B OntheBanach-Stonetheoremforalgebrasofholomorphicgerms

DOI 10.1007/s13398-016-0289-z O R I G I NA L PA P E R

On the Banach-Stone theorem for algebras of holomorphic germs

Domingo García¹ · Manuel Maestre¹ · Daniela M. Vieira²

Received: 19 November 2015 / Accepted: 10 February 2016 / Published online: 19 February 2016

© Springer-Verlag Italia 2016

Abstract Given two balanced compact subsetsK andL of two Banach spaces XandY respectively such that every continuousm-homogeneous polynomial on X^∗∗and onY^∗∗is approximable, for allm∈N, we characterize when the algebras of holomorphic germsH(K) andH(L)are topologically algebra isomorphic. This happens if and only if the polynomial hulls ofK andLon their respective biduals are biholomorphically equivalent.

Keywords Holomorphic germs·Holomorphic functions·Algebras·Banach spaces Mathematics Subject Classification 46G20·46E25

1 Introduction

In 1932, Banach [4] proved that two compact metric spaces K andLare homeomorphic if and only if the Banach spacesC(K)andC(L)are isometrically isomorphic. Stone [16]

Dedicated to Professor José Bonet on his 60th birthday.

The D. García and M. Maestre authors were supported by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013. The D. M. Vieira was supported by FAPESP Proc. 2014/07373-0 (Brazil).

B

1 Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain

2 Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, São Paulo CEP 05508-090, Brazil

generalized this result to arbitrary compact Hausdorff topological spaces, the well-known Banach-Stone theorem. This result found a large number of extensions, generalizations and variants in many different contexts, and some of them can be found in [12].

Recently, Carando and Muro [7] studied this theorem for the case of holomorphic functions of bounded typeHb(U), when everym-homogeneous polynomial on the bidual of one of the spaces involved is approximable (see below for the definition). More precisely, they proved that if the algebrasHb(U)andHb(V), forU andVbalanced open subsets of some Banach spacesXandY respectively, are algebra isomorphic then their dual spacesX^∗andY^∗are isomorphic, the converse being true when the domains are the whole spaces, generalizing a result of [6]. This is a Banach-Stone type theorem for holomorphic functions of bounded type.

In [19], Vieira proved another Banach-Stone theorem for the algebras of holomorphic functionsHb(U)andHb(V). This result holds wheneverU andV are balanced andm- polynomially convex subsets (see definition below) of reflexive Banach spacesX andY, respectively, for which all continuousm-polynomials are approximable (for allm).

Other algebras of holomorphic functions are studied by Vieira in [18,20].

Also, in [19] is obtained the same result for the algebras of holomorphic germsH(K) andH(L), where K andL are balanced andm-polynomially convex compact subsets of reflexive Banach spacesXandY, respectively, such that every continuousm-homogeneous polynomial on X and onY is approximable, for allm ∈ N. In this paper we give a full characterization of those algebra isomorphisms.

The result in [19] concerningH(K)andH(L)rest on the characterizations of homomor- phisms betweenHb(U)andHb(V)obtained in [19].

To obtain the result for nonreflexive Banach spaces we use new characterizations of homomorphisms betweenHb(U)andHb(V)obtained in [7].

We refer to [11] or [15] for background information on infinite dimensional complex analysis.

2 Preliminaires

Throughout this paperXandY will always denote complex Banach spaces. We denote by X^∗∗the bidual ofX, andJ_X :X−→ X^∗∗denotes the canonical inclusion. We also denote byB_X the open unit ball ofX and, for a givenx ∈ X,r > 0,B_X(x,r)denotes the open ball of radiusrcentered atx. LetP(X)denote the normed space of all continuous complex polynomials P : X −→ C, endowed with the normP =sup{|P(x)| : x ≤ 1}. For eachm∈N,P(^mX)denotes the subspace ofP(X)of allm-homogeneous polynomials, and Pf(^mX)denotes the subspace ofP(^mX)of allm-homogeneus polynomials of finite type, i. e.

finite linear combinations of products ofmcontinuous linear functionals onX. We recall that anm-homogeneous continuous polynomial on a Banach spaceXis calledapproximableif it is in the norm-closure ofPf(^mX).

