Holomorphic convexity and analytic structures in Banach algebras
JAN-ERIK BJORK
Institut Mittag-Leffler and University of Stockholm
Introduction
I n this p a p e r e v e r y B a n a c h algebra is c o m m u t a t i v e and semi-simple with a unit element. I f B is a B a n a c h algebra t h e n M B denotes its maximal ideal space and 0B its Shilov b o u n d a r y . Since B is semLsimple t h e Gelfand t r a n s f o r m identifies B with a subalgebra of C(MB). I f B t h e n becomes a uniformly closed subalgebra of C(MB) we say t h a t B is a uniform algebra. Notice t h a t B is a uniform algebra precisely when it is complete in its spectral radius norm.
I f B is a uniform algebra and if A is a closed point-separating subalgebra of C(MB) such t h a t A C B, t h e n we simply say t h a t A is a subalgebra of B. I n Section 1 and 2 we analyze this situation. H e r e follow two results which are typical applications of this material.
TI~EOREM 1.1. Let X be a reduced analytic space and let A be a point-separating subalgebra of ~ ( X ) . Suppose that K is a compact ~(X)-convex set in X such that the set ]~A = {x E X : If(x)] < JflK for all f in A } is compact in X . Then K = KA and ~I A(~) = K, where A ( K ) is the uniform algebra on K generated by the restriction algebra A IK.
I n T h e o r e m 2.1. the following notations are us3d. D is t h e closed unit disc in C 1 and A(D) is t h e usual disc-algebra on D. An element f C A(D) is smooth if the restriction of f to T is continuously differentiable. I{ere T is the unit circle.
TI~EORE~ 2.1. Let A be a subalgebra of A(D) such that A contains a dense subalgebra of smooth functions. Then M A ~ D.
L e t us r e m a r k here t h a t the results above also apply to B a n a c h algebras. F o r if B is a B a n a c h algebra and if B c is the uniform closure of B in C(MB), then it is well known t h a t MB ---- MBc. I f ~A is a closed subalgebra of B which separates points in MB, t h e n Ao is a subalgebra of B e.
40 ARKIV F6P~ ~r Vol. 9 No. 1
Using a f a m o u s principle due t o Shilov (See [8, p. 214]), it follows t h a t w h e n MB is i d e n t i f i e d w i t h a subset o f M A, t h e n aA C MB. This implies t h a t MA can be i d e n t i f i e d w i t h M A . I n p a r t i c u l a r we h a v e t h e basic principle below.
SI~ILOV'S PI~INCn~LE. Let B be a Banach algebra and let A be a closed subalgebra of B which separates points in M B. Then MA ---- MB if and only i f MA~ = M B.
I n Section 3 we p r o v e a r e s u l t a b o u t B a n a c h algebras g e n e r a t e d b y a n a l y t i c f u n c t i o n s .
TI~Ol~E~ 3.1. Let X be a compact subset of a reduced analytic space Y. Let B be a Banach algebra on X such that X ~ .M m Suppose also that there exists an open neighborhood U of X in Y and a point separating subalgebra A of ~(r such that A x = { f e C ( X ) : f - ~ ] [ X for some f e A } , is a dense subalgebra of B. I f now J is a closed ideal of B such that A X A J is a dense subset of J, while H u l l (J) --~ {x C X : j(x) -~ 0 for all j in J } is a compact subset of the interior of X , then J has a finite codimension in B.
I n Section 4 we s t u d y c o m p a c t h o l o m o r p h i c a l l y c o n v e x sets in C ~. I f K is a c o m p a c t set in C ~ we let ~ ( K ) be t h e a l g e b r a o f g e r m s of a n a l y t i c f u n c t i o n s on K , a n d H ( K ) is t h e u n i f o r m closure of @(K) in C(K). W e s a y t h a t K is a holomorphic set if K ---- MB(K). I t is well k n o w n t h a t if K is a n i n t e r s e c t i o n o f o p e n d o m a i n s o f h o l o m o r p h y , t h e n K is a h o l o m o r p h i c set. W e give an e x a m p l e w h i c h shows t h a t t h e converse is false.
F i n a l l y Section 5 contains some m a t e r i a l a b o u t integral e x t e n s i o n s o f B a n a c h algebras, b a s e d on results f r o m Section 1.
V e r y o f t e n we e m p l o y t h e L o c a l M a x i m u m P r i n c i p l e ( a b b r e v i a t e d L M P ) in t h e proofs. T h e L M P states t h a t if B is a u n i f o r m algebra while V is a subset of M B ~ 0 8 , t h e n V c HullB(bV), w h e r e b V is t h e t o p o l o g i c a l b o u n d a r y o f V in M B.
W e r e f e r to [5] for basic facts a b o u t u n i f o r m algebras.
1. Convexity and analytic structures in uniform algebras
L e t X be a c o m p a c t H a u s d o r f f space a n d let B be a u n i f o r m algebra on X . W h e n X is i d e n t i f i e d w i t h a closed subset o f M B we h a v e a s c X . I f K is a closed subset of M B we p u t ttUllB(K ) ~- {y C M B : ]f(Y) l <-- lf[~: for all f in B}.
I f S is a closed subset of JIB we let B(S) be t h e u n i f o r m algebra on S w h i c h is g e n e r a t e d b y t h e r e s t r i c t i o n algebra BIS. F i n a l l y X B d e n o t e s t h e C h o q u e t b o u n d a r y of B. H e r e XB is dense in 0 B.
I f t h e set M B ~ X is n o n e m p t y , t h e n we p u t A = M B ~ X a n d let bzl be its topological b o u n d a r y in M B. T h e n t h e L M P shows t h a t z] C HullB(bA) a n d if B 1 ~ B ( H u l l B (bA)), t h e n MB, ---- Hull B (hA) while 0B1 C hA. I n t h e results w h i c h follow we show t h a t
~BI
is c o n t a i n e d in a smaller set t h a n bA u n d e r c e r t a i n assumptions.HOLOlVIOR]?HIC C O N V E X I T ~ A ~ D A N A L Y T I C S T R U C T U R E S I N B A N A C t t A L G E B R A S 41
Definition 1.1. L e t B be a u n i f o r m a l g e b r a o n X. W e s a y t h a t B has a n analytic structure a t a p o i n t x E X if t h e r e is a n o p e n n e i g h b o r h o o d W o f x in X a n d a h o m e o m o r p h i s m q~ f r o m W o n t o a n a n a l y t i c subset V o f t h e open polydise D n in C", such t h a t t h e f u n c t i o n s f o r are h o l o m o r p h i c on V for all f in B.
N o t i c e t h a t in D e f i n i t i o n 1.1 t h e a n a l y t i c set V m a y be 0-dimensional a t r This h a p p e n s preeisely w h e n x is an isolated p o i n t in X . I n t h e n o n t r i v i a l case w h e n dim (V) > 0 a t r t h e m a x i m u m principle for h o l o m o r p h i c f u n c t i o n s on
V implies t h a t x c a n n o t belong to a B.
T~EOREM 1.2. Let B be a uniform algebra on X and let W be an open subset of X such that B has an analytic structure at each point in W. I f now the set A = M B ~ O B is non empty and if we put B1 = B(KullB(bA)), then OB, C ( b A ~ W ) .
N e x t we i n t r o d u c e some c o n c e p t s f r o m [10]. I f W is a locally closed subset o f MB, t h e n we p u t / / ~ ( W ) = { f E C ( W ) : t o each x E W t h e r e is a n o p e n neigh- b o r h o o d U / o f x in MB a n d a sequence (bn) in B such t h a t lim If -- b, lwn v I = 0}.
