In this paper we give a partial answer to this question, i.e., we show that the result remains true for balancedHb(U )-domains of holomorphy on Tsirelson’s space

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Indag. Mathem., N.S., 18 (2), 269–279 June 18, 2007

Spectra of algebras of holomorphic functions of bounded type

by Daniela M. Vieira¹

Caixa Postal 6065 CEP 13083-859,Campinas-SP Brasil

Communicated by Prof. J.J. Duistermaat at the meeting of September 25, 2006

ABSTRACT

We prove that ifUis a balanced_Hb(U )-domain of holomorphy in Tsirelson’s space then the spectrum of_Hb(U )is identified withU. We derive theorems of Banach–Stone type for algebras of holomorphic functions and algebras of holomorphic germs.

INTRODUCTION

LetEbe a Banach space and letUbe an open subset ofE. In [2], it is proved that ifE is Tsirelson’s space, then the spectrum of_H_b(U ) is identified withU, when U=E. In [11], J. Mujica generalizes this result for absolutely convex open subsets of Tsirelson’s space, and asks if the result can be improved for a more general class of open subsets ofE, for instance, polynomially convex open subsets. In this paper we give a partial answer to this question, i.e., we show that the result remains true for balancedHb(U )-domains of holomorphy on Tsirelson’s space. In Section 1 we defineHb(U )-convex open subsets and present properties and examples of such sets. We also give some auxiliary results before proving the main result. Most of them are generalizations toU-bounded sets of known results for compact sets. In Section 2 we present the main result and a corollary on finitely generated ideals of the algebraHb(U ). In Section 3 we present theorems of Banach–Stone type for the

MSC:46G20, 46E25, 32E20

E-mail: danim@ime.unicamp.br (D.M. Vieira).

1 Research supported by FAPESP, Process no. 04/07441-3, Brazil.

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algebras_Hb(U )and_Hb(V ), and also for algebras of holomorphic germs_H(K)and H(L), improving results from [14].

I wish to thank Prof. J. Mujica for so many helpful comments concerning this work.

1. PRELIMINARIES

We refer to [7] and [10] for background information on infinite-dimensional complex analysis.EandF will always denote Banach spaces. LetP(E;F )denote the Banach space of all continuous polynomials fromEintoF._P(^mE;F )denotes the Banach space of all continuousm-homogeneous polynomials fromE intoF. Pf(^mE;F )denotes the subspace ofP(^mE;F )generated by all polynomials of the formP (x)=ϕ(x)^mb, for allx∈E, whereϕ∈Eandb∈F. Such polynomials are calledof finite type. WhenF =C, we writeP(E),P(^mE)andPf(^mE)instead of P(E;C),_P(^mE;C)and_Pf(^mE;C), respectively.

LetU be an open subset ofE. We say that a subsetA⊂UisU-boundedifAis bounded and there existsε >0such thatA+B(0, ε)⊂U.

We will denote by Hb(U;F ) the vector space of all holomorphic mappings f:U−→F which are bounded on every U-bounded subset. Such mappings are calledholomorphic mappings of bounded type. IfF =C, we writeHb(U )instead of Hb(U;C). We denote byτ_bthe topology on_H_b(U;F )of the uniform convergence on all U-bounded subsets. Hb(U;F ) is a Fréchet space for this topology, and likewise_H_b(U )is a Fréchet algebra. IfU is balanced, it follows from the Cauchy inequalities that the Taylor series of eachf ∈Hb(U;F ) at the origin converges uniformly on eachU-bounded subset. In particular, ifρ_U denotes the restriction of mappings toU, thenρU(P(E;F ))isτb-dense inHb(U;F ).

We denote by Sb(U ) the spectrum of the algebra Hb(U ), i.e., the set of all continuous complex homomorphisms (and by that we mean linear and multiplicative) ofHb(U ). Every point ofU can be associated with an element ofSb(U )as follows: for eachz∈U fixed, let δ_z:Hb(U )−→Cbe defined by δ_z(f )=f (z), for allf ∈Hb(U ). Each δ_z is calledevaluation at z. It is clear that δ_z∈Sb(U ), for allz∈U, and the mappingδ:U−→Sb(U )is used in order to identifyU with the subsetδ(U ) ofSb(U ). Note that δ is injective because the continuous linear forms already separate the points ofE.

