Homomorphisms between Algebras of Holomorphic Functions

Abstract

For two complex Banach spaces 𝑋 and 𝑌, in this paper, we study the generalized spectrum ℳ_𝑏(𝑋,𝑌) of all nonzero algebra homomorphisms from ℋ_𝑏(𝑋), the algebra of all bounded type entire functions on 𝑋, into ℋ_𝑏(𝑌). We endow ℳ_𝑏(𝑋,𝑌) with a structure of Riemann domain over ℒ(𝑋^∗,𝑌^∗) whenever 𝑋 is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set ℳ_𝑏,∞(𝑋,𝐵_𝑌) of all nonzero algebra homomorphisms from ℋ_𝑏(𝑋) into ℋ_∞(𝐵_𝑌) of bounded holomorphic functions on the open unit ball of 𝑌 and ℳ_∞(𝐵_𝑋,𝐵_𝑌) of all nonzero algebra homomorphisms from ℋ_∞(𝐵_𝑋) into ℋ_∞(𝐵_𝑌).

1. Introduction

The study of homomorphisms between topological algebras is one of the basic issues in this theory. Two are the main topological algebras that we come across when we deal with holomorphic functions on infinite dimensional spaces (see Section 2 for precise definitions): ℋ_𝑏(𝑋), the holomorphic functions of bounded type (which is a Fréchet algebra), and ℋ_∞(𝐵_𝑋), the bounded holomorphic functions on the open unit ball (which is a Banach algebra). Here, as a first step in the study of the set of homomorphisms between ℋ_∞(𝐵_𝑋) spaces, we mainly focus on algebras of holomorphic functions of bounded type and homomorphisms between them; 𝐿∶ℋ_𝑏(𝑋)→ℋ_𝑏(𝑌) (i.e., continuous, linear, and multiplicative mappings). These were already considered in [1]. There the focus was to study the homomorphisms as “individuals,” seeking properties of single ones. We have here a different interest: we treat them as a whole, considering the set ℳ_𝑏(𝑋,𝑌)=ℳℋ_𝑏(𝑋),ℋ_𝑏(𝑌)=Φ∶ℋ_𝑏(𝑋)⟶ℋ_𝑏(𝑌)algebrahomomorphisms⧵{0}. (1) We will call this set the generalized spectrum or simply the spectrum. Our main aim is to study ℳ_𝑏(𝑋,𝑌) and to define on it a topological and a differential structure.

This problem has the same flavor as considering ℳ_𝑏(𝑋), the spectrum of the algebra ℋ_𝑏(𝑋) (i.e., the set of nonzero continuous, linear, and multiplicative Φ∶ℋ_𝑏(𝑋)→ℂ). This was studied in [2, 3], where a structure of Riemannian manifold over the bidual 𝑋^∗∗ was defined on it (see also [4, Section 3.6] for a very neat and nice presentation and [5–7] for similar results). Our approach is very much indebted to that in [2] and we get up to some point analogous results, defining on ℳ_𝑏(𝑋,𝑌) a Riemann structure over ℒ(𝑋^∗,𝑌^∗) (note that 𝑋^∗∗=ℒ(𝑋^∗,ℂ)). We will also be interested in the fibers of elements of ℒ(𝑋^∗,𝑌^∗).

The outline of the paper is the following. In Section 3, for two complex Banach spaces 𝑋 and 𝑌, we study the generalized spectrum ℳ_𝑏(𝑋,𝑌) of all nonzero algebra homomorphisms from ℋ_𝑏(𝑋) to ℋ_𝑏(𝑌). We endow it with a structure of Riemann domain over ℒ(𝑋^∗,𝑌^∗) whenever 𝑋 is symmetrically regular. In Section 4, we focus on the sets (fibers) of elements in ℳ_𝑏(𝑋,𝑌) that are projected on the same element 𝑢 of ℒ(𝑋^∗,𝑌^∗). The size of these fibers is studied and we prove that they are big by showing that they contain big sets. Following the philosophy of [2], all about ℳ_𝑏(𝑋,𝑌) is a step to study in Section 5 the spectrum ℳ_𝑏,∞(𝑋,𝐵_𝑌) of all nonzero algebra homomorphisms from ℋ_𝑏(𝑋) to ℋ_∞(𝐵_𝑌) of bounded holomorphic functions on the open unit ball of 𝑌. Finally, in Section 6, we deal with the generalized spectrum ℳ_∞(𝐵_𝑋,𝐵_𝑌) of all nonzero algebra homomorphisms from ℋ_∞(𝐵_𝑋) to ℋ_∞(𝐵_𝑌).

2. Definitions and Preliminaries

Unless otherwise stated capital letters such as 𝑋,𝑌,… will denote complex Banach spaces. The dual will be denoted by 𝑋^∗ and the open ball of center 𝑥₀ and radius 𝑟>0 by 𝐵_𝑋(𝑥₀,𝑟). If 𝑥₀=0 and 𝑟=1 we just write 𝐵_𝑋. The space of continuous, linear operators from 𝑋 to 𝑌 will be denoted by ℒ(𝑋,𝑌); this is a Banach space with the norm ‖𝑢‖=sup_{𝑥∈𝐵𝑋}‖𝑢(𝑥)‖. The adjoint operator of 𝑢∈ℒ(𝑋,𝑌) will be denoted by 𝑢^∗∈ℒ(𝑌^∗,𝑋^∗).

We will denote the canonical inclusion of a space into its bidual by 𝐽_𝑋∶𝑋↪𝑋^∗∗.

Given two complex locally convex spaces 𝐸 and 𝐹, a mapping 𝑃∶𝐸→𝐹 is a continuous 𝑚-homogeneous polynomial if there is a continuous 𝑚-linear mapping 𝐿∶𝐸×⋯×𝐸→𝐹 such that 𝑃(𝑥)=𝐿(𝑥,…,𝑥)forevery𝑥. (2) Throughout the paper every polynomial and multilinear mapping will be assumed to be continuous.

An 𝑚-linear mapping 𝐿 is called symmetric if 𝐿(𝑥_𝜎1,…,𝑥_𝜎𝑚)=𝐿(𝑥₁,…,𝑥_𝑚) for every permutation 𝜎 of {1,…,𝑚}. Each 𝑚-homogeneous polynomial has a unique symmetric mapping (which we denote by 𝑃^∨) satisfying (2). If 𝑃∶𝑋→𝑌 is a continuous 𝑚-homogeneous polynomial between Banach spaces, the following expressions define norms for 𝑚-linear mappings and for 𝑚-homogeneous polynomials, respectively: ‖𝐿‖=sup𝐿𝑥₁,…,𝑥_𝑚∶𝑥_𝑗,‖𝑃‖𝑃(𝑥∶‖𝑥‖}.≤1,𝑗=1,…,𝑚=sup{‖)‖≤1 (3) The polarization formula gives [4, Corollary 1.8] ‖𝑃‖≤𝑃^∨≤𝑚^𝑚‖𝑃‖.𝑚! (4) Given an 𝑚-linear mapping 𝐿, the notation 𝐿(𝑥^(𝑘),𝑦^{(𝑚−𝑘)}) will mean that 𝑥 is repeated 𝑘 times and 𝑦 is repeated 𝑚−𝑘 times.

A mapping 𝑓∶𝑈⊆𝐸→𝐹 is holomorphic on the open set 𝑈 if for every 𝑥₀ in 𝑈, there exist (𝑃_𝑚𝑓(𝑥₀))_𝑚, each one of them an 𝑚-homogeneous polynomial, such that the series 𝑓(𝑥)=^∞_𝑚=0𝑃_𝑚𝑓𝑥₀𝑥−𝑥₀ (5) converges uniformly in some neighborhood of 𝑥₀ contained in 𝑈. This is called the “Taylor series expansion” of 𝑓 at 𝑥₀.