Definition 1.2. L e t W be a locally closed set in M B a n d let V be a locally closed subset of W. T h e n V is ealled a Bw-analytic variety if to each p o i n t x E V, t h e r e is an o p e n n e i g h b o r h o o d U o f x in W a n d a f a m i l y {fi}iei in C(U) such t h a t t h e following holds. F i r s t l y UI'I V = {x E U :fi(x) = 0 for all i}, a n d s e c o n d l y each f i I U ~ V belongs to H B ( U ~ V ) . T h e f u n c t i o n s in C(U) satisfying this c o n d i t i o n are d e n o t e d b y t t B ( U , U fl V).
Suppose n e x t t h a t U is an o p e n subset o f MB while V is a locally closed subset of bU. T h e n we s a y t h a t V is a U-analytic variety at bU if V is a B v u v- a n a l y t i c v a r i e t y . I t is easily seen t h a t a f i n i t e u n i o n of U - a n a l y t i c varieties at b U is a U - a n a l y t i c v a r i e t y a t b U.
T~tEOR~t 1.3. Let B be a uniform algebra on a compact space X and suppose that the set M B ~ X = A is non empty. Let V be a relatively open subset of bA, such that V is a A-analytic variety. Then A ~ Hull~(bA~ V).
T ~ g o R ~ t 1.4. Let B be a uniform algebra on X and let W be an open subset of X . Let K be a compact subset of W such that KullB(K )V1 W - ~ K. I f now the set M B ~ X = A is non empty, while b A f l W is a A-analytic variety, then K = HullB(K ) and K = MB(tc).
N o t i c e here t h a t if A is e m p t y in T h e o r e m 1.4 t h e n K = HullB(K ) follows because e a c h closed c o m p o n e n t of HullB(K ) i n t e r s e c t s K .
Suppose n e x t t h a t A is a u n i f o r m a l g e b r a a n d let B be a u n i f o r m a l g e b r a on M A such t h a t A c B. T h e n we s a y t h a t A is holomorphically dense in B i f t h e following holds. T h e r e exists a f i n i t e decreasing sequence o f closed sets M A =
U 0 ~ U1 D . . . . D U, ~ Un+l = O, such t h a t U~+ 1 is a B u - a n a l y t i e v a r i e t y for each i = 0 , . . . , n - - l, a n d each r e s t r i c t i o n a l g e b r a B[(U~U~+~) is c o n t a i n e d in HA(U,~U~+I).
T~EORE~ 1.5. Let A be a uniform algebra and let .B be a uniform algebra on M A such that A c B. I f A is holomorphically dense in B, then M A = MB and
OA : OB.
42 ARKIV FOR MATEMATIK. VO1. 9 N O . 1
Finally we show t h a t Theorem 1.3 and 1.4 can be used to s t u d y analytic structures.
T~EOREM 1.6. Let B be a uniform algebra on X and let W be an open subset of X such that B has an analytic structure at each point in W. I f now the set A ~ M B ~ X is non empty, then b A f l W is a A-analytic variety.
:Notice here t h a t Theorem 1.6 and 1.3 together imply Theorem 1.2. I n a series of subsections we prove the results above.
1.a. Proof of Theorem 1.1.
We explain here w h y Theorem 1.1 is a consequence of Theorems 1.4 through 1.6.
For consider the situation in Theorem 1.1 and choose a compact neighborhood K1 of the set /~A in X. I f A1 ---- A(K1) it is obvious t h a t A1 has an analytic structure in the interior of KI. Hence HullA~(KA) = ~A and then an application of Theorems 1.6 and 1.4 implies t h a t J~A --~ M~(~A).
I t remains to prove t h a t K--~/~A. L e t B be the uniform algebra on KA generated by ~(X)[I~ A. Since K is @(X)-convex we see t h a t if the set W ~ - - / ~ A ~ K is non empty, then a Bfl W e O. On the other h a n d 0 ~ ( ~ ) c K , so if we can prove t h a t A(/~A) is holomorphically dense in B, then Theorem 1.5 implies t h a t aA(~A ) --~ 0B, which proves t h a t W must be empty. This fact will be proved in Section 1.b.
We also remark t h a t Theorem 1.1 essentially is contained in the work by H.
l~ossi in [13]. We will employ results from [13] in the proof of Theorem 3.1.
1.b. Remarks about analytic spaces
L e t X be a reduced analytic space and let A be a point-separating subalgebra of ~(X). The following assertions are well-known. See [6] or [9].
L e t l%eg (X) denote the open set of all regular points in X and let Sing (X) be the singular set. Then Sing (X) is a nowhere dense analytic subset of X. I f x E t~eg (X) there is a unique integer s > 0 such t h a t there is some open neigh- borhood of x in X which is biholomorphic with the open polydisc in C s. The number s is denoted b y dim (x), and the case dim (x) ~- 0 occurs when x is an isolated point in X.
I f x E l ~ e g ( X ) and if s = d i m ( x ) , then we say t h a t x E l % e g ( A ) if A gives local coordinates at x. The condition t h a t x E ]~eg (A) is tested by looking at the Jacobians JF(x)----det (Of,/aui(x)), where u l , . . . , u8 determine local coordinates at x, while F = ( f l . . . f s ) runs over s-tuples from A. Then x C g e g (A) if and only if Jr(x) r 0 for some F.
H O L O M O P I ~ H I e C O N V E X I T Y A N D A N A L Y T I C S T R U C T U R E S I N SAI~A(DH A L G E B I ~ A S 4 3
The definition of l~eg (A) implies t h a t all isolated points in X belong to t~eg (A) while Sing (X) i'l l~eg (A) is empty. The discussion above s h o w s : t h a t the set Sing (A) = XN`l~eg (A) is an analytic subset of X. Since A separates points in X it is an easy exercise to verify t h a t Sing (A) is nowhere dense in X.
I f we p u t S 1 = Sing (A), t h e n S 1 is a reduced analytic space while A IS 1 is a subalgebra of ~($1). Then we can define the set l~eg (A IS1) and p u t S 2 ~- S l ~ l ~ e g (A [$1). Here S 2 is an analytic subset of X and S 2 is nowhere dense in
Inductively we get a sequence S 1 ~ S~ ~ . . . such t h a t Si+ 1 ~ SiN`lteg (A IS/).
I f x E X and if n = dim (x), then it follows t h a t S~+~ does not contain x. More generally, if K is a compact subset of X , then there is an integer v such t h a t S , ~ K is empty.
Clearly the assertions above imply t h a t if K is a compact set in X such t h a t K ~ MA(~), t h e n A ( K ) is holomorphicMly dense in the uniform algebra on K generated by @(X)[K.
1.c. Convexity and analytic phenomena
Here we collect essentially well-known facts from the theory of functions locally approximable in a uniform algebra. The theory is developed in [10] and we also refer to [2].
The following result appears in [2, Theorem 5].
PROPOSITION 1.1.c. Let A be a uniform algebra and let B be a uniform algebra on M A such that A c B. Let .F be a closed A-convex subset of M A such that B I ( M AN`F ) contains a uniformly dense subalgebra contained in H A(M A ~ F ) . Then the (possibly empty) set A ~ M B N ` M A is contained in H u l l s ( F ) . Here H u l l s ( F ) f l M A = F and b A ~ F .
