_~,g(w) x) A dw~ A . . . h dw,~ A dz~ h . . . A dz~
1
(2:xi) 2nL
• (R(w_:)r g(w) x} A dwl A . . . h dw~ A dzi A . . . A dzn, b y T h e o r e m 3.6. I f we t r a n s f o r m t h e t u p l e ( w i - a i . . . wn-an, z i - a I . . . zn-a~) b y t h e m a t r i x (uu) , w h e r e u u = 1 if i = j, u u = - 1 if j = n + i, a n d u u = 0 o t h e r w i s e , t h e n we o b t a i n t h e t u p l e (w 1 - z 1 ... wn - z n , zi - a l . . . z , - a m ) . Also, since d e t (uu) = 1 i t follows f r o m P r o p o - sition 3.12 t h a t(2~i) 2~ xv . . . .
_ ( 2 ~ i ) 2 ~ 1 f ~ x ~ {R(~-~)r "" Adwn/kdZlA "'" Adzn - (2xei)2n R(z-a)/(z) (R(w-z)g(w)x) A d w i A ... AdWn A d z l A ... Adzn
i f v
-- (2~i)~ (R~_:)f(z)g(z)x)AdziA..,Adz~=(/g)(~z)'x.
H e n c e , ] ( ~ ) g ( ~ ) = (/g)(:r a n d t h e p r o o f is c o m p l e t e .
If 9~(Sp (a, X)) denotes the algebra of functions defined and analytic in some neighbor- hood of Sp (a, X) - t h a t is, the inductive limit of the spaces ~(U) over neighborhoods U descending on S p ( a , X ) - t h e n since Proposition 3.11 implies t h a t the m a p / - ~ / ( a ) com- mutes with restriction, we have the following corollary to Theorem 4.3:
C o R 0 L L X a Y 4.4. T h e m a p I-~ 1(~) ol Theorem 4.3 de/ines a h o m o m o r p h i s m o/9~( Sp(a, X)) into (a)".
The following invariance law for the functional calculus follows directly from Proposi- tion 3.8:
P~OPOS~TIO~ 4.5. L e t X a n d Y be B a n a c h spaces a n d u: X ~ Y a bounded linear m a p . L e t a 1 .. . . . a n E L ( X ) a n d bl, ... , bn E L ( Y ) be c o m m u t i n g tuples related by u a ~ = b ~ u /or i = 1 ... n.
I / t is analytic i n a neighborhood o / S p (a, X)U Sp (fl, Y ) , then u / ( ~ ) = / ( f l ) u .
Note t h a t at this point we have all of the conclusions of the usual Shilov-Arens- Calderon Theorem. In fact, if A is a commutative Banaeh algebra with identity and a 1 .. . . . a n E A , then we m a y consider a 1 ... an to be operators on A via the regular repre- sentation. I t turns out t h a t Sp (a,A) is then just the usual spectrum of an n-tuple in a Banaeh algebra (of. [15]). In this case, the functional calculus of Theorem 4.3 reduces to the usual functional calculus in a commutative Banach algebra. If h is a complex homo- morphism of A, then Proposition 4.5--with u = h , Y=C, and b ~ = h ( a O - - i m p l i e s t h a t h(/(o:)) = l ( h ( a l ) , ..., h(an)).
Our next Proposition gives a powerful relationship between the functional calculi for two different tuples of operators.
PROPOS~TIO~ 4.6. L e t a I . . . an, bl . . . bn be a c o m m u t i n g tuple o / o p e r a t o r s on X a n d set ~ = (a 1 .. . . , an), fl = (bl . . . . , bn), a n d ~ - f l = (a 1 - b 1 .. . . . a n - b n ) . I1 / is analytic i n a neighbor- hood o / S p (~, X) U Sp (fl, X ) then l( o~) - l(fl) acts as the zero operator on HP( X , a - f l ) /or each p.