LetU be an open subset of a Banach space X. We say that a set A⊂U isU-bounded if Ais bounded and there exists ε > 0 such that A+ B(0, ε) ⊂ U. Every open set U admits a fundamental sequence ofU-bounded sets,(A_j)j∈N, such that eachU-bounded set is contained in someA_j. For example, we can takeA_j= {x ∈X : x ≤ j,d(x,X\U)≥

1 j}.

We will denote byHb(U)the algebra of all holomorphic functions f :U −→Cwhich are bounded on everyU-bounded set. Such functions are called holomorphic functions of

bounded type. We denote byτb the topology onHb(U)of the uniform convergence on all U-bounded sets. Then(Hb(U), τb)is a Fréchet algebra.

LetA⊂Xbe a bounded set. We define A _P(X):=

z∈X^∗∗ : |A B(P)(z)| ≤sup

A

|P|, for allP∈P(X) ,

where A B(P)denotes de Aron-Berner extension of P, see [2]. We also denote A_P(X) = A _P(X)∩X.

A compact subsetK of a Banach spaceXis called polynomially convex ifK_P(X)= K. Changing_P(X)to_P(^mX)then it is saidm-polynomially convex.

IfUis an open subset ofX, let(An)n∈Nbe a fundamental sequence ofU-bounded sets.

Then we define

U_P(X) :=

n∈N

(A_n) _P(X).

In [7], it is shown thatU_{P (X)}is an open subset ofX^∗∗. Moreover, the following extension result is proved.

Theorem 1 [7, Proposition 3.3]Let U be a bounded and balanced open subset of a symmet- rically regular Banach space X . Then there exists an A B-extension operator from_Hb(U)to Hb(U_{P (X)}).

By anA B-extension operatorwe mean a continuous homomorphism such that, for every x ∈U, there existsr>0 such thatA B(f)coincides with the Aron-Berner extension of fon B_X∗∗(J_X(x),r). A Banach spaceXis said to be (symmetrically) regular if every continuous (symmetric) linear mappingT :X→X^∗is weakly compact. Recall thatTis symmetric if

<T x,y>=<x,T y>for allx,y∈X.

In the next proposition we present some properties of the setK_P(X) , whenKis a compact subset of a Banach spaceX.

Proposition 1 Let X be a Banach space, let K ⊂X be a compact set and let U_k=K+¹_kB_X, for all k∈N. Then

1. JX K_P(X)

=K_{P (X)}

2. K_{P (X)}is a compact subset ofUk

P(X)

3. K_{P (X)}= _k∈NU_k

P(X).

Proof 1. By the Hahn-Banach theorem, it follows thatK_{P (X)}⊂^w∗(K), wheredenotes the absolutely convex hull andw^∗is the weak-star topologyw(X^∗∗,X^∗). So ifz∈K_{P (X)}, then there exists a net(x_α)⊂(K)such thatx_α−→^w∗ z. But since(K)is norm-compact, there existsx∈Xand a subnet(x_β)such thatx_β −→^· x. Then it follows thatz=JX(x).

The other inclusion is obvious.

2. SinceK is compact, it follows by [15, Proposition 11.1] thatK_P(X)is compact. Then the assertion follows by the previous item.

3. Recall thatUk = K + ¹_kBX. SinceK_P(X) ⊂ (Uk) _P(X), for allk ∈N, it follows that K_{P (X)} ⊂ _k∈NUk

P(X). To show the opposite inclusion, let us fix P ∈ P(X)and denote byc = sup_K|P|and byck = sup_Uk|P|. Observe that(ck)k∈N is a bounded decreasing sequence. We will show that c = limk→∞ck. Givenε > 0, since P is uniformly continuous onK, there existsk ∈Nsuch that ifx−y< ¹_k then|P(x)−

P(y)|< ε. So, letx ∈K andx_k∈B_X(0,¹_k). Theny=x+x_k∈U_kandx−y< ¹_k hence|P(y)|< |P(x)| +ε. This shows thatc ≥inf_k∈Nc_k = lim_k→∞c_k. The other inequality is obvious. Now letz∈ _k∈NU_k

P(X). Then|A B(P)(z)| ≤c_k, for allk∈N, which implies that|A B(P)(z)| ≤c=sup_K|P|, and thenz∈K_{P (X)}.