We explain here w h y Proposition 1.1.c follows from Theorem 5 in [2]. Firstly Theorem 5 is stated in the case when B is generated over A b y a single function f E C(MA) for which fI(MAN.F) E HA(MAN`F). B u t the proof easily extends to the general case. The conclusion in Theorem 5 is t h a t if y ~ A, then there is a point
~(y) E F such t h a t a ( y ) = a(:z(y)) for all a E A. Here ~ is the induced map from M s into M A given b y the inclusion of A into B.
Since :~ is continuous while :z(A) ~ F it follows t h a t bA c F. Now the LMP implies t h a t A c I-Iul] s (F). Since F is A-convex it follows trivially t h a t F = H u l l , (F) rl MA, and t h e n bA C F follows.
PROPOSITIO~ 1.2.c. Let A be a uniform algebra and let U be an open subset of M A ~ ~A. Let x C b U and suppose there exists an open neighborhood W of x in bU such that W is an Avuw-analytic variety. Then it follows that
U C Hull~ (bUN`W).
Proof. P u t A I = A ( U U b U ) . Then the LMP shows t h a t a A , ~ b U and
44 AI~KIV :F61~ ~r V o l . 9 2qo. 1
because ~A1 is t h e closure of Z~, we conclude t h a t if ZA1 n w is e m p t y , t h e n 0.4, c (b U ~ W).
To prove t h a t 2:A1fl W is e m p t y we consider a point x E W. Now we can choose a closed neighborhood /2 of x in M A such t h a t /2 N b U c W a n d where /2 N W = {y e -(20 0 :fi(y) = 0 for all i E I}. H e r e F = {f~},eI is a f a m i l y in
H~(/2 n 9 , -(2 n w).
N e x t we consider t h e u n i f o r m algebra B on -(22 O 0 which is g e n e r a t e d b y F a n d A](/2 A ~). Using t h e L M P a n d we see t h a t -(2 n 0 c Hull n (b/2 fl U U (-(2 O W)).
So if y c / 2 ~ U, t h e n there is a so called J e n s e n measure m carried on (b/2O 9 ) U (-(21"1 W) satisfying log ]b(y)]
_<flog
Ibldm for all b E B . Since this holds for all f~ we conclude t h a t m carries no mass on t h e set -(2 O W. This implies t h a t y C Hull n (b/2 fl 9).Since x belongs to t h e closure of s A U we conclude t h a t x E H u l l s (b/2 fl 9 ) c HUllA1 (b/2 A 9). Since x E ZA~ it follows t h a t x C bD O U, b u t this contradicts t h e f a c t t h a t -(2 is a neighborhood of x in MA.
W e h a v e n o w p r o v e d t h a t 0A1C ( b U ~ W ) a n d t h e n U c H u l l A l ( b U ~ W ) : HullA(b U ~ W) follows.
PROI'OSlTIO~ 1.3.c. Let A be a uniform algebra and let Z be a closed A-analytic varie 0 in M A. I f F is a closed set in MA, then HullA ( F U Z) ~ I{ullA (F) U Z.
Proof. L e t us p u t D ~-- H u l l A ( F U Z ) ~ ( H u l l A (F) U Z), a n d suppose t h a t D is n o n e m p t y . I f A 0 --~ A (Hull A ( F U Z)), t h e n D is a n open subset of MAo~0A, while Z is an A0-analytic v a r i e t y .
The L M P shows t h a t D c HullA, (bD), where bD is t h e topological b o u n d a r y of D in MA.. Clearly W ~ ( b D ~ I ~ u l l A (F)) c Z, a n d hence it is obvious t h a t W is a n (A0)wuo-anMytic v a r i e t y . I t follows from Proposition 1.2.c t h a t D c HullA. (b U ~ W) c Hull A (F), a contradiction.
PROt'OSlTION 1.4.C. Let A be a uniform algebra and let B be a uniform algebra on M A such that A C B. Suppose that V is a closed BMA-analytic variety in M A and that B I ( M A ~ , V ) contains a uniformly dense subalgebra of functions from H A ( M A ~ V ). Then V = Hull A (V).
Proof. Consider a point x C V. W e can f i n d a closed A-convex neighborhood /2 of x in MA a n d a f a m i l y F : ( f l } in HB(/2,-(2O V) for which -(2n V { y C / 2 : f i ( y ) : O for all i}.
The hypothesis on B implies t h a t e v e r y function in B is A-holomorphic of t h e second k i n d in MA~, V. As a consequence t h e elements in F are A-holomorphic of t h e t h i r d kind i n / 2 ~ V. See [11] for a general concept of A-holomorphic functions.
N e x t we consider A 1 ~ A (HullA (V)) a n d p u t U ~-- Hull A ( V ) ~ V . T h e n b U c V holds a n d t h e discussion above shows t h a t t h e hypothesis of Proposition 1.2.c is satisfied for each point in bU, except t h a t we n o w deal with A-holomorphic functions of the t h i r d kind on / 2 ~ V. B u t using R i c k a r t ' s general t h e o r y in [11] it follows t h a t Proposition 1.2.c remains true. H e n c e U c HullA,(b U ~ V) ~- HullA1 (0), which implies t h a t U is e m p t y .
IqOLOlYIOI%PtIIC C O N V E X I T Y A N D A N A L Y T I C STI:tUCTUI%ES I N B A N A ( J H ALGEBI%AS 4 5
1.d.
Proof of Theorems
1.3--1.6.Proof of Theorem
1.3. P u t S ---- A UbA
a n d i n t r o d u c e t h e u n i f o r m a l g e b r aB(S).
T h e n aB(S)CbA
a n d A is a n o p e n s u b s e t o fMB(s)=
H u l l s(bA).
U s i n g P r o p o s i t i o n 1.2.c it follows t h a tA C KullB(s)(bA~V)=
lqullB(bA~V).
Proof of Theorem
1.4. I n t r o d u c e t h e u n i f o r m a l g e b r a A = B ( H u l l B(K)).
T h e n M ~ = H u ] I B ( K ) while 0 A c K . Since H u l l B ( K ) gl W = K we see t h a t K is a closed A - a n M y t i c v a r i e t y in MA. T h e n P r o p o s i t i o n 1.4.c a p p l i e d t o t h e case B - - ~ A , shows t h a t K - - - - H u l l A ( K ) . Since a A c K it follows t h a t K = M A = H u l l B (K), a n d as a c o n s e q u e n c e K =MB(K).
Proof of Theorem
1.5. l~irstly P r o p o s i t i o n 1.4.c shows t h a t U 1 is A - c o n v e x in M A. T h e n P r o p o s i t i o n 1.1.c shows t h a t t h e set A =M B ~ M A
is c o n t a i n e d in Hul]B(U1). T h i s m e a n s t h a t if A1 =A(U1)
a n dB 1 -- B(U1),
t h e n A ----M~I~MA ~
while U1 = MA 1.N e x t we n o t i c e t h a t
BI[(UI~U2)
c o n t a i n s a dense s u b a l g e b r a o f f u n c t i o n s inH~I(UI~U2).