In this paper we will show that under certain hypotheses onE and U, all the elements ofSb(U )are evaluations at some point ofU, and in this sense we say that Sb(U )is identified withδ(U ).

In the following we give some definitions, examples and auxiliary results to the main result.

LetXbe a subset ofE,Abe a subset ofX, andF⊂C(X). Then theF-hull of Ais the following set:

A_F=

x∈X: f (x)sup

A

|f|, for allf ∈F .

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Definitions 1.1. LetEbe a Banach space and letUbe an open subset ofE. We say thatUis:

(1) Pb(E)-convexifA_P_(E)∩U isU-bounded, for everyU-bounded subsetA; (2) stronglyPb(E)-convexifA_P(E)⊂U and isU-bounded, for everyU-bounded

subsetA;

(3) Hb(E)-convexifA_H_b_(E)∩U isU-bounded, for everyU-bounded subsetA; (4) stronglyHb(E)-convexifA_H_b_(E)⊂Uand isU-bounded, for everyU-bounded

subsetA;

(5) _H_b(U )-convexifA_H_b_{(U )}isU-bounded, for everyU-bounded subsetA. The following lemma shows that the notions of (strongly) Pb(E)-convex and (strongly)_Hb(E)-convex set coincide.

Lemma 1.2. LetAbe a bounded subset ofE. ThenA_H_b(E)=A_P(E).

Proof. Since _P(E) ⊂ Hb(E), we have that A_H_b(E) ⊆ A_P(E). Now let us suppose that there existsa∈A_P_(E)such thata /∈A_H_b_(E). Letf ∈Hb(E)be such that|f (a)|>sup_A|f|. SinceA_P_(E)is bounded andP(E)is dense inHb(E)for the τb topology, given ε >0 there exists P ∈P(E)such that sup_A

P(E)|f −P|< ^ε₂. In particular we have that sup_A|P|sup_A|P −f| +sup_A|f|< ^ε₂ +sup_A|f|. Finally we get that |f (a)| |f (a)−P (a)| + |P (a)| < ^ε₂ +sup_A|P| < ε + sup_A|f|,for allε >0, which is a contradiction. ₂

The next lemma shows that the last condition on Definition 1.1.2 (and 1.1.4) is superfluous.

Lemma 1.3. If A_H_b_(E)⊂U, for every U-bounded subsetA, thenU is strongly Hb(E)-convex.

Proof. We follow ideas of [10, Lemma 54.8]. LetAbe a U-bounded subset. We must show thatA_H_b_(E) is U-bounded. Since it is clear thatA_H_b_(E) is bounded, it remains to show that there existsε >0such thatA_H_b(E)+B(0, ε)⊂U. Letε >0be such thatA+B(0, ε)isU-bounded. Then(A+B(0, ε))ˆHb(E)⊂U. Lety∈A_H_b_(E), t∈B(0, ε)and0< θ <1. Then for eachf∈Hb(E)we have that

f (y+θ t )^∞

m=0

θ^mP_t^m(f )(y)^∞

m=0

θ^msup

A

P_t^m(f )(1−θ )⁻¹ sup

A+B(0,ε)

|f|, where the second inequality follows becauseP_t^m(f )∈Hb(E)andy∈A_H_b_(E). The third inequality follows by applying [10, Corollary 7.3], witht∈B(0, ε)andr=1. Next we apply the above inequality tofⁿ, taken-th roots and letn→ ∞to get that |f (y+θ t )|sup_A+B(0,ε)|f|, that is, y+θ t∈(A+B(0, ε))ˆ_Hb(E)⊂U. By lettingθ→1we have thaty+t∈U, and the conclusion follows. 2

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Lemma 1.4. Let_F_⊂_Hb(E)be a family with the property that the functionx → f (λx)is an element of_F, for everyf ∈F and_|λ|1. LetA⊆Ebe a balanced subset. ThenA_Fis balanced.

Proof. Letf ∈F. For eachλ∈Csuch that|λ|1, letf_λ∈Fbe such thatf_λ(x)= f (λx), for allx∈E. Lety∈A_F. Then_|f (λy)| = |fλ(y)|sup_A|fλ|sup_A|f|, proving thatλy∈A_F, and henceA_F is balanced. 2

It is clear that strongly_Hb(E)-convex open subsets are always_Hb(E)-convex.

Also, it is easy to see thatA_H_b_{(U )}⊂A_H_b_(E)∩U, and hence we have thatHb(E)- convex open subsets are alwaysHb(U )-convex. The next proposition shows that ifU is balanced, then all these notions coincide. Moreover, this proposition is the heart of the proof of the main result.