We thank Daniel Carando for pointing out item(1)of the previous proposition.

LetXbe a Banach space, andK ⊂Xa compact set. We define the algebra h(K)=

{H(U) : U ⊃K is open inX}.

Let f₁, f₂∈h(K)andU₁,U₂be open subsets ofXwithK ⊂U₁andK ⊂U₂, such that f1 ∈H(U1)and f2∈H(U2). We say that f1and f2areequivalent(and we write f1∼ f2) if there is an open setW ⊆XwithK ⊂W ⊆U1∩U2such that f1= f2onW. Then∼is an equivalence relation inh(K)and we denoteH(K)=h(K)/∼. The elements ofH(K) are called holomorphic germs. Finally, we endowH(K)with the locally convex inductive topology of the locally convex algebras(H(U), τω), whereUvaries among the open subsets ofXsuch thatK ⊂U, and we denote

(H(K), τ_ω)=li m−→^U⊃K(H(U), τ_ω).

It has been proved [14, Theorem 7.1] that_H(K)is a locally m-convex topological algebra.

We recall that a topological algebra is called locally m-convex if there exists a basis for the neighborhoods of the origin consisting of convex sets withV² ⊂V.

LetU_n =K+_n¹B_X, for alln∈N. Then

(H(K), τω)=li m−→^n∈NHb(Un). (1) Actually we are going to use (1) as definition ofτω.

We will denote byinthe canonical inclusionin:Hb(Un) →H(K), for eachn∈N.[f] will denote the elements of the algebraH(K), i.e.,[f] ∈H(K)if and only if there exists n∈Nsuch that f ∈Hb(U_n).

We refer to [5,11] or [14] for background information on the algebras of holomorphic germs.

One of the keys of the main result of this paper is the following proposition.

Proposition 2 Let X be a Banach space such that every polynomial on X^∗∗is approximable, and let K ⊂ X be a balanced compact set. Then the algebras H(K)andH(K_{P (X)})are topologically algebra isomorphic.

Proof Given[f] ∈H(K)there existsk∈Nsuch that f ∈Hb(Uk). Since every polynomial onX^∗∗is approximable, thenXmust be symetrically regular [7]. Now considerA B(f) ∈ Hb

(U_k) _P(X)

, given by Theorem1. Then we defineT :H(K)−→H K_{P( X)}

byT([f])= [A B(f)]. It is not difficult to see thatTis well defined, linear and injective. To show thatTis surjective, let[g] ∈H K_{P (X)}

. Then there existsk∈Nsuch thatg∈Hb K_{P (X)}+_k¹B_X∗∗

. Letl ≥ k be such thatK_P(X) + ¹_lB_X∗∗ ⊂ (U_k) _P(X). Since every polynomial on X^∗∗ is approximable, we have thatg∈Hw^∗u K_{P (X)}+ ¹_lBX^∗∗

. Forz∈K_P(X) +¹_lBX^∗∗, we can apply Lemma 2.1 of [3] to the restriction ofgto a suitable ball, to obtain thatdⁿ(g)(z)is aw^∗- continuous polynomial, for everyn. By [21, Theorem 2] we can conclude thatg=A B(f), where f =g◦J_X|Ul.

It remains to show thatTis an algebra isomorphism. By the definition ofT, we have that the following diagram commutes, for allk ∈N

H(K) −−−−→^T H K_{P (X)}

ik⏐⏐ ⏐⏐^jk

Hb(Uk) −−−−→^{A B} HbUk

P(X)

whereikandjkdenote the canonical inclusions. Now, sinceA Bis continuous, it follows that T is continuous. To see thatT⁻¹is continuous, just observe thatT⁻¹([g])=[g◦J_X|Ul].