Proo/. Let U ~ Sp (~, X) U Sp (fl, X). If / E 9~(U) and k E F P ( X , ~ -fl), then
(recall F ~ ( X , ~ - f l ) i s a direct sum of ( ; ) e o p i e s o f X ) . N o w , b y C o r o l l a r y l . 1 5 , ( R z _ a - R z _ p ) / ( z ) k is cohomologous to zero as an element of F V ( H n ( ~ o ( U , X ) , 8z), ~ - f l ) . Also, the integral commutes with the coboundary operator determined b y ~ - f t . Hence, (](a) - / ( f l ) ) k
T H E A N A L Y T I C - F U N C T I O N A L C A L C U L U S F O R S E V E R A L C O M M U T I N G O P E R A T O R S 33 is cohomologous to zero in F ' ( X , a - f l ) . I t follows t h a t / ( ~ ) - [ ( f l ) is t h e zero o p e r a t o r on H~(X, ~ - f l ) .
COROLLARY 4.7. I[ ~=(a 1 ... an) is a commuting tuple o/ operators on X , U a domain containing Sp ( ~, X), and / ~ ~( U), then/or each z ~ U the operator/(a) acts as the scalar opera- tor ](z) on H~(X, z - ~) /or each p.
This serves as an effective tool in pinpointing the action of the o p e r a t o r / ( ~ ) on X - - a s we shall see in t h e n e x t theorem.
T h E OR~EM 4.8. Let ~ = (a 1 .... , a~) be a commuting tuple in L(X), U a domain containing S p ( a , X ) , a n d / 1 .... , / m e ~ t ( U ) . L e t / : U - > C m be de/ined by/(z) =(/~(z) ... /m(Z)) and let/(:r be the tuple o/operators (/1( ~), ...,/m(~)). Then Sp (/(~), X ) = / ( S p (g, X)).
Proo/. I t follows f r o m T h e o r e m 3.2 of [15] t h a t t h e s p e c t r u m o f / ( ~ ) = (/1(~) ... /re(a)) is t h e projection on t h e last m-coordinates of t h e s p e c t r u m of (a I ... an, ]l(a) ... /re(a)) = a O ] ( ~ ) . Hence, it suffices to prove t h a t t h e s p e c t r u m of a | is {(z,w)E(3n+m:
zE S p ( ~ , X ) , w=/(z)}. To prove this it is sufficient to prove t h a t if zE S p ( ~ , X ) t h e n ( z - cr162 is non-singular if a n d only if/(z) ~=0. W e prove this b y i n d u c t i o n on m.
W e assume t h a t m ~> 0 is an integer such t h a t t h e following t w o s t a t e m e n t s are t r u e of a n y m - t u p l e / 1 ... / m E ~ ( U ) a n d a n y zE U:
(1) if
ge2(U)
t h e n g(a) acts as t h e scalar operator g(z) on H(X, ( z - a ) @ / ( ~ ) ) ; a n d (2) ( z - a ) | is non-singular on X if a n d only i f / ( z ) =~0.L e t [1, ..., [m+l be an ( m + l ) - t u p l e in 9~(U). B y L e m m a 1.3 of [15], there is an exact sequence
. . . . HP(X, (z-~)@/(~))--->H'(X, (z-a)@/'(a))--->H~+l(X, ( z - a ) Q ] ( ~ ) )
[m+l(~!HP+l(X," (z -- ~ ) ( ~ / ( ~ ) ) - > ...
w h e r e / ' = (/1, ...,/re+l) a n d / = (/1 .... ,/m)- I f g eg~(U) t h e n g(a) acts as g(z) on H ' ( X , (z - ~) | [(a)) for each p. I t follows f r o m t h e above sequence t h a t g(a) acts as g(z) on H~(X, ( z - ~)|
]'(:r as well. Also, since/m+l(~) acts as/m+l(z) on H~(X, (z-:r174 for each p it follows t h a t H~(X, ( z - ~ ) | if /,~+l(z)~0 a n d HP(X, ( z - a ) | HP(X, (z-a)|
if/m+l(Z) = 0 . Hence, since s t a t e m e n t (2) above holds f o r / 1 .... , ]~, it continues t o hold for /1 ... /~+1.
Since (1) a n d (2) clearly hold if m = 0 , t h e y hold for all m b y induction. This completes t h e proof.
3 - 7 0 2 9 0 2 Acta mathematica. 125. I m p r i m ~ le 18 S e p t e m b r e 1 9 7 0 .