Proposition 1.5. Let U ⊂E be a balanced open subset. Then the following conditions are equivalent.

(1) U is strongly_P_b(E)-convex.

(2) U is stronglyHb(E)-convex.

(3) U is_Pb(E)-convex.

(4) U is_H_b(E)-convex.

(5) U isHb(U )-convex.

Proof. The implications(1)⇔(2)and(3)⇔(4)were proved in Lemma 1.2. The implications(2)⇒(4)⇒(5)were commented above.

(5)⇒(4)We show thatA_H_b_{(U )}=A_P_(E)∩U=A_H_b_(E)∩U, for everyU-bounded subsetA, and then conclude thatU is _Hb(E)-convex. Lety∈A_P(E)∩U and we show thaty∈A_H_b_{(U )}. Letf ∈Hb(U )fixed. SinceUisHb(U )-convex, the setB= A_H_b_{(U )}∪ {y}isU-bounded, and sinceUis balanced, givenε >0, there isP∈P(E) such thatsup_B|f−P|<^ε₂. Then_|f (y)||f (y)−P (y)| + |P (y)|<^ε₂+sup_A|P|. On the other hand:

sup

A

|P|sup

A

|f −P| +sup

A

|f|<ε 2 +sup

A

|f|.

And finally we get that _|f (y)|< ε+sup_A|f|, for all ε >0, which implies that y∈A_H_b_{(U )}.

(4)⇒(2)LetA be an U-bounded subset. By Lemma 1.3, it suffices to prove thatA_H_b_(E)⊂U. First we assume thatAis balanced. Letx∈A_P_(E). DefineD1= {λ∈C: |λ|1}andD= {λ∈D₁: λx∈U}. ThenDis a disk centered at the origin becauseUis balanced, andDis an open subset ofD₁becauseUis open. Letε >0 be such thatA_P(E)∩U+B(0, ε)⊂U. Letλ∈D1,λx∈U, and letμ∈D1be such that|μ−λ|x< ε. Thenλx∈A_P_(E)∩Ubecausex∈A_P_(E)andA_P_(E)is balanced by Lemma 1.4. Furthermoreμx−λx< ε, henceμx∈A_P_(E)∩U+B(0, ε), and thereforeμx∈U. This implies that any point on the boundary ofDbelongs toD,

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andD is an open and closed subset ofD1, and thereforeD=D1. It follows that x=1x∈U. Since this holds for anyx∈A_P_(E), we have proved thatA_P_(E)⊂U.

IfAis not balanced, we considerB=ba(A), the balanced hull ofA. If follows by [5, Lemma 1.3(b)] thatB is a balanced U-bounded subset. Then we apply the arguments above and get thatA_H_b_(E)⊂B_H_b_(E)⊂U. 2

Next we give some examples of balanced_H_b(E)-convex open subsets.

Example 1.6. LetP ∈P(^mE;F )and letU= {x∈E: P (x)<1}. ThenU is a balancedHb(U )-convex open set.

Proof. ClearlyU is a balanced open set. LetAbe anU-bounded subset ofU. Let ε >0 denote the distance from Ato the boundary of U, and letr=sup_x_∈_Ax. Ifx∈Aand1λ <1+^ε_r, then λx−x = |λ−1|x< ε, henceλx∈U, and thereforeP (x) = P ((¹_λ)λx) =λ⁻^mP (λx)< λ⁻^m. Taking in the right-hand side the infimum over allλsuch that1λ <1+^ε_r , we conclude thatP (x) c:=(1+^ε_r)⁻^m<1for everyx∈A.

Let us show thatA_H_b_(E)⊂U. Lety∈A_H_b_(E)andϕ∈F. Thenϕ◦P ∈Hb(E) and hence|ϕ◦P (y)|sup_A|ϕ◦P|. Now

P (y)= sup

ϕ∈B_F

ϕP (y) sup

ϕ∈B_F

sup

x∈A

ϕP (x) sup

x∈A

P (x)=c <1, and hencey∈U.

This shows thatA_H_b_(E)isU-bounded, because if0< c <1, then every bounded subset of{x∈E: P (x)c}isU-bounded. HenceU is stronglyHb(E)-convex by Lemma 1.3. FinallyU is_H_b(U )-convex by Proposition 1.5. ₂

Corollary 1.7. Let P ∈P(^mE)and let U = {x ∈E: |P (x)|<1}. Then U is a balancedHb(U )-convex open set.