3 Banach-Stone theorems

LetXandY be Banach spaces,K ⊂ XandL⊂Y be compact sets. We say thatK andL are biholomorphically equivalent if there exist open setsU ⊂ X andV ⊂Y with K ⊂U andL⊂Vand a biholomorphic mappingϕ:U −→V such thatϕ(K)=L.

We will need the next theorem, due to Grothendieck (see [13]).

Theorem 2 [13]Let F be a Hausdorff locally convex space which is the union of an increas- ing sequence of Fréchet spaces(Fn)n∈Nand assume that each inclusion in : Fn −→ F is continuous. Let T : E −→ F be a continuous linear mapping of a Fréchet space E into F . Then there exists n ∈ Nand a continuous linear mapping Tn : E −→ Fn such that i_n◦T_n=T .

We will denote, as before,Un = K +_n¹BX, for alln ∈NandVm = L+_m¹BY for all m∈N. The mappingsinand jmwill be the canonical inclusionsin:Hb(Un) →H(K)and

j_m:Hb(V_m) →H(L), for eachn,m∈N.

Remark 3 Let X andY be Banach spaces. LetK ⊂ X andL ⊂ Y be compact sets. Let us observe that if given a continuous homomorphism A : H(K) −→ H(L)and an open setU with K ⊂ U such that we can find an open set V with L ⊂ V and a continuous homomorphismB:Hb(U)−→Hb(V)satisfying that the following diagram commutes

H(K) −−−−→^A H(L)

i⏐⏐ ⏐⏐^j

Hb(U)−−−−→^B Hb(V)

whereiand jare the corresponding canonical injections, then, given any other open setW such thatL⊂W⊂Vand definedB(f):=B(f)|W for f ∈Hb(U), whereB(f)|W is the restriction ofB(f)toW, the diagram

H(K) −−−−→^A H(L)

i⏐⏐ ⏐⏐^j

Hb(U) −−−−→^B Hb(W) commutes too.

Next we present the main theorem of the article, which is a Banach-Stone type theorem for algebras of holomorphic germs.

Theorem 4 Let X and Y be Banach spaces such that every continuous m-homogeneous polynomial on X^∗∗and on Y^∗∗is approximable, for all m∈N. Let K ⊂ X and L ⊂Y be balanced compact sets. Then the following conditions are equivalent.

1. H(K)andH(L)are topologically algebra isomorphic.

2. K_{P (X)}andL _P(Y)are biholomorphically equivalent.

3. H(K_{P (X)})andH(L _P(Y))are topologically algebra isomorphic.

Proof (1)⇒ (2) Let T : H(K) −→ H(L) be a topological algebra isomorphism. By Theorem2( see also [9, Theorem 3.1]), for eachk∈Nthere existm_k∈Nand a continuous homomorphismT_k :Hb(U_k) −→ Hb(V_mk)such thatT ◦i_k = j_mk ◦T_kand actually, by Remark3, the sequence(m_k)can be chosen to be strictly increasing. Then the following diagram is commutative.

H(K) −−−−→^T H(L)

ik⏐⏐ ⏐⏐^jmk

Hb(U_k) −−−−→^Tk Hb(V_mk)

SinceTkis multiplicative, it follows from [7, Lemma 4.6] that there is a holomorphic mapping

ϕk :

Vm_k

P(Y) −→Uk

P(X)such thatTk(f) = ¯f ◦ϕk, for all f ∈Hb(Uk), where the bar denotes the extension given by Theorem1. Applying the same argument forS=T⁻¹, beginning withHb(V_mk), we find an integern_k, that again by Remark3, it can be chosen strictly bigger thannk−1and a holomorphic mappingψk :Un_k

P(X)−→

Vm_k

P(Y)such that

H(L) −−−−→^S H(K)

j_mk⏐⏐ ⏐⏐^ink

Hb(Vmk) −−−−→^Sk Hb(Unk)

is commutative, whereSk : Hb(Vm_k) −→ Hb(Un_k)is such that Sk(g) = ¯g◦ψk, for all g∈Hb(Vm_k).