A n e w a p p l i c a t i o n of P r o p o s i t i o n 1.4.c p r o v e s t h a t U 2 is Al-Convex in M B a n d t h a tM~,~MA1 C
HUI1B~ (U2). T h i s m e a n s t h a tMB~M4 C
HullB (U~), a n d b y i n d u c t i o n we see t h a tMB~MA c
Hull~ (U~) for all v ~ 1. F i n a l l y v = n ~ - 1 gives M B = M ~ .I t r e m a i n s t o p r o v e t h a t 0B ---- OA. So let us p u t F = H u l l B (OA) a n d s u p p o s e t h a t t h e set
M B ~ F
is n o n e m p t y . SinceBI(Mx~U1 ) c HA(MA~U1)
it follows f r o m C o r o l l a r y 2.4. in [10], t h a t OB c a n n o t i n t e r s e c tMA~(F U U~).
H e n c e O B c ( F ( 3 U1) a n d i f
y C M A ~ F ,
t h e n y E g u l l ~ ( F ( 3 U~) while y does n o t b e l o n g t o H u l l B (F). Since U 1 is a closed B - a n a l y t i c v a r i e t y inM B
it follows f r o m P r o p o s i t i o n 1.3.c t h a t y E U~. H e n c eMA'~F ~ U~
holds.Since
M A ~ F
is a n o p e n s u b s e t ofM B
w h i c h is c o n t a i n e d in U1, t h e c o n d i t i o n t h a tBI(U~U.~)cH4(UI~U2)
a g a i n i m p l i e s t h a t 0 B c a n n o t i n t e r s e c tMA~(F t_] U2).
T h e n t h e s a m e a r g u m e n t as a b o v e p r o v e s t h a tM A ~ F c U2.
I n d u c t i v e l y we see t h a t
M A ~ F c U~
for all v > 1, so w h e n v = n + 1 we c o n c l u d e t h a t F = M A. B u t this m e a n s t h a t 0 B = 0 A.Proof of Theorem
1.6. W e m a y consider W as a r e d u c e d a n a l y t i c s p a c e whileB[W
is a p o i n t s e p a r a t i n g s u b a l g e b r a of @(W). S u p p o s e n o w t h a tx e bA n W
is s u c h t h a t x also b e l o n g s t o I~eg (B[W). L e t s = d i m (x).I f s = 0 t h e n x is a n i s o l a t e d p o i n t in W a n d h e n c e also in
bA.
T h e n {x}is a A - a n a l y t i c v a r i e t y . I f s > 0 we choose f~ , . . . , f~ in B w h i c h m a p a n o p e n n e i g h b o r h o o d U o f x in W h o m e o m o r p h i c a l l y o n t o t h e o p e n p o l y d i s e D ~ in C ~.
I f g E B we p u t g(zl . . . z~) = g(~(z)), w h e r e ~(z) is t h e u n i q u e p o i n t in U for w h i c h zl = f i ( 5 ( z ) ) for i = 1 . . . s. T h e n ~
e~(D ")
holds.L e t us choose 0 ~ r ~ 1 a n d p u t K = { y s U : ]f~(y)[ _ < r for i = 1 . . . s}.
N e x t we let /2 be a n o p e n n e i g h b o r h o o d o f x in
3I~
such t h a t If~la < r f o r all i.W e claim t h a t /2 f~ U is a B g - a n a l y t i c v a r i e t y .
F o r let y s be given. W e c a n choose z ~ U s u c h t h a t f i ( y ) = f ~ ( z ) for
46 ARKIV FOR MATElgATIK. Vol. 9 :NO. 1
all i. T h e n we t a k e some g 6 B such t h a t g(y) =- 1 a n d g(z) ---- 0. Since g 6 ~ ( D ") we can f i n d a sequence o f p o l y n o m i a l s P~ 6 C[Z 1 . . . Z,] such t h a t lira
] P ~ - glD(0 ---- 0, w h e r e D(r) is t h e closed polydisc o f radius r in C ~.
I t follows t h a t lim P~(f~ . . . . ,f,) exists u n i f o r m l y in Y2 a n d t h e limit f u n c t i o n h belongs to HB(Y2 ). H e r e h - = g holds on Y2Cl U while h(y)----limP~(fl(y) . . . f,(y))
=- lim P~(fl(z) . . . . , f,(z)) -~ g(z) ---- O. H e n c e h 1 =- g - - h belongs to HB(Y2 ) a n d hi----0 on f 2 N U while hi(y)----1. This p r o v e s t h a t Y2 N U is a BQ-analytie v a r i e t y .
W e h a v e n o w p r o v e d t h a t t h e set R e g ( B I W ) = S O is a A-analytic v a r i e t y a t bY2. N o w we consider t h e general case. Using t h e n o t a t i o n of Section 1.b we h a v e a decreasing sequence S 1 D S 2 D . . . , w h e r e S ~ + I - - ~ S i ~ ] % e g ( B ] S 3 . I f x 6 W we choose a r e l a t i v e l y c o m p a c t o p e n n e i g h b o r h o o d V of x in W. T h e n S , N V is e m p t y for some n > 1. Using t h e same a r g u m e n t as a b o v e we can p r o v e t h a t t h e sets U~ = ( S ~ S , : + I ) f] V are A-anMytie varieties a t bA. B u t t h e n V--~
(S o N V) U U~ O . . . U U,~_ 1 also b e c o m e s a A-analytic v a r i e t y a t bA. Since this holds for all points in W, we conclude t h a t W is a A-analytic v a r i e t y at bA.
2. P r o o f o f T h e o r e m 2 . 1 .
F i r s t l y we n e e d some f a c t s f r o m t h e so called ~)arc-theory~) dealing w i t h u n i f o r m algebras p r e s e n t e d on s m o o t h J o r d a n arcs in C". W e r e f e r to [3, 15, 18] for details.
T h e following result is c o n t a i n e d in [15, L e m m a 5 - - 8 ] .
L~M~A 2.1. Let A be a uniform algebra on a compact space X . Let f C A be such that there is an open set W in X which f maps homeomorphicaUy onto an open Jordan arc J in C, 1. Suppose also that W ~- {x C X : f(x) E J } and that J is a relatively open subset of f ( X ) , which borders the outer component A~o of the set
C l e f ( X ) . I f we now identify X with a closed subset of M A then the following condition holds for each p o i n t z C J : There exists an open disc A(z) such that f maps the open set U - = {y E M A : f ( y ) E A ( z ) } homeomorphicaUy onto a subset of A(z), and here f ( U ) N A | is empty.
L e t us n o w consider t h e u n i f o r m algebra A in T h e o r e m 2.1. Clearly ~A C T holds. I f D A ~ T we can choose a closed i n t e r v a l I o f T such t h a t 0A C I . T h e n A ] I = A ( I ) b e c o m e s a p r o p e r subMgebra o f C(I) w h i c h is g e n e r a t e d b y con- t i n u o u s l y differentiable functions. B u t this c o n t r a d i c t s [15, A n Application, p. 186].
W e conclude t h a t 0~ = T holds.
I f
f
is ~ s m o o t h f u n c t i o n in A we s a y t h a t a p o i n t a 6 T is a regular pea]c p o i n t of f if t h e following c o n d i t i o n holds: F i r s t l y f ' ( a ) --/: 0, w h e r e f ' 6 C(T) is t h e c o n t i n u o u s d e r i v a t i v e d e t e r m i n e d b y f i T . S e c o n d l y {a} = {x 6 T : f ( x ) = - f ( a ) } a n d f(a) belongs t o t h e b o u n d a r y o f t h e o u t e r c o m p o n e n t of t h e set C l e f ( T ) .W i t h t h e n o t a t i o n s a b o v e we h a v e t h e following result.