We close this section with an extension of the Shilov idempotent theorem (ef. [14])i T h a t it is a true extension stems from two facts. First, our spectrum Sp (~, X) is in general smaller--hence more likely to be d i s c o n n e c t e d - - t h a n the spectrum of ~ = (al, ...,an) as computed in terms of some enveloping c o m m u t a t i v e Banach algebra of operators. Second, Sp (~, X) is likely to be computable in situations where it is virtually impossible to tell when an operator equation (z 1 - a ~ ) b 1 +... § can or cannot be solved.
THEOREM 4.9. I / a t = ( a I . . . a,) is a commuting tuple o/ operators on X , and i/
Sp (~, X ) = K 1 U K s where K 1 and K1 are disjoint compact sets, then there are closed subspaces X x and X s o / X such that:
(1) X = X 1 0 X s ;
(2) X 1 and X s are invariant under any operator which commutes with each ai; and (3) Sp ( ~ , X 1 ) = K s and S p ( ~ , X s ) = K s .
Proo/. L e t U 1 and U s be disjoint open sets in C" containing K1 and K 2 respectively.
I f gv, is the characteristic function of UI, then Zv, E ~[(U 1 U Us). Hence, there exists an idempotent pE(~)" such t h a t Xv,(~)=p, I f X l = I m p and X ~ = K e r p , then (1) and (2) above clearly hold for X1, X s. Condition (3) follows from Theorem 4.8 applied to the tuples (zlgv ... zngv,) and (zlgv, , ..., zngv,).
COROLLARY 4.10. I / Z is an isolated point o / S p ( a , X ) , then X = X I | where each z~-a~ is quasi.nilpotent on X 1 and z ~Sp (~, X2).
5. Spectral hull
The functional calculus allows us to derive relationships between our notion of spectrum and notions based on Banach algebra theory.
Notation 5.1. L e t X be a Banach space and A a Banach algebra of operators on X.
I f ~ = ( a l , ..., an) is a tuple of operators in the center of A and wEC n, then we shall say w E SpA (a) if the equation
(wl - al) bl +... + (w, - an) bn = id (5.1) fails to have a solution for b 1 ... bnEA. We pointed out in [15] t h a t S p A ( a ) = S p ( ~ , A ) , if al .... , an are considered operators on A via multiplication.
I f A is an algebra of operators, then A ' will denote the algebra of all operators t h a t c o m m u t e with each element of A. I f (a) denotes the Banach algebra generated b y al .... , a n in L ( X ) , then for a n y Banach algebra A with al .... , anE center (A) we have ( a ) c A c (~)"
and Sp (~, X) c Sp(~.(a) c SpA (a) c Sp(~ (~) (cf. [15], w 4).
T H E ANALYTIC-FUNCTIONAL CALCULUS FOR S E V E R A L COMMUTING OPERATORS 3 5
L e t ~(a) denote the norm closure of the algebra of operators of the f o r m / ( ~ ) f o r / analytic in a neighborhood of Sp (~, X). Note t h a t (a), (a) ~, and ~{(a) are c o m m u t a t i v e algebras and ( a ) c ~ ( ~ ) c ( ~ ) ' c (~)'. Hence, we have
Sp (~, X) c Sp(~), (a) = Sp~),, (~) c S p ~ ) (~) c S p ~ (a). (5.2) There are examples where each of the containments in (5.2) are proper. E x a m p l e s where Spa(a)(~)~=Sp(a)(~) abound; one such example is the single operator /-~z/ on C(F), where F is the unit circle. I n w 4 of [15] we gave an example where Sp (a, X) +Sp(a),(~).
We shalI reproduce this example here and show how it can be modified to obtain examples in the other two cases.