Corollary 1.8. LetA∈L(E₁, . . . , E_m;F )andE=E₁× · · · ×E_m. Then U=

(x₁, . . . , x_m)∈E: A(x₁, . . . , x_m)<1 is a balancedHb(U )-convex open set.

Proof. By [10, Theorem 3.6] it follows that A, viewed as a mapping from E to F, is a homogeneous polynomial of degree m. Then the result follows by Example 1.6. ₂

Corollary 1.9. LetU= {(x, λ)∈E×C:λx<1}. ThenUis a balancedHb(U )- convex open set.

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2. THE MAIN RESULT

In [13], B. Tsirelson constructed a reflexive Banach spaceX, with an unconditional Schauder basis, that does not contain any subspace which is isomorphic toc0or to any p. R. Alencar, R. Aron and S. Dineen proved in [1] thatPf(^mX)is norm-dense inP(^mX), for allm∈N. Inspired by this result, we will say that a Banach spaceEis aTsirelson-like spaceifEis reflexive and_Pf(^mE)is norm-dense in_P(^mE), for all m∈N.

The following theorem is the main result of this paper.

Theorem 2.1. LetEbe a Tsirelson-like space, and letU be a balanced_H_b(U )- convex open subset ofE. Then the spectrum ofHb(U )is identified withU. Proof. SinceU is balanced and_H_b(U )-convex, it follows by Proposition 1.5 that U is strongly Hb(E)-convex. Now we follow the ideas of [11, Theorem 1.1].

LetT:Hb(U )−→C be a continuous homomorphism. Then there exists C >0 and anU-bounded subsetA⊂Usuch that

T (f )Csup

A

|f|, for allf ∈Hb(U ).

SinceT is multiplicative, we have that|T (f )|ⁿ= |T (fⁿ)|Csup_A|f|ⁿfor every n∈N. Taking n-th roots and making n→ ∞ we conclude that actually C =1. Letr >0 such that A⊂B(0, r). In particular, we have that|T (f )|sup_A|f| sup_B(0,r)|f|, for all f ∈E. Hence we have thatT|E∈E=E, so there exists a uniquea∈Esuch thatT (f )=f (a), for all f ∈E, and henceT (P )=P (a), for all P ∈Pf(^mE), for allm∈N. Since Pf(^mE) is norm-dense in P(^mE), for all m∈N, it follows thatT (P )=P (a), for allP ∈P(E). Then we have that|P (a)| =

|P (f )|sup_A|P|, for allP ∈P(E), which implies thata∈A_P_(E)=A_H_b_(E)⊂U. SinceUis balanced, we have thatP(E)isτ_b-dense inHb(U ), and then we conclude thatT (f )=f (a), for allf ∈Hb(U ), proving the theorem. 2

Definition 2.2. LetEbe a Banach space and letUbe an open subset ofE. We say thatU is aHb(U )-domain of holomorphyif there are no open setsV andW inE with the following properties:

(1) V is connected and not contained inU; (2) ∅ =W⊂U∩V;

(3) for eachf ∈Hb(U )there existsf˜∈H(V )such thatf˜=f onW.

The following corollary is the announced result for balancedHb(U )-domains of holomorphy.

Corollary 2.3. LetEbe a Tsirelson-like space, and letU be a balancedHb(U )- domain of holomorphy inE. Then the spectrum of_Hb(U )is identified withU.

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Proof. By [6, Theorem 1] or [8, Theorem 1], we have that U is _Hb(U )-convex.

Then apply Theorem 2.1. ₂

The following result is a consequence of Corollary 2.3. It says that, under the hypotheses of Corollary 2.3, every proper finitely generated ideal ofHb(U )has a common zero.

Theorem 2.4. LetEbe a Tsirelson-like space. LetU⊂Ebe a balancedHb(U )- domain of holomorphy. Then givenf1, . . . , fn∈Hb(U )without common zeros, we can findg₁, . . . , g_n∈Hb(U )such thatⁿ_i₌₁f_ig_i=1.

Proof. The proof of [11, Theorem 1.5] applies. ₂

3. THEOREMS OF BANACH−STONE TYPE

In [3], S. Banach proved that two compact metric spaces X and Y are home- omorphic if and only if the Banach algebras _C(X) and _C(Y ) are isometrically isomorphic. M.H. Stone, in [12], generalized this result to arbitrary compact Hausdorff topological spaces, the well-known Banach–Stone theorem.