H O L O M O R P H I C C O N V E X I T Y A N D A N A L Y T I C S T R U O T U I : t E S I N B A N A C K A L G E B R A S 47
LEMMA 2.2.
Let f be a smooth function in A and let a E T be a regular peak point of f. Then there exists an open disc z]l(a ) such that the set D • ~l(a) is an open subset of M A.
Proof.
Clearly t h e r e exists an o p e n are W in T such t h a t a E W while f m a p s W h o m e o m o r p h i c a l l y o n t o a n o p e n J o r d a n arc J in C 1. I n a d d i t i o n J is a r e l a t i v e l y open subset o ff(T),
a n d W ---- {x E T : f ( x ) E J}, while J b o r d e r s t h e o u t e r c o m p o n e n t A~ o f t h e setC l e f ( X ) .
Since f is a n a l y t i c a n d n o n c o n s t a n t in
D ~ T ,
we see t h a tf ( D ~ T )
is c o n t a i n e d in t h e interior o ff(D).
I t follows t h a t J is a r e l a t i v e l y open subset of t h e topological b o u n d a r y o ff(D).
Sincef ( D ~ T ) f'l A+
is e m p t y , we see t h a t J also b o r d e r s a b o u n d e d c o m p o n e n t V o f t h e setC l e f ( T ) .
B e c a u s ef'(a) --~ 0
it follows easily t h a t t h e r e is a n o p e n discA(f(a))
a n d a n open discA(a)
such t h a tf(A(a) n D)
containsA(f(a)) r (F U J).
N o w we a p p l y L e m m a 2.1 t o t h e u n i f o r m algebra
A I T
a n d t h e p o i n tf(a) E J.
F o r t h e n we can choose a n o p e n disc
Ao(f(a)),
w h e r eAo(f(a))c A(f(a)),
such t h a t f m a p s t h e set U 0 = {y E M A : f ( y ) E A o ( f ( a ) ) } h o m e o m o r p h i c a l l y o n t o a subset ofAo(f(a)).
Sincef(Uo)fl A~
is e m p t y , we see t h a tf(Uo)cf(A(a)f'l D).
This m e a n s t h a t U 0 C A(a) A D, a n d since a E U 0 we easily get t h e desired disc
~l(a).
LEMMA 2.3.
Let A be as in Theorem 2.1. Then there exists a smooth function f in A which determines a regular peak point.
Proof.
L e t f be a n o n c o n s t a n t s m o o t h f u n c t i o n in A. L e t us p u t E - ~ {a E T : f ( a ) beloI~gs t o t h e o u t e r c o m p o n e n t A~ of C l e f ( T ) } . Sincef(D) F1 Ao~
is e m p t y , it follows t h a t
f(E)F] f ( D ~ T )
is e m p t y . H e n c ef ( D ~ T )
is c o n t a i n e d in t h e p o l y n o m i a l l y c o n v e x hull of t h e setf(E).
This implies t h a tf(E)
is n o t s i m p l y c o n n e c t e d . I f f ' z 0 on E , t h e n it is easily v e r i f i e d t h a tf(E)
is t o t a l l y d i s c o n n e c t e d a n d s i m p l y c o n n e c t e d .W e conclude t h a t
f'(a) :/: 0
for some a E E . N e x t we consider t h e set F -~{x E T : f ( x ) = f ( a ) } . Since F sa T we k n o w a l r e a d y t h a t A ( F ) =
C(F).
Since a is an isolated p o i n t in F we can choose g C A such t h a tg(a)
= 1 while lgI < 1/2 onF ~ { a }
a n d g is s m o o t h .I~ow it is easily v e r i f i e d t h a t we can choose some e > 0 a n d a suitable >>direction~>
e ~, such t h a t if we p u t
h -~ f - t - e%g,
t h e n a is a r e g u l a r p e a k p o i n t of h.Proof of Theorem
2.1. T r i v i a l l y A has an a n a l y t i c s t r u c t u r e inD ~ T .
I fA - ~ M 4 ~ D
is n o n e m p t y , t h e n T h e o r e m 1.2 shows t h a t A C Hull~(hA f'l T)
holds. N o w L e m m a 2.2 shows t h a tbA f'l T
does n o t c o n t a i n a n y r e g u l a r p e a k p o i n t d e t e r m i n e d b y a s m o o t h f u n c t i o n in A. H e n c e L e m m a 2.3 implies t h a tbAD T :/= T.
B u t t h e n we k n o w t h a t
A(bA N T) = C(bAN
T), a n d t h e nbAfl T
is A - c o n v e x in MA, a c o n t r a d i c t i o n .48 ARKIV FOR MATEMATIK. Vol. 9 No. 1
3. P r o o f o f T h e o r e m 3 . 1 .
I f B c is t h e u n i f o r m closure o f B in C(X), t h e n it is obvious t h a t Bc = A ( X ) . Since X = M s it follows t h a t X = MA(x), a n d because A s e p a r a t e s p o i n t s in Y we conclude t h a t X is A - c o n v e x in Y. L e t us p u t Jo --~ { f C A : f i x C J}. Since H u l l (J) does n o t i n t e r s e c t bX we can f i n d a n o p e n n e i g h b o r h o o d W o f X in Y such t h a t H u l l (J0) does n o t i n t e r s e c t ( W ' ~ X ) .
B y a s t a n d a r d a r g u m e n t we can f i n d an a n a l y t i c p o l y h e d r o n /~ ~ - { z E W : ]fi(z)I < 1, i = 1 , . . . ,s}, w h e r e f l , . . . , L e A a n d X c F while ~ c c W. I t follows t h a t if A 0 --~ A iN, t h e n each c o m p a c t subset o f ]7 has a c o m p a c t Ao-convex hull in ]7. N e x t follows a n i m p o r t a n t step in t h e proof.
L]~M~A 3.1. Let ~t be the closure of A in the Frdchet space ~ ( I ' ) . Then ~4IX is contained in B, and if T is the map which sends f C ~4 into f I X E B, then T is a continuous map from the Frdchet algebra ~1 into B.
Proof. Using T h e o r e m 6.8 in [13] a p p l i e d to t h e a n a l y t i c space /', it follows t h a t t h e r e exists a h o m e o m o r p h i s m 4 f r o m r o n t o a Stein space M such t h a t if H = { g e C ( M ) : g = f o 4 -1 for some f e A } , t h e n H = @ ( M ) .
N o w 4 ( X ) is a c o m p a c t ~@(M)-convex subset of M. Since H o : {g E C(M) : g : f o r -1 for some f C A}, is a dense s u b a l g e b r a o f ~ ( M ) , we can f i n d a n O k a - Weft d o m a i n A in M such t h a t 4 ( X ) c A, while A is d e f i n e d b y f u n c t i o n s f r o m H 0. See [6, p. 211]. More precisely t h e r e are elements g l , . . . , gN in H 0 such t h a t t h e m a p G : y ---> (gl(Y), 9 9 9 gN(Y)), is a b i h o l o m o r p h i c m a p f r o m A o n t o a closed a n a l y t i c subset [2 of t h e open polydise D ~ in C N.
I f K ~-- G ( 4 ( X ) ) , t h e n K is t h e j o i n t s p e c t r u m d e t e r m i n e d b y t h e e l e m e n t s bl . . . bN, w h e r e bl = gl o 4 -1IX. T h e n a n a p p l i c a t i o n o f t h e Symbolic Calculus shows t h a t if g E ~ ( M ) , t h e n g o 4 - 1 [ X d e t e r m i n e s a n e l e m e n t o f B. I t follows t h a t A I X c B .