L e t D be a compact polydisc and U an open polydisc with compact closure such t h a t 0 6 int D c D c U c @ s. We set V = U \ D . L e t C(V) be the space of continuous functions on the closure V of V and CI(V) be the subspace of C(V) consisting of functions with uni- formly continuous first partial derivatives on V. We give C(V) the sup norm. For / 6 CI(V) we define II/H to be the sum of the sup norms of / and its first partial derivatives. We set
x =c1(V)|
We define five operators on X as follows:
at(l,
y)=(zl/, zig), as(/, g)=(zs/, zsg), as(/, g) = (0, ~//~Zx), ad(/, g) = (0, D//~zs), and as(~, g) = (0,/). Note t h a t (a 1 .... , as) is a com- muting tuple of bounded linear operators on X. Note also t h a t (al, as)' contains all operators of the form (/, g)-+(h/, h(j) (hECI(U)) as well as as, ad, and a 5.Since 0 q V the equation Zlh I + zsh s = 1 can be solved for hi, h s E CI(V). I t follows t h a t 0 ~ Sp(~,.a,). (as, a~). However, 0 E Sp( ... ),, (al, as) , for if we could solve a 1 b 1 ~-a s b s = 1 with bl, b 2 E (al, as)" then this equation would remain valid on X1/Xo, where X 1 = k e r a s N ker a~ = {(/, g)EX: / is analytic on V} and X 0 = k e r a 5 = {(0, g)EX}: however, X I / X o is isomorphic to the space of continuous functions on U which are analytic on U (cf. [15], w 4). Since 0 E U and a l l = z I/, a J = z j we have a contradiction. Hence,
Sp( ... ),, (a 1, as) ~= Sp( ... ). (al, as).
A similar argument (which appears in [15]) shows t h a t Sp (a,X) =~ Sp(~). (a) for a = (a I ... as).
I t is simpler to obtain an example where Sp~(a)(~)~=Sp(a),(~). We let V and C(V) be as above. L e t X = C ( V ) and define a ~ e L ( X ) ( i = 1 , 2) b y a i / = z t / . I t can be shown t h a t (al, as)" consists of the o p e r a t o r s / - + h / , where h e C ( V ) . I t follows t h a t 0r as).
However, ~(al, as) is the algebra of operators of the f o r m / - + g / , where g is continuous on and analytic on V. Any such g can be uniquely extended to be analytic on U (cf. [9], I).
Since 0 ~ U we have 0 ~Sp~(a,.~,)(a~, as).
I t turns out t h a t Sp~a) (~) and Sp~(a) (a) are determined b y the geometry of Sp (~, X) as a subset of C n.
I f K c C n is compact, t h e n the polynomial hull of K is (zECn: ]p(z)] ~<supw~KlP(W)]) for all polynomials p ) (cf. [9]).
THEOREM 5.2. For any commuting tuple :r 1 ... an), Sp(~)(zt) is the polynomial hull o/ Sp (~, X).
Proo/. Since (~) is the closure of the image of the m a p p-->p(ot) from the algebra P of polynomials into (a), the spectrum of a is just the set of z E C n for which the complex homo- morphism p-+p(z) of P extends to a complex homomorphism of (~). I t is easily seen t h a t this set is exactly the polynomial hull of Sp (~, X).
Definition 5.3. I f K c C ~ is compact then the spectral hull of K is the set of all wEC n such t h a t the equation
(z 1-wl)]l(z ) § ... § (z n-wn)/n(z ) = 1 (5.3)
fails to have a solution for/1 ... /n analytic in a neighborhood of K.
THEOREM 5.4. For any commuting n-tuple a=(a 1 ... an) o/ operators, Sp~(a)(~)is the spectral hull o / S p (~, X).
Proo/. I t suffices to show t h a t for w EC n equation (5.3) can be solved for /1 ... /hE O~(Sp (~, X)) if and only if equation (5.1) can be solved for b 1 ... bn Eg~(a).
I f /1 .... ,/hE 9~(Sp(~,X)) satisfy equation (5.3) then clearly the operators bl =]1(~), .... , b~=/n(a), given b y the functional calculus, satisfy equation (5.1).
Conversely, if b 1 ... bnEg~(:r satisfy (5.1), then there exist functions gl, ...,gnE 9~(Sp (~, X)) such t h a t I] (wl - al) (gl (a) - bl) § § (Wn - - an) (gn (~) -- bn)II < 1. i t follows t h a t (w1--al)g1(6~)§ § is invertible in 9~(~), where h ( z ) = ( w l - z l ) g l ( z ) §
.... § ( W n - - Z n ) g n ( Z ). However, it follows from Theorem 4.8 t h a t h cannot vanish on Sp (a, X) if h(~) is invertible. Hence, h -1E ~(Sp (a, X)) and /1 =h-lgl ... /n =h-lgn is a solution of (5.3).
A compact set K ~ C n is polynomially convex if it is equal to its polynomial hull.
Similarly, we call K spectrally convex if it is equal to its spectral hull.