If 𝐸 and 𝐹 are Fréchet spaces, then 𝑓∶𝐸→𝐹 is Gâteaux holomorphic if for every 𝑥,𝑦∈𝐸 the function ℂ→𝐹, 𝑡↦𝑓(𝑥+𝑡𝑦) is holomorphic on some neighborhood of 0. It is known [4, page 152-153] that 𝑓 is holomorphic if and only if it is Gâteaux holomorphic and continuous.

In the case that 𝑓∶𝑋→𝑌, 𝑋 and 𝑌 being complex Banach spaces, 𝑓 is holomorphic on 𝑋 if and only if it is Fréchet differentiable on 𝑋 and in that case 𝑃₁(𝑥)=𝑑𝑓(𝑥) for every 𝑥. Also, by the Cauchy inequalities [4, Proposition 3.2] we have, for every 𝑚, sup_{𝑥∈𝐵𝑋()0,𝑅}𝑃_𝑚𝑥₀(𝑥)≤sup_{𝑥∈𝐵𝑋𝑥0,𝑅}𝑓(𝑥). (6) A holomorphic function 𝑓∶𝐸→𝐹 is of bounded type if it sends bounded subsets of 𝐸 to bounded sets of 𝐹. We denote by ℋ_𝑏(𝐸,𝐹) the space of holomorphic functions of bounded type from 𝐸 to 𝐹. If 𝐹=ℂ we simply write ℋ_𝑏(𝐸). Given 𝑈⊆𝐸 we write 𝑓_𝑈=sup_𝑥∈𝑈𝑓(𝑥). (7) With this notation for 𝑋 a Banach space ℋ_𝑏(𝑋) is a Fréchet algebra endowed with the topology 𝜏_𝑏 of uniform convergence on the bounded sets, whose seminorms are (for 𝑅>0) 𝑞_𝑅𝑓=𝑓_{𝐵𝑋(0,𝑅)}=sup𝑓(𝑥)∶‖𝑥‖<𝑅. (8) Let 𝑀 be a differential manifold on a complex Banach space 𝑋 and 𝐹 a locally convex space. A mapping 𝑓∶𝑀→𝐹 is said to be holomorphic (of bounded type) if 𝑓∘𝜑⁻¹∶Ω→𝐹 is holomorphic (of bounded type) for every chart (𝜑,Ω) of 𝑀.

Given 𝑥∈𝑋, we write 𝛿_𝑥 for the evaluation mapping at 𝑥; that is, 𝛿_𝑥(𝑓)=𝑓(𝑥) for all holomorphic 𝑓.

Let 𝐿∶𝑋×𝑋→ℂ be a continuous bilinear form. Fix 𝑥∈𝑋 and for 𝑤∈𝑋^∗∗ let (𝑦_𝛽) be a net in 𝑋 weak-star convergent to 𝑤. Since 𝐿(𝑥,−)∈𝑋^∗, then there exists lim_𝛽𝐿(𝑥,𝑦_𝛽)∶=𝐿(𝑥,𝑤). Now, fix 𝑤∈𝑋^∗∗ and for 𝑧∈𝑋^∗∗ let (𝑥_𝛼) be a net in 𝑋 weak-star convergent to 𝑧. Since 𝐿(−,𝑤)∈𝑋^∗, then there exists 𝐿(𝑧,𝑤)∶=lim_𝛼𝐿𝑥_𝛼,𝑤=lim_𝛼lim_𝛽𝐿𝑥_𝛼,𝑦_𝛽. (9) Following this idea Aron and Berner showed in [8] that every function 𝑓∈ℋ_𝑏(𝑋) admits an extension to the bidual, called the Aron-Berner extension, 𝑓∈ℋ_𝑏(𝑋^∗∗). By [9, Theorem 3], for every 𝑚-homogeneous polynomial 𝑃, we have ‖𝑃‖=‖𝑃‖.

A Banach space 𝑋 is symmetrically regular if for all continuous symmetric bilinear form 𝐿∶𝑋×𝑋→ℂ it follows that 𝐿(𝑧,𝑤)=lim_𝛽lim_𝛼𝐿𝑥_𝛼,𝑦_𝛽. (10)

We refer the reader to [4, 10, 11] for the general background on the theory of holomorphic functions on infinite dimensional spaces.

We are going to work with the set ℳ_𝑏(𝑋,𝑌) of nonzero algebra homomorphisms between spaces of holomorphic functions of bounded type defined in (1). Observe that an idempotent element 𝑔 in ℋ_𝑏(𝑌) satisfies that 𝑔(𝑌) is a subset of {0,1} and 𝑔(𝑌) is a connected set. Hence, either 𝑔≡0 or 𝑔≡1. So, for any Φ∈ℳ_𝑏(𝑋,𝑌), we should have Φ(𝟏_𝑋)=𝟏_𝑌.

3. The Differential Structure of ℳ_𝑏(𝑋,𝑌)

Our aim in this section is to endow ℳ_𝑏(𝑋,𝑌) with a structure of Riemann domain over ℒ(𝑋^∗,𝑌^∗). On a first step, for each linear operator 𝑢∈ℒ(𝑋^∗,𝑌^∗), we define, inspired by [12, Lemma 1], 𝑢∶ℋ_𝑏(𝑋)⟶ℋ_𝑏(𝑌),𝑓⟼𝑓∘𝑢^∗∘𝐽_𝑌. (11) It is plain that 𝑢 is an algebra homomorphism and 𝑢|_𝑋^∗=𝑢. This defines a natural inclusion: 𝑖∶ℒ𝑋^∗,𝑌^∗⟶ℳ_𝑏(𝑋,𝑌),𝑢⟼𝑢. (12) On the other hand there is also a projection: 𝜋∶ℳ_𝑏(𝑋,𝑌)⟶ℒ𝑋^∗,𝑌^∗,Φ⟼𝑥^∗⟼𝑑Φ𝑥^∗(0). (13) These clearly satisfy 𝜋(𝑖(𝑢))=𝑢 for every 𝑢.

Given 𝑧∈𝑋^∗∗ we consider the mapping 𝜏^∗_𝑧∶ℋ_𝑏(𝑋)→ℋ_𝑏(𝑋) defined by 𝜏^∗_𝑧(𝑓)(𝑥)=𝑓(𝐽_𝑋𝑥+𝑧). It is a well-known fact that fixed 𝑓 in ℋ_𝑏(𝑋) the mapping 𝑇∶𝑋^∗∗→ℋ_𝑏(𝑋) given by 𝑇(𝑧)=𝜏^∗_𝑧(𝑓) is a holomorphic function of bounded type. See [4, Proposition 6.30] or more in general the proof of [3, Theorem 2.2].

With this, for each Φ∈ℳ_𝑏(𝑋,𝑌) and each 𝑢∈ℒ(𝑋^∗,𝑌^∗), we can define Φ^𝑢∈ℳ_𝑏(𝑋,𝑌) by Φ^𝑢𝑓𝑦=Φ𝜏^∗_{𝑢∗𝐽𝑌𝑦}𝑓𝑦=Φ𝑥⟼𝑓𝐽_𝑋𝑥+𝑢^∗𝐽_𝑌𝑦𝑦, (14) for all 𝑓∈ℋ_𝑏(𝑋) and 𝑦∈𝑌. Let us see that Φ^𝑢 is well defined. To do that we need to check that the function 𝑦⟼Φ𝜏^∗_{𝑢∗𝐽𝑌𝑦}𝑓𝑦 (15) belongs to ℋ_𝑏(𝑌). We see first that it is holomorphic. The following function of two variables 𝜏𝑌×𝑌⟶ℂ,𝑦,𝑧⟼Φ^∗_{𝑢∗𝐽𝑌𝑧}𝑓𝑦, (16) is holomorphic on each variable. Then Hartogs’ theorem gives that it is holomorphic and hence it is so when restricted to the diagonal {(𝑦,𝑦)∶𝑦∈𝑌}. This gives that (15) is holomorphic.