In [14], we present similar results for algebras of holomorphic functions of bounded type, using results on the spectrum of such algebras. More specifically, letEandF be reflexive spaces, one of them a Tsirelson-like space, and letU⊂E andV ⊂F be absolutely convex opens subsets. Then it is shown that the algebras Hb(U ) and Hb(V )are topologically isomorphic, if and only if there is a special type of biholomorphic mapping betweenUandV. To show these results we use the characterization of the spectra ofHb(U )withUdue to J. Mujica, [11, Theorem 1.1].

In this section we generalize this result to balanced_Hb(U )-domains of holomorphy, using the characterization of the spectrum of Hb(U ), Corollary 2.3 of this paper.

LetEand F be Banach spaces, andU⊂E and V ⊂F be open subsets ofE andF, respectively. We denote byHb(V , U )the set of all holomorphic mappings ϕ:V −→E, withϕ(V )⊂U, such thatϕmapsV-bounded subsets intoU-bounded subsets.

Next theorem is the result announced, and improves [14, Corollary 14].

Theorem 3.1. LetEandF be reflexive Banach spaces, one of them a Tsirelson- like space. LetU⊂EandV ⊂F be balancedHb-domains of holomorphy. Then the following conditions are equivalent.

(1) There exists a bijective mappingϕ:V −→Usuch thatϕ∈Hb(V , U )andϕ⁻¹∈ Hb(U, V ).

(2) The algebrasHb(U )andHb(V )are topologically isomorphic.

Proof. The proof of [14, Corollary 14] applies. ₂

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In [14, Theorem 16] it is shown that ifK⊂EandL⊂F are absolutely convex compact subsets of Tsirelson-like spaces, then the algebras_H(K) and_H(L) are topologically isomorphic if and only ifKandLare biholomorphically equivalent.

The key to the proof of such result is a theorem of Banach–Stone type between algebras of holomorphic functions of bounded type [14, Corollary 14]. We are going to present a generalization of this result to greater class of compact sets, using Theorem 3.1. But before we need some preparation.

LetEbe a Banach space, and letK⊂Ebe a compact subset. We define_H(K) to be the algebra of all functions that are holomorphic on some open neighborhood of K. The elements of H(K) are called germs of holomorphic functions. We endow_H(K)with the locally convex inductive limit of the locally convex algebras (H(U ), τ_ω), where U varies among the open subsets of E such that K ⊂U. If U_n=K+B(0,¹_n), for alln∈N, then it is easy to see that

H(K), τ_ω =lim_−→

n∈N

Hb(U_n).

We refer to [4,7] or [9] for background information on algebras of germs of holomorphic functions.

Definition 3.2. LetEbe a Banach space, letK be a compact subset ofEand let m∈N. We say thatKisP(^mE)-convexifK=K_P₍^m_E).

Before we present examples of balancedP(^mE)-convex compact sets, we shall need the next lemma, which is inspired in [10, Proposition 11.1]. IfAis a subset of a Banach space, we denote by(A)the closed, absolutely convex hull ofA. Lemma 3.3. LetEbe a Banach space and letAbe a bounded subset ofE. Then A_P_f₍^m_E)⊂(A), for allm∈N.

Proof. Lety /∈(A). By the Hahn–Banach theorem, there existsϕ∈E such that

|ϕ(y)|>sup_x_∈_(A)|ϕ(x)|. Hence |ϕ^m(y)|>sup_x_∈_(A)|ϕ^m(x)|sup_A|ϕ^m|, which shows thaty /∈A_P_f₍^m_E). 2

Example 3.4. Every absolutely convex compact subset of a Banach space E is P(^mE)-convex, for allm∈N.

Proof. LetK⊂Ebe an absolutely convex compact set. Since_P_f(^mE)⊂P(^mE), we have thatK_P₍^m_E)⊂K_P_f₍^m_E)⊂(K)=K, where the last inclusion follows by Lemma 3.3. 2

Example 3.5. LetEbe a Banach space, and L⊂Ebe a compact, balanced and P(^mE)-convex set. LetP ∈P(^mE). Then K= {x ∈L: |P (x)|1}is compact, balanced andP(^mE)-convex.