T o p r o v e t h e c o n t i n u i t y o f T we use a result in [14, p. 161], w h i c h asserts t h a t if {g.} is a sequence in @(D) for w h i c h lira g~ ~ 0 holds u n i f o r m l y on c o m p a c t subsets o f ~ , t h e n t h e r e exists a f i x e d o p e n n e i g h b o r h o o d U of K in C N a n d e l e m e n t s G ~ e ~ ( U ) such t h a t G ~ ] K = g ~ I K while lim [G~lu~-O. T h e n t h e c o n t i n u i t y principle for t h e S y m b o l i c Calculus implies t h a t if u~----g~ o 4-11X = Gn o 4 -1, we h a v e lim []u.[[ = 0.
T h e n e x t r e s u l t follows f r o m t h e c o n t i n u i t y o f T in L e m m a 3.1.
L~M~A 3.2. Let us put J ~- { f e A : f i X e J}. Then J is a closed ideal in the Frdchet space A.
Proof of Theorem 3.1. T h e set S ~ H u l l (J) is a c o m p a c t a n a l y t i c subset of Y, a n d hence it is finite. Since H u l l (Jo)13 ( / ~ X ~ is e m p t y while J o [ / ' C J , we can easily f i n d e l e m e n t s gx . . . gN in J such t h a t t h e set Z ~ {y E / ~ : gx(Y)
9 .----gx(Y) ~ 0}, is t h e finite set S.
ttOLOlYIORPHIC C O N V E X I T Y A I ~ D A N A L Y T I C S T R U C T U R E S I N BAI~ACt~ ALGEBRAS 49
Next we consider the closed ideal I which g~ . . . gN generate in A. Since has been identified with q@(M), where M is the Stein space from L e m m a 3.1, we can apply standard sheaf theory. For g~ . . . . , gs determine a coherent sheaf, called I, of .4-ideals. The sections of this sheaf are the elements of I. See [6, p. 244] and [15].
Since the cohomology groups vanish for all coherent sheaves of el-modules, we conclude t h a t the factor space A / I is isomorphic with the sections of the quotient sheaf R = A / I . Here R has a non trivial stalk in the finite set S only. At each point s E S the complex vector space R~ is finite dimensional, because {s} is an isolated point in the analytic set determined b y I.
I t follows t h a t I has ~ finite eodimension in .4, and a fortiori J has a finite codimension in A. T h e n we can write A = 5 d- Cf~ -t- . . . + Cf,,,, where fl . . . . ,f,~ E~I. Let bl = f i I X be the corresponding elements in B.
Since J is a closed ideal in B it follows t h a t J -~ Cb 1 d- 9 9 9 -4- Cb,, is a closed subspgce of B. This subspace contains the dense subset Ax, a n d hence J has a finite codimension in B.
Finally we remark t h a t the w o r k in [16] contains material related with Theorem 3.1.
4. H o l o m o r p h i c sets in C ~
Let A be a uniform algebra. I f W is an open subset of MA we denote by
@A(W) the algebra of all functions on W which are locally approximable in W by functions from A. I f K is a compact subset of M A we p u t q~A(K) = { r E C(K) : K an open neighborhood W of K in MA and some g CHA(W ) such that glK = f } . Finally HA(K) is the uniform closure of HA(K ) in C(K). We say t h a t K is a holomorphic set if K = MxA(~ ).
Since each element of A determines an element in HA(K ) via its restriction to K, we get induced map ~ from MuA(K) into MA satisfying f ( y ) - ~ f ( n ( y ) ) for all yEMHA(K I and all f in A.
The result below is essentially well-known and a proof when K is a compact set in (P occurs in [17].
PROPOSITIOn; 4.1. Let K be a compact set in M.4 and put K I = ~(MHAK)), I f now f C It~(K) is such that there is some g E HA(K1) satisfying g[K = f, then f(y) = g(~(y)) for all y E MuAK ). So in particular g is determined by f.
Pro@ Suppose firstly t h a t g E~@A(K~) a n d define ~ ( y ) = g(n(y)) for all y E MHAK). The continuity of ~ shows t h a t ~ is locally approximable by functions from A on MuA~ 3. So if B is the uniform algebra on MuA(~) generated b y H.4(K) and ~, t h e n 0B = 0nA(~)c K holds. See [10, Corollary 2.4].
50 Al%KIV FOR !~IATEI~IATIK. VO], 9 l~o. 1
W h e n g~ ~ glK is c o n s i d e r e d as a n e l e m e n t on K . Since a B c K it follows t h a t ~ = g l on w h e n g E ~A(K1).
T h e general case follows if we let (g~) b e a
o f HA(K ) we see t h a t ~ = g l M~AK). This p r o v e s t h e r e s u l t sequence in @A(K1) such t h a t lim [gn - - gig, ~ O. F o r t h e n fn ---- gn]K belong to H a ( K ) while lim lfn - - f I K -~ 0.
This implies t h a t f(y) = l i m f n ( y ) = lim gn(a(y)) ~ g(z~(y)) for all y E MH~(K).
W e n o t i c e t h a t P r o p o s i t i o n 4.1 i m m e d i a t e l y implies t h a t if K 1 -~ K , t h e n K is a h o l o m o l ~ h i e set. This was a l r e a d y p r o v e d in [2, T h e o r e m 10]. I n general K 1 is c o n t a i n e d in t h e u n i q u e smallest h o l o m o r p h i c se~ B(K) w h i c h contains K . H e r e B(K) is called t h e barrier of K. See [2].
Suppose n e x t t h a t K is a h o l o m o r p h i c set in MA. I f T is a closed subset o f K we define t h e algebra @K(T) ---- {f E C(T) : IK some o p e n n e i g h b o r h o o d U o f T in K a n d some g E@Ha(K)(U ) satisfying g l T = f } . F i n a l l y HK(T ) is t h e u n i f o r m closure o f ~ ( T ) )in" C(T). W i t h t h e s e n o t a t i o n s we s a y t h a t T is a K-holomorphic set if M~:( T = T holds.
T h e result below is a d i r e c t c o n s e q u e n c e of C. E . R i c k a r t ' s t h e o r y .
PROPOSITION 4.2. Let T be a compact subset of the holomorphic set K. I f now T is a holomorphic set, then T is K-holomorphic.
W e do n o t k n o w if t h e c o n v e r s e is t r u e , i.e. i f t h e c o n d i t i o n t h a t T is K - h o l o - m o r p h i c implies t h a t T is a h o l o m o r p h i e set. B u t t h e following result shows t h a t t h e c o n v e r s e is s o m e t i m e s t r u e .
P~OPOS[TION 4.3. Let K be a compact holomorphic set in C ~ and let W be a relatively open subset of K. Let S be a closed subset of W which is ~K(W)-convex.
Then S is a K-holomorphic set and if g E H ( S ) , then g can be uniformly approxi- mated on S by functions from @K(W). I f finally B(S) is a compact subset of W, then S is a holomorphic set.
Proof. T h e assertion t h a t S is a K - h o l o m o r p h i c set is a consequence of T h e o r e m 12 in [2]. N e x t we let B be t h e u n i f o r m algebra on S g e n e r a t e d b y ~K(W)]S.
B e c a u s e S is @K(W)-eonvex in W, while W is a n o p e n subset o f M~CK)= K , it follows f r o m t h e w o r k in [12] t h a t S = M s.