T~EOR]~M 5.5. I / S p ( ~ , X ) is polynomially convex, then S p A ( ~ ) = S p ( ~ , X ) /or any closed subalgebra A ~ L(X) with a 1 ... an Ecenter (A).
-1/ Sp (~, X) is spectraUy convex, then SpA (~) = Sp (~, X) /or any closed subalgebra A c L(X) such that 9~(a)~center (A).
Proo]. I f a I ... an E center (A) then (a) ~ A and Sp (a, X) ~ SpA (a) ~ Sp~ (a). B y The- orem 5.2, if Sp (a, X) is polynomially convex then Sp (:r X ) = SpA (:r Sp~ (a).
T H E A N A L Y T I C - F U N C T I O N A L C A L C U L U S F O R S E V E R A L C O M M U T I N G O P E R A T O R S 37 If 9~(~)c center (A) then Sp (a, X ) c SpA(~)c Sp~(a)(a) and Theorem 5.3 implies the three are equal if Sp (a, X) is spectrally convex.
There are several conditions t h a t ensure t h a t a set K c C n is spectrally convex. For example, K is spectrally convex if K has trivial cohomology relative to the sheaf of germs of analytic functions. Hence, K is spectrally convex if it is an ~(U)-convex subset of a domain of holomorphy U (cf. [9]).
We close with a few comments concerning the algebra 9~(~). I t follows from Theorem 4.8 t h a t 9~(~) is closed under the application of analytic functions. Hence, 9~(a) m a y be viewed as an analytic functional completion of the algebra (a). Warning: although it is true t h a t Sp2(~)(~) is the spectral hull of Sp (~, X), this m a y not be the maximal ideal space of 9~(~). If A is the maximal ideal space of 9/(a) and a~', ..., an are the Gelfand trans- forms of the elements a 1 ...
an,
then the map ~: A-+ C n (~ = (a~', ..., a~')) maps A onto the spectral hull of Sp (a, X). However, a~', ..., a~" m a y fail to separate points of A. We give an example to show what can happen.Example
5.6. L e t r 1 > r 2 > ... be a sequence of positive numbers converging to zero and, with n > l , set Sk={zECn: Iz] = ( I z l 1 2 + . . . +Iznl~)~=r~}.
We set K = { 0 } U (U~=~S~) and X =C(K).
The operators a 1 .... , an are defined by(a,/)(z)=z~/(z).
If ] is a function defined and analytic in a neighborhood of a set of the form (zEC~: r - e < Iz] < r + e } then / has a unique extension to a function analytic on
(zEcn:
IzI < r + e ) provided n > l (cf. [9]). I t follows t h a t the spectral hull of the set K above is Since the equation (Z 1 - - W l ) / l ( Z ) ~ - . . . ~- (Z n - -
Wn)/n(Z)
= 1 can be solved for/1 .... ,/~ EC(K)
ifw = ( w 1 .... , Wn)~K,
we have S p ( ~ , X ) c S p ( a ) , ( a ) c K . On the other hand, ifwEK
then the above equation cannot be solved for ]1,-..,/nEC(K)=X;
hence, the map ( w - ~ ) n - l :Fn-I(X, w-o~)=(~)nx--~X=Fn(X, w - a )
fails to be onto, and the complexF(X, w - ~ )
is not exact (cf. Def. 1.2). I t follows t h a t Sp (~, X ) = K and Sp~(~)(~) is the spectral hull of K which is (z e C~: I z I 4 r l ) . Note, however, t h a t the maximal ideal space A of 9~(~) is the one point compactification of the disjoint union of the sets (z E Ca: I zl ~< r~}.This follows from the fact t h a t 9~(K)~ | ~((z: I zI ~< r~}). The map a: A-~Sp~(~)(~)=
{~: I~l <~r~} is just the map induced on A b y the inclusions
(z: Iz] ~r~}~(z: Izl <~ri}.
The inverse image of zero under this map is an infinite set. Hence, & is not even a light map.Note one other thing about this example. The algebra 9~(~) contains a non-trivial projection corresponding to each of the sets (z: I z I = rl}. The version of the Shilov idempo- t e n t Theorem given in 4.9 detects these projections since each (z: I z[ =r~} is a component of Sp (~, X). However, Sp~{~) (~) = {z: [z [ = r~) is connected and does not indicate the existence of these projections.
References
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Received September 10, 1969