Since both diagrams are commutative, it follows thati_k =i_nk◦S_k◦T_k, for allk ∈N. Then for f ∈ X^∗⊆Hb(U_k)we have that[f] = [S_k(T_k(f))], and therefore by the Identity Principle we have the equality f =S_k(T_k(f))onU_nk. Hence f¯=S_k(T_k(f))onU_nk

P(X), that is,f¯= ¯f◦ϕk◦ψkonU_nk

P(X), for all f ∈X^∗. This shows that for eachz∈U_nk

P(X), we have thatz(f) = (ϕk ◦ψk)(z)(f), for all f ∈ X^∗, that is,ϕk◦ψk : Unk

P(X) −→

U_k

P(X)is the inclusion mapping.

Observe now thatT([f])= [T₁(f)] = [T_k(f)], for all f ∈X^∗. SinceV_mkis connected, we conclude thatT₁(f)=T_k(f)onV_mk, and then f ◦ϕ1=T₁(f)=T_k(f)= f ◦ϕk, for all f ∈ X^∗. Henceϕ1=ϕkon

V_mk

P(Y). By the same arguments we prove thatψ1 =ψk

on eachUn_k

P(X).

Next we are going to show thatϕ1(L _P(Y))⊂K_{P (X)}. By Proposition1(2), we have that L _P(Y)⊂

Vmk

P(Y)and henceϕ1(L _P(Y))⊂ϕ1( Vmk

P(Y))=ϕk( Vmk

P(Y))⊂Uk

P(X), for allk ∈N. Then it follows thatϕ1(L _P(Y))⊂ _k∈NU_k

P(X) = K_{P( X)}, where the last

equality follows by Proposition1(3). And by the same arguments we show thatψ1(K_{P( X)})⊂ L _P(Y).

If we setV = ϕn⁻₁¹

(U_n1) _P(X)

,U = ψ₁⁻¹ (V_mn

1) _P(Y)

,ϕ = ϕn1|V : V −→ U and ψ=ψ1|U :U −→V, then we have thatϕandψare bijective holomorphic functions such thatϕ⁻¹=ψ, andϕ(L _P(Y))=K_P(X) .

(2)⇒(3) Since the compact setsK_P(X) andL _P(Y)are biholomorphically equivalent, by the same arguments of [18, Theorem 16], we can show that the algebrasH K_P(X)

and H(L _P(Y))are topologically isomorphic.

(3)⇒(1) Follows by Proposition2.

By using the notion of polynomially convex we can state the following corollary.

Corollary 1 Let X and Y be reflexive Banach spaces such that every continuous m- homogeneous polynomial on these spaces is approximable, for all m∈N, and let K⊂ X and L ⊂Y be balanced and polynomially convex compact sets. Then the following conditions are equivalent.

1. The algebrasH(K)andH(L)are topologically isomorphic.

2. The compact sets K and L are biholomorphically equivalent.

Corollary1is a generalization of Theorem 3.8 of [19].

Finally we are going to give examples of Banach spaces satisfying conditions of Theorem 4.

Examples 5 1. In [17], Tsirelson constructed a reflexive Banach spaceX, with an uncon- ditional Schauder basis, that does not contain any subspace which is isomorphic toc₀or to anyp, 1≤ p≤ ∞. R. Alencar, R. Aron and S. Dineen proved in [1] that_Pf(^mX)is norm-dense inP(^mX), for allm∈N. This space is known as Tsirelson space.

2. In [8], Casazza et al. constructed a nonreflexive Banach spaceX that does not contain any subspace which is isomorphic toc0 or to anyp, (see also [11, Example 2.43]).

This space is known as Tsirelson-James space. In [10, Lemma 19], it is shown that every continuousm-homogeneous polynomial onX^∗∗is approximable.

The Tsirelson space, Examples5(1), is the main example of a space satisfying conditions of Corollary1. We see that Theorem4improves [19, Theorem 3.8] not only in the geometric aspect of the compact setK, but is valid for a greater class of Banach spaces, as Examples5 shows.

Acknowledgements Part of this paper was written during the visit of D. M. Vieira to the Universidad de Valencia, and she wants to thank the Departamento de Análisis Matemático of the Universidad de Valencia and its members for their kind hospitality.

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