We see now that Φ^𝑢(𝑓) is of bounded type. Given 𝑅>0 there exists 𝑆=𝑆(𝑅)>0 such that Φ(𝑔)_{𝐵𝑌(0,𝑅)}≤𝑔_{𝐵𝑋(0,𝑆)}, (17) for all 𝑔∈ℋ_𝑏(𝑋). Hence sup_𝑦<𝑅Φ𝜏^∗_{𝑢∗𝐽𝑌𝑦}𝑓𝑦≤sup_{‖𝑧‖<𝑅}Φ𝜏^∗_{𝑢∗𝐽𝑌𝑧}𝑓_{𝐵𝑌(0,𝑅)}≤sup_{‖𝑧‖<𝑅}𝜏^∗_{𝑢∗𝐽𝑌𝑧}𝑓_{𝐵𝑋(0,𝑆)}≤𝑓_{𝐵𝑋(0,𝑆+‖𝑢‖𝑅)}<∞, (18) since ‖𝐽_𝑋𝑥+𝑢^∗𝐽_𝑌𝑧‖≤‖𝑥‖+‖𝑢^∗‖‖𝑧‖<𝑆+‖𝑢‖𝑅 for all 𝑥∈𝐵_𝑋(0,𝑆) and all 𝑧∈𝐵_𝑌(0,𝑅).

On the other hand it is easy to see that Φ^𝑢 is an algebra homomorphism, which finally gives Φ^𝑢∈ℳ_𝑏(𝑋,𝑌).

As a second step we show that 𝜋Φ^𝑢=𝜋(Φ)+𝑢. (19) Indeed, for each 𝑥^∗∈𝑋^∗ and 𝑦∈𝑌, we have Φ^𝑢𝑥^∗𝑦=Φ𝑥⟼𝑥^∗𝐽_𝑋𝑥+𝑢^∗𝐽_𝑌𝑦𝑦=Φ𝑥⟼𝑥^∗(𝑥)+𝑢𝑥^∗𝑦𝑦=Φ𝑥^∗+𝑢𝑥^∗𝑦. (20) Thus, 𝜋Φ^𝑢𝑥^∗=𝑑Φ^𝑢𝑥^∗0)Φ𝑥(=𝑑^∗+𝑢𝑥^∗(0)=𝑑Φ𝑥^∗(0)+𝑢𝑥^∗=(𝜋(Φ)+𝑢)𝑥^∗. (21) Finally, given Φ∈ℳ_𝑏(𝑋,𝑌) and 𝜀>0, we consider the set 𝑉_Φ,𝜀=Φ^𝑢∶𝑢∈ℒ𝑋^∗,𝑌^∗,‖𝑢‖<𝜀. (22) Now we assume that 𝑋 is symmetrically regular. We proceed as in the definition of the Riemann domain structure of the spectrum of ℋ_𝑏(𝑋) and we omit the details (see [4, Section 6.3] for a complete and detailed explanation of the procedure). First of all, if Ψ∈𝑉_Φ,𝜀, then Ψ=Φ^𝑢, with ‖𝑢‖<𝜀. Hence, for any 𝑣∈ℒ(𝑋^∗,𝑌^∗), Ψ^𝑣𝑓𝑦=Φ^𝑢𝑣𝑓𝑦=Φ^𝑢𝜏^∗_{𝑣∗𝐽𝑌𝑦}𝑓𝑦=Φ𝜏^∗_{𝑢∗𝐽𝑌𝑦}∘𝜏^∗_{𝑣∗𝐽𝑌𝑦}𝑓𝑦=Φ𝜏^∗_{(𝑢∗+𝑣∗)𝐽𝑌𝑦}𝑓𝑦=Φ^𝑢+𝑣𝑓𝑦. (23) Therefore, for 𝛿=𝜀−‖𝑢‖, we have 𝑉_Ψ,𝛿⊂𝑉_Φ,𝜀 and {𝑉_Φ,𝜀}_𝜀>0 is a neighborhood basis at Φ.

Also, for Φ≠Ψ∈ℳ_𝑏(𝑋,𝑌), we have that if 𝜋(Φ)=𝜋(Ψ), then 𝑉_Φ,𝜀∩𝑉_Ψ,𝛿=∅, for all 𝜀,𝛿>0 and if 𝜋(Φ)≠𝜋(Ψ), then 𝑉_Φ,𝜀∩𝑉_Ψ,𝜀=∅ for 𝜀=‖𝜋(Φ)−𝜋(Ψ)‖/2. This gives that the topology generated by {𝑉_Φ,𝜀}_𝜀>0 is Hausdorff.

Let us note that for each Φ in ℳ_𝑏(𝑋,𝑌) the subset 𝑉_Φ={Φ^𝑢∶𝑢∈ℒ(𝑋^∗,𝑌^∗)} is the connected component containing Φ.

Summing all this up we have proved the following result.

Proposition 1. If 𝑋 is a symmetrically regular Banach space and 𝑌 is any Banach space, (ℳ_𝑏(𝑋,𝑌),𝜋) is a Riemann domain over ℒ(𝑋^∗,𝑌^∗) and each connected component of (ℳ_𝑏(𝑋,𝑌),𝜋) is homeomorphic to ℒ(𝑋^∗,𝑌^∗).

Our aim now is to show that each function 𝑓∈ℋ_𝑏(𝑋) can be extended, in some sense, to a function on ℳ_𝑏(𝑋,𝑌) of bounded type. We do it with the following sort of Gelfand transform: 𝑓∶ℳ_𝑏(𝑋,𝑌)⟶ℋ_𝑏(𝑌),Φ⟼Φ𝑓, (24) and showing that this, when restricted to each connected component, is holomorphic of bounded type. To do that we need the following lemma.

Lemma 2. If 𝑋 and 𝑌 are complex Banach spaces and 𝐺 is an element of ℋ_𝑏(𝑋×𝑌), then the mapping 𝑔 defined by 𝑔(𝑥)(𝑦)=𝐺(𝑥,𝑦) for (𝑥,𝑦)∈𝑋×𝑌 belongs to ℋ_𝑏(𝑋,ℋ_𝑏(𝑌)). Conversely, given 𝑔 in ℋ_𝑏(𝑋,ℋ_𝑏(𝑌)) the mapping 𝐺(𝑥,𝑦)=𝑔(𝑥)(𝑦) belongs to ℋ_𝑏(𝑋×𝑌).