I f zl , 9 9 9 , z~ are t h e c o o r d i n a t e f u n c t i o n s in C ~, t h e n ziIS = b~ are e l e m e n t s in B. H e r e S is t h e j o i n t s p e c t r u m of b 1 , . . . , b ~ . I t follows t h a t if g E ( ~ ( S ) , t h e n glS E B. This implies t h a t H(S) c B which p r o v e s t h e second assertion.
Suppose n e x t t h a t B(S) is a c o m p a c t subset of W. I f g E H ( K ) it is obvious t h a t glS EH(B(S)). I t follows t h a t if f E ~ K ( W ) , t h e n fIB(S) is locally a p p r o x i m a b l e on B(S) b y f u n c t i o n s f r o m H(B(S)). Since B(S) ~ MH(B(S) ) it follows f r o m [10, Corollary 2.4] t h a t Ifls(,s) _~ ]fit for all f in ~K(W), w h e r e T ~ OH(s(S)).
B u t n o w T h e o r e m 11 in [2] shows t h a t T c S. T h e n t h e c o n d i t i o n t h a t S is (OK(W)-convex implies t h a t S = B(S). H e n c e S is a h o l o m o r p h i c set.
H O L O M O R I ? H I C C O N V E X I T Y A:ND A ~ A L Y T I C S T R U C T U R E S I ~ B A N A C H A L G E B R A S 5 1
A n e x a m p l e
W e give a n e x a m p l e o f a c o m p a c t h o l o m o r p h i c set in C n which is n o t a hull, i.e. it c a n n o t be r e p r e s e n t e d as t h e i n t e r s e c t i o n of o p e n d o m a i n s o f h o l o m o r p h y . This e x a m p l e gives a n e g a t i v e answer t o t h e q u e s t i o n raised in [7, p. 515].
F i r s t l y we discuss c o m p a c t R e i n h a r d t sets in C 2. I f K is a c o m p a c t R e i n h a r d t set in C 2 it can be r e p r e s e n t e d as a c o m p a c t subset of R ~ D {(]z ] , ]w I) :
(z, w) e {~2}.
A set o f t h e t y p e Z ( a l , a s , b l ) = { ( z , w ) C C 2 : a 1 < ] z l _ < a s a n d I w [ = b l } is called a Z - s e g m e n t . I n a similar w a y we get t h e W-segments W ( a l , b l , b~) and f i n a l l y t h e squares Q(a 1 , a 2 , b 1 , b s ) = { ( z , w ) e C 2 : a 1 < IzI _ < a s a n d bl .~<
Iwl
_< bs}.I f K is a c o m p a c t set o f t h e f o r m W ( a l , b 1 , bs) tJ Z ( a l , a s , bl) 13 Z ( a l , a2, bs), w h e r e a I < a 2 a n d b 1 < bs, t h e n it is easily v e r i f i e d t h a t ~(K) = Q(a I , a s , b 2 , bs).
Using this f a c t it follows t h a t i f K 0 = W ( a I , 0 , bs) [.J Z ( a I , a 3 , O) [.J Z ( a 1 , a 2 , b2) [J Z ( a 2 , a a , b ) for some 0 < b < b s , t h e n u ( K 0 ) = Q ( a 1 , a s , O , b 2 ) U Z ( a ~ , a a , b ) t j Z ( a 2 , a 3 , 0 ) . I t follows t h a t u(K0) = K1 is n o t a h o l o m o r p h i c set, while ~(K1) = Q(a 1 , a ~ , O , b s ) U Q ( a 2 , a a , O , b ) is t h e b a r r i e r o f K .
T h e e x a m p l e . L e t K = W ( 0 , 0 , 1 ) t J Z ( 0 , 1 , 0 ) U T , ~There T - = I , ] T n a n d T n = Z(2 -n-1 , 2 - " , 1 - - 1In) for all n > 1. I f n o w V is a d o m a i n o f h o l o m o r p h y c o n t a i n i n g K , t h e n t h e p r e c e e d i n g discussion easily implies t h a t V contains all p o i n t s ( z , w ) for w h i c h 2 - ~ - I _ < ]z I _<2 -~ a n d [w I__< 1 - - 1In. I t follows t h a t K is n o t a hull.
B u t using t h e f a c t t h a t t h e closed sets Tn are h o l o m o r p h i c sets a n d closed c o m p o n e n t s of K , it follows easily t h a t ~(K) = K , a n d h e n c e K is a h o l o m o r p h i e Set.
I n t h e e x a m p l e K is d i s c o n n e c t e d , a n d this is n e c e s s a r y because t h e following assertion can easily be verified.
P~OPOSITION 4.4. L e t K be a connected compact R e i n h a r d t set i n C ~. T h e n K is a holomorphie set i f a n d o n l y i f K is a hull.
F i n a l l y we show h o w to c o n s t r u c t a c o n n e c t e d h o l o m o r p h i c set w h i c h is n o t a hull. L e t ( z , w , t ) be t h e c o o r d i n a t e s in C 3. P u t K l = { ( z , w , t ) : ( z , w ) C K a n d t = 1}, w h e r e K is t h e set f r o m t h e e x a m p l e above. T o e a c h n > 1 we let J~ be a closed arc in C 3 which s t a r t s f r o m t h e p o i n t p ~ = ( 2 - ~ - 1 , 1 - - l / n , 1 ) a n d has q ~ = (2 - ~ - 1 , I - 1/(n-~- 1 ) , 1 ) as a n e n d p o i n t . As we pass f r o m p~ to q~ t h e t-variable winds a r o u n d t h e u n i t circle, i.e. we m a y t a k e J~ = {p(x) : p ( x ) = (2 - n - 1 , 1 + x / n ( n ~ - 1 ) , e 2~i~) as 0 > x < 1}.
N o w t h e set S = K I tJ U (J~ : n > 1), is c o n n e c t e d . B e c a u s e K is n o t a hull we see t h a t S is n o t a h u l l . B u t S i s a h o l o m o r p h i e s e t . F o r if n > 1 is given, t h e n we can d e t e r m i n e b r a n c h e s of t h e f u n c t i o n (2~i) -1 log (t) in such a w a y t h a t we get an e l e m e n t f C ~ ( S ) satisfying f = 1 on t h e set S~ = {(z, w , t) : (z, w) C T n a n d t = 1}, while f = 0 on K ~ S ~ . I n a d d i t i o n f = 0 on Jm, w h e n m = / : n ,
52 ARKIV FOR 3IATEMATIK. Vol. 9 No. 1
a n d f , m a p s J~ o n t o t h e closed u n i t i n t e r v a l . Using t h e existence o f t h e f a m i l y {f,} in H(S), it follows easily t h a t S = MH(S).
F i n a l l y we r e f e r to [19] for m o r e results a b o u t h o l o m o r p h i c sets.
Remark. F o r a v e r y i n t e r e s t i n g t h e o r y , p a r t l y b a s e d on h o l o m o r p h i e sets in 13 ~, we r e f e r t o 1%. H a r v e y ' s p a p e r ~The t h e o r y of h y p e r f u n c t i o n s on t o t a l l y real subsets o f a c o m p l e x m a n i f o l d w i t h a p p l i c a t i o n s to e x t e n s i o n problems~> in A m e r . g. M a t h . 91 (1969), 8 5 3 - - 8 7 3 .