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Remark 3.6. If K is a _P(^mE)-convex compact set, then it is clear that K is polynomially convex. But the converse is not true. Indeed, it is easy to see that ifK=K_P₍^m_E), thenK is balanced. Now letK be a convex compact set, which is not balanced. ThenKis polynomially convex by [10, Examples 24.2(a)], but is not P(^mE)-convex, for anym∈N.

The next theorem will be useful to prove the main result of this section.

Theorem 3.7. LetEbe a Banach space and letK be a compact, balanced and P(^mE)-convex subset ofE, for somem∈N. LetUbe an open subset ofEsuch that K⊂U. Then there exists an open setV ⊂Ewhich is balanced andHb(V )-convex, such thatK⊂V ⊂U.

Proof. We begin with a slight modification of [10, Lemma 24.7]. If(K)⊂U, then we takeV =(K)+B(0, ε), where ε is such that (K)+B(0, ε)⊂U. If (K)is not contained inU, then for eacha∈(K)\U there isP ∈P(^mE)such thatsup_K|P|<1<|P (a)|. Since(K)\U is compact, we can find polynomials P1, . . . , P_k∈P(^mE)such that

(K)\U⊂ k

j=1

x∈E: P_j(x)>1 .

Now it is easy to see that{x∈(K): |P_j(x)|1, forj =1, . . . , k} ⊂U. Next we follow the arguments of [10, Theorem 28.2], finding a positive numberδ >0such thatV =((K)+B(0, δ))∩ {x∈E: |Pj(x)|<1, forj =1, . . . , k} ⊂U. NowV is balanced and_H_b(V )-convex, by Corollary 1.7. ₂

LetEandF be Banach spaces, and letK⊂EandL⊂F be compact subsets.

We say thatKandL arebiholomorphically equivalentif there exist open subsets U ⊂E and V ⊂F with K ⊂U and L ⊂V and a biholomorphic mapping ϕ:V −→U such that ϕ(L)=K. The next theorem is the announced result for algebras of holomorphic germs, and generalizes [14, Theorem 16].

Theorem 3.8. LetE andF be Tsirelson-like spaces. LetK⊂Eand L⊂F be balanced compact subsets, such thatKisP(^mE)-convex, andLisP(^kF )-convex, for somem, k∈N. Then the following conditions are equivalent.

(1) KandLare biholomorphically equivalent.

(2) The algebras_H(K)and_H(L)are topologically isomorphic.

Proof. (1)⇒(2)The proof of [14, Theorem 16] applies.

(2)⇒ (1) We claim that H(K) is the inductive limit of a sequence of Fréchet spaces Hb(V_n), where each V_n is balanced and Pb(E)-convex (and the same for H(L)). Indeed, let Un =K +B(0;_n¹), for every n ∈ N. Applying Theorem 3.7, for eachn∈N there exists a balanced _Hb-convex open subset Vn

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such thatK⊂Vn⊂Un. Since_H(K)=lim

−→n∈NHb(Un)and the inclusion_Hb(Un) → Hb(V_n) is continuous, we have that H(K)=lim

−→n∈NHb(V_n), and our claim is proved. Next we apply the same arguments of (2)⇒(1) of [14, Theorem 16], replacing [14, Corollary 14] there by Theorem 3.1 here. 2

CONCLUDING REMARKS

We go back to the introduction of this paper, where we say that we give a partial answer to Mujica’s question. He asks if, in Tsirelson-like spaces, _S_b(U ) can be identified withU, whenUis a polynomially convex open subset. On the one hand, we think that maybe this question could be reformulated toPb(E)-convex open subsets. Then, as showed in Lemma 1.2,Pb(E)-convex open sets are alwaysHb(E)- convex, and henceHb(U )-convex open subsets, so in this sense Theorem 2.1 gives a partial answer to Mujica’s question. On the other hand, it is clear thatPb(E)-convex open subsets are always polynomially convex, but we don’t know if the converse holds in general.

We still don’t know whether it is possible to remove the hypothesis that U is balanced on our results. It is known that if E is separable and has the bounded approximation property, then the spectrum of (H(U ), τ₀) is identified with U if and only ifU is a domain of holomorphy (see [10, Theorem 58.11]). We also ask whether it is possible to state an analogous result for the algebraHb(U ).

ACKNOWLEDGEMENTS

The author wishes to thank the referee for the careful examination of the paper and for his/her helpful comments.

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(Received March 2005)