Proof. Let 𝐺 be in ℋ_𝑏(𝑋×𝑌), and let ∑^∞_𝑚=0𝑃_𝑚𝑓 be the Taylor series expansion of 𝐺 at (0,0). We have 𝐺𝑥,𝑦=^∞_𝑚=0𝑃_𝑚𝑓=𝑥,𝑦_{∞𝑚𝑚=0𝑘=0}𝑚𝑘^∨𝑃_𝑚𝑓(𝑥,0)^(𝑘),0,𝑦^{(𝑚−𝑘)}. (25) Note that if we take 𝑅,𝑆>0 by the polarization formula (4) and Cauchy inequalities (6), we have ^∞𝑚_𝑚=0𝑘=0𝑚𝑘sup_{‖𝑥‖≤𝑅}sup𝑃_𝑦≤𝑆^∨_𝑚𝑓(𝑥,0)^(𝑘),0,𝑦^{(𝑚−𝑘)}≤^∞𝑚_𝑚=0𝑘=0𝑚𝑘sup_{‖𝑥‖≤𝑅}sup_𝑦≤𝑆𝑚^𝑚𝑃𝑚!^𝑚𝑓‖𝑥‖^𝑘𝑦^𝑚−𝑘≤^∞_𝑚=0(𝑒(𝑅+𝑆))^𝑚×sup_{‖𝑥‖()≤2𝑒𝑅+𝑆}sup_{𝑦()≤2𝑒𝑅+𝑆}1(2𝑒(𝑅+𝑆))^𝑚𝐺𝑥,𝑦=2sup_{‖𝑥‖≤2𝑒(𝑅+𝑆)}sup_{𝑦≤2𝑒(𝑅+𝑆)}𝐺𝑥,𝑦<∞. (26) By the properties of convergent double series of nonnegative numbers we obtain that, for each fixed 𝑘, the series 𝑄_𝑘(𝑥)(𝑦)∶=∑^∞_𝑚=𝑘(^𝑚_𝑘)𝑃^∨_𝑚𝑓((𝑥,0)^(𝑘),(0,𝑦)^{(𝑚−𝑘)}) converges absolutely and uniformly on any product of balls in 𝑋 and 𝑌 with finite radii. Hence 𝑄_𝑘∶𝑋→ℋ_𝑏(𝑌) is a continuous 𝑘-homogeneous polynomial and actually 𝑔=∑^∞_𝑘=0𝑄_𝑘 is an entire function from 𝑋 to ℋ_𝑏(𝑌) that, by above inequalities, is of bounded type.
Conversely, consider 𝑔 in ℋ_𝑏(𝑋,ℋ_𝑏(𝑌)) and define 𝐺∶𝑋×𝑌→ℂ by 𝐺(𝑥,𝑦)=𝑔(𝑥)(𝑦). By definition, for each 𝑥∈𝑋, 𝐺(𝑥,−) belongs to ℋ_𝑏(𝑌). If we fix now 𝑦∈𝑌, we have that 𝛿_𝑦 is a continuous linear form on ℋ_𝑏(𝑌) and 𝐺(𝑥,𝑦)=𝛿_𝑦(𝑔(𝑥)), implying that 𝐺(−,𝑦) is the composition of holomorphic mappings. Thus 𝐺(−,𝑦) is holomorphic for every 𝑦∈𝑌. By Hartogs’ theorem, 𝐺∈ℋ(𝑋×𝑌). Finally, for fixed 𝑅,𝑆>0 we have that 𝑔(𝐵_𝑋(0,𝑅)) is a bounded subset of ℋ_𝑏(𝑌). Hence sup_{𝑦‖𝑥‖≤𝑅,≤𝑆}𝐺𝑥,𝑦=sup_𝑦≤𝑆sup_{‖𝑥‖≤𝑅}𝑔(𝑥)𝑦<∞, (27) and 𝐺 is bounded on the bounded subsets of 𝑋×𝑌.

Proposition 3. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓∈ℋ_𝑏(𝑋) consider its extension 𝑓∶ℳ_𝑏(𝑋,𝑌)→ℋ_𝑏(𝑌) defined in (24). We have that 𝑓 is a holomorphic function of bounded type. That is, 𝑓∘(𝜋∣_𝑉Φ)⁻¹∈ℋ_𝑏(ℒ(𝑋^∗,𝑌^∗),ℋ_𝑏(𝑌)) for every Φ.

Proof. The point is to prove that the function ℒ𝑋^∗,𝑌^∗⟶ℋ_𝑏(𝑌),𝑢⟼Φ^𝑢𝑓 (28) is holomorphic of bounded type. For that, we introduce an auxiliary mapping 𝑀∶ℒ(𝑋^∗,𝑌^∗)×𝑌^∗∗×𝑌→ℂ, defined by 𝑀𝑢,𝑧,𝑦=Φ𝑥⟼𝑓𝑥+𝑢^∗(𝑧)𝑦. (29) As above, we only need to check that it is separately holomorphic to conclude that 𝑀 is holomorphic. First we fix 𝑢∈ℒ(𝑋^∗,𝑌^∗) and 𝑧∈𝑌^∗∗ and denote 𝑀_𝑢,𝑧(𝑦)∶=𝑀(𝑢,𝑧,𝑦). We have 𝑀_𝑢,𝑧=Φ(𝜏^∗_{𝑢∗(𝑧)}(𝑓)) and this belongs to ℋ_𝑏(𝑌). Now we fix 𝑧,𝑦 and take 𝑀_𝑧,𝑦(𝑢)∶=𝑀(𝑢,𝑧,𝑦)=𝛿_𝑦(Φ(𝜏^∗_{𝑢∗(𝑧)}(𝑓))). This mapping is holomorphic (of bounded type) since it is the composition of the linear mapping ℒ(𝑋^∗,𝑌^∗)→𝑋^∗∗ defined by 𝑢↦𝑢^∗(𝑧) with the holomorphic mapping of bounded type: 𝑋^∗∗𝛿⟶ℂ,𝑣⟼_𝑦∘Φ𝜏^∗_𝑣𝑓. (30) Finally we fix 𝑢,𝑦 and denote 𝑀_𝑢,𝑦(𝑧)∶=𝑀(𝑢,𝑧,𝑦)=𝛿_𝑦(Φ(𝜏^∗_{𝑢∗(𝑧)}(𝑓))). Again, this is the composition of a linear mapping 𝑌^∗∗→𝑋^∗∗ defined by 𝑧→𝑢^∗(𝑧) with the same holomorphic mapping (30). We conclude that 𝑀 is holomorphic.
Let now 𝑅,𝑆,𝑇>0. Given 𝑇>0 there exists 𝑈>0 such that ‖Φ(ℎ)‖_{𝐵𝑌(0,𝑇)}≤‖ℎ‖_{𝐵𝑋(0,𝑈)} for every ℎ∈ℋ_𝑏(𝑋). Hence sup_{‖𝑢‖‖𝑧‖𝑦≤𝑅,≤𝑆,≤𝑇}𝑀𝑢,𝑧,𝑦=sup_{‖𝑢‖≤𝑅}sup_{‖𝑧‖≤𝑆}Φ𝜏^∗_{𝑢∗(𝑧)}𝑓_{𝐵𝑌(0,𝑇)}≤sup_{‖𝑢‖≤𝑅}sup_{‖𝑧‖≤𝑆}𝜏^∗_{𝑢∗(𝑧)}𝑓_{𝐵𝑋(0,𝑈)}≤𝑓_{𝐵𝑋(0,𝑈+𝑅𝑆)}<∞, (31) and 𝑀 is of bounded type. Since the mapping 𝑌→𝑌^∗∗×𝑌 defined by 𝑦↦(𝐽_𝑌𝑦,𝑦) is obviously holomorphic of bounded type we have that 𝐺∶ℒ𝑋^∗,𝑌^∗×𝑌⟶ℂ (32) defined by 𝐺(𝑢,𝑦)=𝑀(𝑢,𝑦,𝑦) is holomorphic of bounded type. Now a direct application of Lemma 2 gives that the mapping 𝑢↦Φ^𝑢(𝑓) belongs to ℋ_𝑏(ℒ(𝑋^∗,𝑌^∗),ℋ_𝑏(𝑌)).

The above proposition is related to the study of extension of functions of bounded type given in [13].

4. The Size of the Fibers of ℳ_𝑏(𝑋,𝑌)

We focus now on the sets of elements in ℳ_𝑏(𝑋,𝑌) that are projected on the same element 𝑢 of ℒ(𝑋^∗,𝑌^∗). This is called the “fiber” of 𝑢 and is defined by ℱ(𝑢)=Φ∈ℳ_𝑏(𝑋,𝑌)∶𝜋(Φ)=𝑢. (33) Our aim in this section is to find out how big these fibers can be. To begin with, each fixed 𝑢∈ℒ(𝑋^∗,𝑌^∗) defines a set 𝐴_𝑢=𝑔∈ℋ_𝑏𝑌,𝑋^∗∗∶𝑑𝑔(0)=𝑢^∗∘𝐽_𝑌. (34) On the other hand every 𝑔∈ℋ_𝑏(𝑌,𝑋^∗∗) defines a composition homomorphism Φ_𝑔∈ℳ_𝑏(𝑋,𝑌) given by Φ_𝑔(𝑓)=𝑓∘𝑔. This gives an inclusion 𝑗∶ℋ_𝑏𝑌,𝑋^∗∗⟶ℳ_𝑏(𝑋,𝑌),𝑔⟼Φ_𝑔, (35) which maps the set 𝐴_𝑢 into the fiber ℱ(𝑢). Let us check that 𝑗 is injective. Given 𝑔₁,𝑔₂∈ℋ_𝑏(𝑌,𝑋^∗∗)⧵{0}, 𝑔₁≠𝑔₂, there exists 𝑦₀∈𝑌 such that 𝑔₁(𝑦₀)≠𝑔₂(𝑦₀) and since 𝑋^∗ separates points of 𝑋^∗∗ we can find 𝑥^∗₀∈𝑋^∗ with 𝑔₁(𝑦₀)(𝑥^∗₀)≠𝑔₂(𝑦₀)(𝑥^∗₀). Thus Φ_𝑔1(𝑥^∗₀)(𝑦₀)≠Φ_𝑔2(𝑥^∗₀)(𝑦₀) and this gives Φ_𝑔1≠Φ_𝑔2.