5. I n t e g r a l e x t e n s i o n s of B a n a c h algebras
L e t B be a B a n a c h a l g e b r a a n d let g E C(MB). W e s a y t h a t g is i n t e g r a l o v e r B if g satisfies a n e q u a t i o n f + big n-1 + . . . + bn = 0 in MB w i t h bl E B.
L e t g be i n t e g r a l o v e r B a n d p u t A = [ B , g] = t h e s u b a l g e b r a of C(MB) g e n e r a t e d b y B a n d g. I t is t h e n possible t o i n t r o d u c e a n o r m on A which m a k e s -4 i n t o a B a n a c h algebra. W e r e f e r t o [20, 22] for details of this c o n s t r u c t i o n .
W h e n B a n d A are as a b o v e we get t h e i n d u c e d m a p n f r o m MA i n t o MB.
I n general 0a # n-I(0B), a n d w e do n o t k n o w i f am D ~-I(OB) is always t r u e . T h e r e s u l t b e l o w was also p r o v e d in [22] u n d e r a slight a d d i t i o n a l a s s u m p t i o n .
T I ~ E O l ~ 5.1. Let B and A be as above. I f ~ is an open map, then 0A = ~-I(0B).
N e x t we i n t r o d u c e some c o n c e p t s m o t i v a t e d b y integral extensions.
Definition 5.1. L e t B a n d A be t w o B a n a c h algebras such t h a t B is a closed s u b a l g e b r a o f A. L e t z be t h e i n d u c e d m a p f r o m M A i n t o M s. W e s a y t h a t a d m i t s a finite string of local charts, if t h e r e is a decreasing sequence o f closed sets M A = Z 0 D Z1 D . . . D Zn D Zn+ ~ = 13 such t h a t t h e following holds: I f p C W~ = Z ~ Z ~ + 1, t h e n t h e r e is a n o p e n n e i g h b o r h o o d U of p in MA s u e h t h a t ~ I ( U f 3 Wi) is injective. I f f i n a l l y a E A, t h e n B c o n t a i n s a sequence {b,} such t h a t limb= = g holds u n i f o r m l y on U 13 Wi, w h e r e g a n d b~ are t h e G e l f a n d t r a n s f o r m s on MA.
I n t h e special ease w h e n A = [ B , g] for some g E C(MB) satisfying Q(g) = 0, w h e r e Q(T) = T ~ q- b i t ~-1 q- . . . q- b,, t h e n t h e m a p z f r o m M A i n t o M B a d m i t s a f i n i t e string o f local charts. F o r let us p u t Qk(T) = OkQ/OT ~ for each k > 1, a n d consider t h e e l e m e n t s gs = Q~(g) in A. I f we t h e n p u t Z~-=
{ y E M ~ : g ~ ( y ) . . . . = g ~ ( y ) = 0}, t h e n it is n o t h a r d t o v e r i f y t h a t M a = Z 0 D Z 1 D . . . D Z= D t3, gives t h e r e q u i r e d string.
N o w we b e g i n t h e s t u d y of Shilov b o u n d a r i e s . T h e following r e s u l t follows f r o m e a s y t o p o l o g i c a l considerations.
L ] ~ A 5.1. Let B ~ A be a pair of Banach algebras such that zr admits a finite string of local charts. Then the fibers ~r-l(x) are totally disconnected for all x E M s.
LSMMA 5.2. Let B C A be a pair of Banach algebras. Let x E X B be such that zt-l(x) is totally disconnected. Then s - l ( x ) C Xa.
Proof. Since x E X B we k n o w t h a t {x} is a n i n t e r s e c t i o n o f p e a k sets in M ~
t I O L O i ~ I O R P I t I C C O N V E X I T Y AN]:) A:b~ALT(TIe STI~,UCTUB, E S I~N" BAI~CAC/:~ A L G E B R A S 5 3
d e t e r m i n e d b y f u n c t i o n s in t h e u n i f o r m closure B~ o f B. I t follows t h a t A 0 ---- A ~ I z - l ( x ) is a u n i f o r m a l g e b r a on ~ - l ( x ) a n d h e n c e M ~ ~ holds. Since z - l ( x ) is t o t a l l y d i s c o n n e c t e d it follows f r o m a w e l l - k n o w n r e s u l t t h a t A o = C ( z - l ( x ) ) . T h i s m e a n s t h a t ~-X(x) is a p e a k i n t e r p o l a t i o n set in MA~ ~--M~. T h e n z - l ( x ) c ZA~ =- 2: A follows.
L E ~ 5.3. Let B ~ A be such that ~ admits a f i n i t e string of local charts.
I f ~ is an open m a p , then O~ ~ Z~-~(OB).
Proof. U s i n g L e m m a 3.1 a n d 3.2 it follows t h a t z~-~(XB) ~ OA. Since OB is t h e closure of XB in M B while ,~ is a n o p e n a n d c o n t i n u o u s m a p it follows t h a t
~-1(~,) c OA-
N o w we o b t a i n a g e n e r a l i z a t i o n o f T h e o r e m 5.1.
T~IEOI~EM 5.2. Let B ~ A be such that ~r admits a f i n i t e string of local charts J~A = Zo ~ ZI ~ . . . ~ Z= ~ O. S u p p o s e also that ~ is an open m a p and that Zi+~ is an A z - a n a l y t i c variety f o r each i. T h e n OA = ~ - ~ ( ~ ) .
Proof. U s i n g L e m m a 5.3 it is sufficient t o p r o v e t h a t 0A ~ K u l l A (~-~(0B)) = S.
So let us a s s u m e t h a t M A ~ S ~-- holds. Since D is o p e n while 0A X A ~ D c Z 1 .
So let y E X A f i D b e g i v e n n e i g h b o r h o o d W o f y in MA
D is n o n e m p t y . N o w we p r o v e t h a t ~a gl D c Z~
is t h e closure of 2A, it is sufficient to p r o v e t h a t a n d p u t ~(y) = x. T h e n we c a n choose a closed such t h a t z m a p s W h o m e o m o r p h i c a l l y o n t o
~(W) while z ( W ) c M B ~ O B . I n a d d i t i o n we m a y a s s u m e t h a t e a c h e l e m e n t in A c a n b e u n i f o r m l y a p p r o x i m a t e d b y f u n c t i o n s f r o m B o n W. Since z is o p e n we see t h a t b(~(W)) c zc(bW). A n a p p l i c a t i o n of t h e L M P to B shows t h a t ]b(y)[
lb(x)t << Ib[~(=(w))< ]b]~ W for all b in B. T h e n t h e a p p r o x i m a t i o n c o n d i t i o n shows t h a t y E H u l l A (bW), a n d h e n c e y c a n n o t belong t o Z~A.
W e h a v e p r o v e d t h a t 0 A f~ D C Z1 a n d h e n c e D c H u l l a (0A) C HullA (S U Z1).
U s i n g P r o p o s i t i o n 1.3.c we d e d u c e t h a t D c Z r Since D is a n o p e n s u b s e t of M A we c a n n o w use t h e local c h a r t s in Z I ~ Z 2 , a n d t h e s a m e a r g u m e n t s show t h a t D c Z 2. I n d u c t i v e l y we g e t D c Z~ for all v > 1, so f i n a l l y D c Z , + I = 0 . H e n c e S = M A w h i c h implies t h a t a a = ~-I(0B).
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Received November 16, 1970 J a n - E r i k BjSrk
D e p a r t m e n t of Mathematics University of Stockholm Box 6701
S-113 85 Stockholm, Sweden