There is also a projection 𝜁∶ℳ_𝑏(𝑋,𝑌)⟶ℋ_𝑏𝑌,𝑋^∗∗,Φ⟼𝑦⟼𝑥^∗⟼Φ𝑥^∗𝑦. (36) The mapping 𝜁(Φ) belongs to ℋ_𝑏(𝑌,𝑋^∗∗) (and hence 𝜁 is well defined). This follows from the fact that 𝑌×𝑋^∗⟶ℂ,𝑦,𝑥^∗⟼Φ𝑥^∗𝑦 (37) is holomorphic. Clearly, (𝜁∘𝑗)(𝑔)=𝑔. Also, note that 𝜁(Φ) determines the values that takes Φ when restricted to 𝑋^∗. This means that when finite type polynomials are dense in ℋ_𝑏(𝑋), 𝜁(Φ) determines Φ. So, the only homomorphisms in ℳ_𝑏(𝑋,𝑌) are the Φ_𝑔’s and we have the following result, which is closely related to [13, Lemmas 4.5 and 4.6] and in [1, Theorem 21].

Proposition 4. Let 𝑋 and 𝑌 be Banach spaces. If finite type polynomials are dense in ℋ_𝑏(𝑋) then for each Φ∈ℳ_𝑏(𝑋,𝑌) there exists 𝑔∈ℋ_𝑏(𝑌,𝑋^∗∗) such that Φ=Φ_𝑔. Also, ℱ(𝑢)={Φ_𝑔∶𝑔∈𝐴_𝑢}.

Let us note that the mapping 𝑗|_𝐴𝑢∶𝐴_𝑢→ℱ(𝑢) is actually injective and, by Proposition 4, if finite type polynomials are dense, surjective. This means that even in the case when finite type polynomials are dense in ℋ_𝑏(𝑋) (i.e., the space ℋ_𝑏(𝑋) is rather small), the fibers are thick. Let us see now that if this is not the case (i.e., when there is a polynomial in 𝑋 that is not weakly continuous on bounded sets), this mapping is no longer surjective and we can find even more homomorphisms inside each fiber.

Proposition 5. If 𝑋 is symmetrically regular and there exists a polynomial on 𝑋 not weakly continuous on bounded sets at a point 𝑥₀, then, for each 𝑢∈ℒ(𝑋^∗,𝑌^∗), there is Φ∈ℱ(𝑢) such that Φ≠Φ_𝑔, for all 𝑔∈𝐴_𝑢.

Proof. It is enough to prove the result for 𝐹(0) because we can change fibers through the mapping 𝐹(0)⟶𝐹(𝑢),Φ⟼Φ^𝑢, (38) Also, if Φ_𝑔∈𝐹(0), then (Φ_𝑔)^𝑢=Φ_ℎ, with ℎ(𝑦)=𝑔(𝑦)+𝑢^∗𝐽_𝑌𝑦.
Let 𝑃 be a polynomial that is not weakly continuous on bounded sets at 𝑥₀ and {𝑥_𝛼} a bounded net weakly convergent to 𝑥₀ such that |𝑃(𝑥_𝛼)−𝑃(𝑥₀)|>1, for all 𝛼. For an ultrafilter 𝒰 containing the sets {𝛼∶𝛼≥𝛼₀}, let Φ be given by Φ𝑓𝑦=lim_𝒰𝑓𝑥_𝛼,∀𝑓∈ℋ_𝑏(𝑋),𝑦∈𝑌. (39) Then Φ is a homomorphism in ℳ_𝑏(𝑋,𝑌) (actually in 𝐹(0)), that is, not of composition type. Indeed, since Φ(𝑓) is a constant function on 𝑌 it follows that 𝑑(Φ(𝑓))(0)=0 for every 𝑓 in ℋ_𝑏(𝑋) and so Φ∈𝐹(0). If Φ=Φ_𝑔 for certain 𝑔∈ℋ_𝑏(𝑌,𝑋^∗∗), then we have 𝑥^∗𝑥₀=lim_𝒰𝑥^∗𝑥_𝛼=𝑥^∗𝑔𝑦,∀𝑥^∗∈𝑋^∗,𝑦∈𝑌. (40) This says that 𝑔(𝑦)=𝐽_𝑋𝑥₀, for all 𝑦. Hence, lim_𝒰𝑃𝑥_𝛼=𝑃𝑔𝑦=𝑃𝐽_𝑋𝑥₀=𝑃𝑥₀, (41) which is a contradiction.

Something more can be said when there is a polynomial that is not weakly continuous on bounded sets. In this case, we can insert in each fiber a big set of homomorphisms that are not of composition type. We do it in detail in the fiber of 0.

First, note that if there is a polynomial not weakly continuous on bounded sets, then there is a homogeneous polynomial 𝑃 which is not weakly continuous on bounded sets at 0. So, the Aron-Berner extension of 𝑃 is not 𝑤^∗-continuous at any point 𝑥^∗∗∈𝑋^∗∗ [14, Corollary 2, Proposition 3] and [15, Proposition 1] (see also [16, Proof of Proposition 2.6]). Thus, we fix, for each 𝑥^∗∗∈𝑋^∗∗ a bounded net {𝑥_𝛼^∗∗} in 𝑋^∗∗ which 𝑤^∗-converges to 𝑥^∗∗, but |𝑃(𝑥_𝛼^∗∗)−𝑃(𝑥^∗∗)|>1, for all 𝛼. For each 𝑥^∗∗∈𝑋^∗∗ we fix also an ultrafilter 𝒰 containing the sets {𝛼∶𝛼≥𝛼₀}. Consider the set 𝐴=𝑔∈ℋ_𝑏𝑌,𝑋^∗∗∶𝑔(0)=0,𝑑𝑔(0)≡0 (42) and define the following mapping 𝐴×𝑋^∗∗⟶𝐹(0),𝑔,𝑥^∗∗⟼Ψ_{𝑔,𝑥∗∗}, (43) where Ψ_{𝑔,𝑥∗∗}𝑓𝑦=lim_𝒰𝑓𝑥_𝛼^∗∗+𝑔𝑦. (44) This mapping is well defined because Ψ_{𝑔,𝑥∗∗}(𝑥^∗)(𝑦)=𝑥^∗(𝑥_𝛼^∗∗+𝑔(𝑦))=𝑥^∗∗(𝑥^∗)+𝑥^∗(𝑔(𝑦)); then 𝜋(Ψ_{𝑔,𝑥∗∗})(𝑥^∗)=𝑑(𝑥^∗∘𝑔)(0)=𝑥^∗∘𝑑𝑔(0)≡0. The mapping is also injective. Indeed, if Ψ_{𝑔,𝑥∗∗}=Ψ_{ℎ,𝑧∗∗}, then Ψ_{𝑔,𝑥∗∗}(𝑥^∗)(𝑦)=Ψ_{ℎ,𝑧∗∗}(𝑥^∗)(𝑦), for all 𝑥^∗∈𝑋^∗ and 𝑦∈𝑌. Then, 𝑥^∗∗𝑥^∗+𝑥^∗𝑔𝑦=𝑧^∗∗𝑥^∗+𝑥^∗ℎ𝑦,∀𝑥^∗∈𝑋^∗,𝑦∈𝑌. (45) Therefore, ℎ(𝑦)=𝑔(𝑦)+𝑥^∗∗−𝑧^∗∗, for all 𝑦 and, evaluating at 0, we obtain that 𝑥^∗∗=𝑧^∗∗ and, hence, 𝑔=ℎ.

Note, also, that the homomorphisms Ψ_{𝑔,𝑥∗∗} are not of composition type. Indeed, if Ψ_{𝑔,𝑥∗∗}=Φ_ℎ, for certain 𝑔∈𝐴, 𝑥^∗∗∈𝑋^∗∗, and ℎ∈ℋ_𝑏(𝑌,𝑋^∗∗), then, for all 𝑥^∗∈𝑋^∗, 𝑥^∗∗𝑥^∗+𝑥^∗∘𝑔=Ψ_{𝑔,𝑥∗∗}𝑥^∗=Φ_ℎ𝑥^∗=𝑥^∗∘ℎ. (46) If this were the case we would have ℎ(𝑦)=𝑥^∗∗+𝑔(𝑦), for all 𝑦. Now, lim_𝒰𝑃𝑥_𝛼^∗∗+𝑔𝑦=Ψ_{𝑔,𝑥∗∗}(𝑃)𝑦=Φ_ℎ(𝑃)𝑦=𝑃𝑥^∗∗+𝑔𝑦. (47) Evaluating at 0 leads to a contradiction.

Let us finish this analysis by studying Λ: the composition of the inclusion 𝑗 defined in (35) with the inclusion of ℳ_𝑏(𝑋,𝑌) into ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)). Then the mapping Λ turns out to be holomorphic (endowing ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) with an appropriate topology). We prepare the proof of this fact with a lemma that is a variant of a classical Dunford result; see also [17, Theorem 3].

Lemma 6. let Ψ∶𝐸→𝑋 be a Gâteaux holomorphic mapping from a complex Fréchet space 𝐸 to a Banach space 𝑋 such that 𝑥^∗∘Ψ is holomorphic for every 𝑥^∗∈𝐴, where 𝐴 is a norming subset of the closed unit ball of 𝑋^∗ (i.e., ‖𝑥‖=sup_{𝑥∗∈𝐴}{|𝑥^∗(𝑥)|} for every 𝑥∈𝑋). Then Ψ is holomorphic on 𝐸.

Proof. Let 𝐾 be a finite dimensional compact subset of 𝐸. By hypothesis, Ψ(𝐾) is a compact subset of 𝑋 and hence it is bounded. Thus, there exists 𝑀>0 such that sup𝑥^∗∘Ψ𝑦∶𝑥^∗Ψ𝑦∈𝐴,𝑦∈𝐾=sup∶𝑦∈𝐾<𝑀. (48) Hence, the family of scalar valued holomorphic functions {𝑥^∗∘Ψ∶𝑥^∗∈𝐴} is bounded on the finite dimensional compact subsets of 𝐸. But as 𝐸 is a Fréchet space, then it is also Baire, and by [11] this family is locally bounded: given 𝑦₀∈𝐸 there exists an open neighborhood 𝑉 of 𝑦₀ such that Ψ𝑦𝑥sup∶𝑦∈𝑉=sup^∗∘Ψ𝑦∶𝑥^∗∈𝐴,𝑦∈𝑉<∞. (49) We have obtained that the Gâteaux holomorphic function Ψ is locally bounded. Then it is holomorphic by [4, Proposition 3.7].

The following proposition gives that Λ is holomorphic. The proof may seem at some points similar to that of Proposition 3, but the fact that in the target space we have now a nonmetrizable locally convex topology makes the whole situation much more delicate. We are going to consider now in ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) the topology 𝜏_𝛽 defined by the following fundamental system of seminorms: 𝑞_𝑅,ℬ(𝑇)=sup𝑇𝑓𝑦∶𝑦∈𝑌,𝑦≤𝑅,𝑓∈ℬ, (50) for 𝑇∈ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)), where 𝑅>0 and ℬ is a bounded subset of ℋ_𝑏(𝑋).

Proposition 7. The mapping Λ∶ℋ_𝑏𝑌,𝑋^∗∗⟶ℒℋ_𝑏(𝑋),ℋ_𝑏(𝑌),𝑔⟼Λ𝑔=Φ_𝑔 (51) is injective and holomorphic if we consider in ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) the 𝜏_𝛽-topology.

Proof. Clearly Λ is well defined. Our first step is to prove that Λ∶ℋ_𝑏(𝑌,𝑋^∗∗)→ℒ_𝜏𝛽(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) is Gâteaux holomorphic. Let 𝑔₁,𝑔₂∈ℋ_𝑏(𝑌,𝑋^∗∗), 𝑓∈ℋ_𝑏(𝑋), 𝑦∈𝑌, and 𝑡∈ℂ. Here we have Λ(𝑔₁+𝑡𝑔₂)(𝑓)(𝑦)=𝑓(𝑔₁(𝑦)+𝑡𝑔₂(𝑦)). If ∑^∞_𝑚=0𝑃_𝑚𝑓 is the Taylor series expansion of 𝑓 at 0 on 𝑋, then 𝑓𝑥^∗∗=^∞_𝑚=0𝑃_𝑚𝑓𝑥^∗∗, (52) for every 𝑥^∗∗∈𝑋^∗∗ and the convergence is absolute and uniform on the bounded subsets of 𝑋^∗∗ [8, 9]. Thus Λ𝑔₁+𝑡𝑔₂𝑓𝑦=^∞_𝑚=0𝑃_𝑚𝑓𝑔₁𝑦+𝑡𝑔₂𝑦=^∞𝑚_𝑚=0𝑘=0𝑚𝑘^∨𝑃_𝑚𝑓𝑔₁𝑦^{(𝑚−𝑘)},𝑡𝑔₂𝑦^(𝑘)=^∞𝑚_𝑚=0𝑘=0𝑚𝑘𝑡𝑃^𝑘∨_𝑚𝑓𝑔₁𝑦^{(𝑚−𝑘)},𝑔₂𝑦^(𝑘)=^∞∞_{𝑘=0𝑚=𝑘}𝑚𝑘^∨𝑃_𝑚𝑓𝑔₁𝑦^{(𝑚−𝑘)},𝑔₂𝑦^(𝑘)𝑡^𝑘, (53) where our last step is simply formal. We are going to concentrate our effort now to show that this formal last equality holds in our setting. If we denote by 𝐵_𝑌(0,𝑅) the closure of 𝐵_𝑌(0,𝑅), then 𝑔_𝑗(𝐵_𝑌(0,𝑅)) is a bounded subset of 𝑋^∗∗ for 𝑗=1,2. Thus there exists 𝑀>0 such that ‖𝑔_𝑗(𝑦)‖≤𝑀, for every 𝑦 in 𝐵_𝑌(0,𝑅) and 𝑗=1,2. We fix 𝑠₀>1 and we take 𝑆>0 such that (1+𝑠₀)𝑒𝑀<𝑆. We have ^∞𝑚_𝑚=0𝑘=0𝑒^𝑚𝑚𝑘𝑀𝑆^𝑚−𝑘𝑠₀𝑀𝑆^𝑘=^∞_𝑚=0𝑒1+𝑠₀𝑀𝑆^𝑚=𝑆𝑆−𝑒1+𝑠₀𝑀. (54) Hence, by the properties of summability of double series of nonnegative numbers, the double series below is convergent in ℝ: ^∞∞_{𝑘=0𝑚=𝑘}𝑒^𝑚𝑚𝑘𝑀𝑆^𝑚−𝑘𝑠₀𝑀𝑆^𝑘<∞, (55) and its sum is again 𝑆/(𝑆−𝑒(1+𝑠₀)𝑀). On the other hand, by using the polarization formula (4), [9, Theorem 3], and Cauchy’s inequalities (6) we get sup𝑃_𝑦≤𝑅^∨_𝑚𝑓𝑔₁𝑦^{(𝑚−𝑘)},𝑔₂𝑦^(𝑘)≤sup_𝑦≤𝑅𝑒^𝑚𝑃_𝑚𝑓𝑔₁𝑦^𝑚−𝑘𝑔₂𝑦^𝑘≤𝑒𝑀𝑆^𝑚𝑓_{𝐵𝑋(0,𝑆)}. (56) By applying now (55) we obtain ^∞∞_{𝑘=0𝑚=𝑘}𝑚𝑘sup𝑃_𝑦≤𝑅^∨_𝑚𝑓𝑔₁𝑦^{(𝑚−𝑘)},𝑔₂𝑦^(𝑘)𝑠^𝑘₀≤𝑆𝑆−𝑒1+𝑠₀𝑀𝑓_{𝐵𝑋(0,𝑆)}. (57) Since the function 𝑦∈𝑌→𝑃^∨_𝑚𝑓(𝑔₁(𝑦)^{(𝑚−𝑘)},𝑔₂(𝑦)^(𝑘)) is the composition of a continuous multilinear mapping and two holomorphic mappings of bounded type, it is holomorphic of bounded type on 𝑌. By using (57), we obtain that the series 𝑇_𝑘𝑓∶=^∞∨_𝑚=𝑘𝑃_𝑚𝑓𝑔₁(⋅)^{(𝑚−𝑘)},𝑔₂(⋅)^(𝑘) (58)𝜏_𝑏-converges in ℋ_𝑏(𝑌); hence 𝑇_𝑘(𝑓) belongs to ℋ_𝑏(𝑌) for every 𝑓 in ℋ_𝑏(𝑋). Actually, if we consider 𝑇_𝑘∶ℋ_𝑏(𝑋)→ℋ_𝑏(𝑌), this is a linear operator. By (57), it is also continuous, since given 𝑅>0 there exists 𝑆>0 such that sup_𝑦≤𝑅𝑇_𝑘𝑓𝑦≤𝑆𝑆−𝑒1+𝑠₀𝑀𝑓_{𝐵𝑋(0,𝑆)}, (59) for every 𝑓∈ℋ_𝑏(𝑋). Then 𝑞𝑇_𝑘=sup_𝑦≤𝑅sup_𝑓∈ℬ𝑇_𝑘𝑓𝑦≤^∞_𝑚=𝑘𝑚𝑘𝑒(1+𝑀)𝑆^𝑚sup_𝑓∈ℬ𝑓_{𝐵𝑋(0,𝑆)}. (60) We have, for 𝑡∈ℂ with |𝑡|≤𝑠₀, again by (57), sup_{|𝑡|≤𝑠0}^∞_𝑘=0𝑞𝑇_𝑘|𝑡|^𝑘=^∞_𝑘=0𝑞𝑇_𝑘𝑠^𝑘₀≤𝑆𝑆−𝑒1+𝑠₀𝑀×sup_𝑓∈ℬ𝑓_{𝐵𝑋(0,𝑆)}. (61) As a consequence the series ∑^∞_𝑘=0𝑇_𝑘𝑡^𝑘 defined on ℂ with values in ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)), 𝜏_𝛽-converges uniformly on the compacts of ℂ. Hence it is an entire function. Since we have proved that all series involved converge absolutely, we can apply the reordering of absolutely convergent double series to conclude that the last formal equality of (53) actually holds and then Λ𝑔₁+𝑡𝑔₂𝑓𝑦=^∞_𝑘=0𝑇_𝑘𝑡^𝑘 (62) is an entire function on ℂ for every 𝑔₁,𝑔₂. This gives that Λ is Gâteaux holomorphic.
We fix now 𝑞=𝑞_𝑅,ℬ, a continuous seminorm of the fundamental system defined in (50) and denote by 𝑍_𝑞 the completion of the normed space (ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌))/Ker𝑞,𝑞). Given 𝑦∈𝑌 and 𝑓∈ℋ_𝑏(𝑋) we define the continuous linear functional 𝛿_𝑓,𝑦∶ℒℋ_𝑏(𝑋),ℋ_𝑏(𝑌)⟶ℂ (63) by 𝛿_𝑓,𝑦(𝑇)=𝑇(𝑓)(𝑦). Clearly the quotient mapping 𝛿_𝑓,𝑦∶𝑍_𝑞→ℂ, defined by 𝛿_𝑓,𝑦(𝑇)=𝑇(𝑓)(𝑦), belongs to 𝑍^∗_𝑞. On the other hand the set {𝛿_𝑓,𝑦∶‖𝑦‖≤𝑅,𝑓∈ℬ} is a norming subset of 𝑍_𝑞 since 𝑞(𝑇)=sup𝛿_𝑓,𝑦(𝑇)∶𝑦∈𝑌,𝑦≤𝑅,𝑓∈ℬ. (64) We can consider Λ∶ℋ_𝑏(𝑌,𝑋^∗∗)→𝑍_𝑞 that remains Gâteaux holomorphic. Thus 𝛿_𝑓,𝑦∘Λ=𝛿_𝑓,𝑦∘Λ∶ℋ_𝑏(𝑌,𝑋^∗∗)→ℂ is Gâteaux holomorphic for every 𝑦∈𝑌 and every 𝑓∈ℋ_𝑏(𝑋). If we show that it is continuous, then it will be holomorphic and, by Lemma 6, we will get that Λ∶ℋ_𝑏(𝑌,𝑋^∗∗)→𝑍_𝑞 is holomorphic for every seminorm 𝑞. Since ℒ_𝜏𝛽(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) is a complete space, we can conclude that Λ∶ℋ_𝑏(𝑌,𝑋^∗∗)→ℒ_𝜏𝛽(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) is holomorphic.
Let 𝑔∈ℋ_𝑏(𝑌,𝑋^∗∗). Now for fixed 𝑓∈ℋ_𝑏(𝑋) and 𝑦₀∈𝑌, we consider 𝑅>‖𝑦₀‖ and we choose 𝑆>0 such that 𝑔𝐵_𝑌(0,𝑅)⊂𝐵_𝑋^∗∗(0,𝑆). (65) As 𝑓 is uniformly continuous on bounded subsets of 𝑋^∗∗, for a given 𝜀>0, there exists 0<𝛿<1 such that if 𝑧₁,𝑧₂∈𝐵_𝑋^∗∗(0,𝑆+1) with ‖𝑧₁−𝑧₂‖<𝛿, then 𝑓𝑧₁−𝑓𝑧₂<𝜀. (66) Let ℎ∈ℋ_𝑏(𝑌,𝑋^∗∗) with sup_{‖𝑦‖<𝑅}‖𝑔(𝑦)−ℎ(𝑦)‖<𝛿. Clearly ℎ(𝑦₀)∈𝐵_𝑋^∗∗(0,𝑆+1). Thus 𝛿_𝑓,𝑦0∘Λ𝑔−𝛿_𝑓,𝑦0∘Λ(ℎ)=𝑓𝑔𝑦₀−𝑓ℎ𝑦₀<𝜀. (67) This gives that 𝛿_𝑓,𝑦0∘Λ is continuous for every 𝑦₀ and 𝑓 and completes the proof.

We recall that the 𝜏_𝑝 topology on ℒ(ℋ_𝑏(𝑋),ℋ_𝑏(𝑌)) is the topology of the pointwise convergence on the points of ℋ_𝑏(𝑋).

Corollary 8. The mapping 𝑗∶ℋ_𝑏𝑌,𝑋^∗∗⟶ℳ_𝑏(),𝑔𝑋,𝑌𝑔⟼𝑗=Φ_𝑔 (68) is injective and holomorphic when ℳ_𝑏(𝑋,𝑌) is endowed with the topology induced by 𝜏_𝑝.

5. Some Properties of ℳ_𝑏,∞(𝑋,𝐵_𝑌)

When Aron et al. undertook in [2] the study of ℳ_𝑏(𝑋), the spectra of the Fréchet algebra of holomorphic functions of bounded type they explicitly stated that it was a step to study the spectra of the Banach algebra ℋ_∞(𝐵_𝑋) of bounded holomorphic functions on the open unit ball of 𝑋 (endowed with the supremum norm). Following this philosophy we now study the spectrum consisting of nonzero continuous homomorphisms Φ∶ℋ_𝑏(𝑋)→ℋ_∞(𝐵_𝑌) that we will denote by ℳ_𝑏,∞(𝑋,𝐵_𝑌).

As above we can define 𝜋∶ℳ_𝑏,∞𝑋,𝐵_𝑌⟶ℒ𝑋^∗,𝑌^∗,Φ⟼𝑥^∗⟼𝑑Φ𝑥^∗(0). (69)

Proposition 9. If 𝑋 is a symmetrically regular Banach space and 𝑌 is any Banach space, then (ℳ_𝑏,∞(𝑋,𝐵_𝑌),𝜋) is a Riemann domain over ℒ(𝑋^∗,𝑌^∗) and each connected component of (ℳ_𝑏,∞(𝑋,𝐵_𝑌),𝜋) is homeomorphic to ℒ(𝑋^∗,𝑌^∗).

Proof. The proof follows almost word by word that of Proposition 1. The basis of neighborhoods of a point Φ in ℳ_𝑏,∞(𝑋,𝐵_𝑌) is given by 𝑉_Φ,𝜀={Φ^𝑢∶𝑢∈ℒ(𝑋^∗,𝑌^∗),‖𝑢‖<𝜀}, for 𝜀>0, where Φ^𝑢𝑓𝑦=Φ𝜏^∗_{𝑢∗𝐽𝑌𝑦}𝑓𝑦=Φ𝑥⟼𝑓𝐽_𝑋𝑥+𝑢^∗𝐽_𝑌𝑦𝑦, (70) for all 𝑓∈ℋ_𝑏(𝑋) and 𝑦∈𝐵_𝑌. The fact that Φ^𝑢(𝑓) is in ℋ_∞(𝐵_𝑌) follows from a similar argument to that in (18) taking 𝑅=1.

Now, as in (24), we can define a Gelfand transform of 𝑓∈ℋ_𝑏(𝑋) by 𝑓∶ℳ_𝑏,∞𝑋,𝐵_𝑌⟶ℋ_∞𝐵_𝑌,Φ⟼𝑓(Φ)=Φ𝑓, (71) and we can see that this is a holomorphic extension of 𝑓 to ℳ_𝑏,∞(𝑋,𝐵_𝑌).

Proposition 10. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓∈ℋ_𝑏(𝑋) consider its extension 𝑓 defined in (71). Then the restriction of 𝑓 to each connected component of ℳ_𝑏,∞(𝑋,𝐵_𝑌) is a holomorphic function of bounded type.

Proof. Like in Proposition 3 it is enough to prove that for each fixed Φ in ℳ_𝑏,∞(𝑋,𝐵_𝑌), the mapping 𝑇∶ℒ(𝑋^∗,𝑌^∗)→ℋ_∞(𝐵_𝑌) defined as 𝑇(𝑢)=Φ^𝑢(𝑓) for 𝑢 in ℒ(𝑋^∗,𝑌^∗) is holomorphic of bounded type.
First we check that it is uniformly continuous on any bounded subset of ℒ(𝑋^∗,𝑌^∗). By definition, there exists 𝑅>0 such that ‖Φ(ℎ)‖=sup_‖𝑧‖<1|Φ(ℎ)(𝑧)|≤sup_{‖𝑥‖<𝑅}|ℎ(𝑥)|, (72) for every ℎ in ℋ_𝑏(𝑋). Let 𝑀>0. Since 𝑓 is uniformly continuous on 𝐵_𝑋^∗∗(0,𝑅+𝑀), given 𝜀>0 there exists 𝛿>0 such that if 𝑧₁,𝑧₂ are in 𝐵_𝑋^∗∗(0,𝑅+𝑀) with ‖𝑧₁−𝑧₂‖<𝛿, then |𝑓(𝑧₁)−𝑓(𝑧₂)|<𝜀. Consider now 𝑢₁,𝑢₂ in ℒ(𝑋^∗,𝑌^∗) with ‖𝑢_𝑗‖<𝑀 for 𝑗=1,2 and ‖𝑢₁−𝑢₂‖<𝛿. We have 𝑇𝑢₁−𝑇𝑢₂=sup_𝑦<1Φ^𝑢1𝑓𝑦−Φ^𝑢2𝑓𝑦=sup_𝑦<1Φ𝜏^∗_𝑢^∗_{1𝐽𝑌𝑦}𝑓−𝜏^∗_𝑢^∗_{2𝐽𝑌𝑦}𝑓𝑦≤sup_𝑦<1sup_‖𝑧‖<1Φ𝜏^∗_𝑢^∗_{1𝐽𝑌𝑦}𝑓−𝜏^∗_𝑢^∗_{2𝐽𝑌𝑦}𝑓(𝑧)≤sup_𝑦<1sup_{‖𝑥‖<𝑅}𝜏^∗_𝑢^∗_{1𝐽𝑌𝑦}𝑓−𝜏^∗_𝑢^∗_{2𝐽𝑌𝑦}𝑓(𝑥)=sup_𝑦<1sup_{‖𝑥‖<𝑅}𝑓𝐽_𝑋𝑥+𝑢^∗₁𝐽_𝑌𝑦−𝑓𝐽_𝑋𝑥+𝑢^∗₂𝐽_𝑌𝑦<𝜀. (73) The last inequality holds because 𝐽_𝑋𝑥+𝑢^∗_𝑗𝐽_𝑌𝑦∈𝐵_𝑋^∗∗(0,𝑅+𝑀) for 𝑗=1,2 and ‖𝐽_𝑋𝑥+𝑢^∗₁𝐽_𝑌𝑦−(𝐽_𝑋𝑥+𝑢^∗₂𝐽_𝑌𝑦)‖≤‖𝑢₁−𝑢₂‖<𝛿.
Now we prove that 𝑇 is Gâteaux holomorphic. Let us denote by ∑^∞_𝑚=0𝑃_𝑚𝑓(𝑥) the Taylor series expansion of 𝑓 at a point 𝑥 in 𝑋. Consider 𝑢₁,𝑢₂ in ℒ(𝑋^∗,𝑌^∗). Since 𝑋 is symmetrically regular we have that 𝑓𝐽_𝑋𝑥+𝑢^∗₁(𝑧)+𝜆𝑢^∗₂(𝑧=_∞𝑚=0)𝑃_𝑚𝑓(𝑥)𝑢^∗₁(𝑧)+𝜆𝑢^∗₂(𝑧)=^∞𝑚_𝑚=0𝑘=0𝑚𝑘^∨𝑃_𝑚𝑓(𝑥)𝑢^∗₁(𝑧)^{(𝑚−𝑘)},𝑢^∗₂(𝑧)^(𝑘)𝜆^𝑘, (74) for every 𝑧 in 𝑌^∗∗ and 𝜆 in ℂ. Now we fix 𝑧 and take 𝑅,𝑀>1 with ‖𝑥‖≤𝑅 and |𝜆|≤𝑀. We have, by using the polarization constants and Cauchy inequalities, that ^∞𝑚_𝑚=0𝑘=0𝑚𝑘^∨𝑃_𝑚𝑓(𝑥)𝑢^∗₁(𝑧)^{(𝑚−𝑘)},𝑢^∗₂(𝑧)^(𝑘)|𝜆|^𝑘≤^∞𝑚_𝑚=0𝑘=0𝑚𝑘𝑒^𝑚𝑃_𝑚𝑓(𝑥)_𝐵𝑋×𝑢^∗₁^𝑚−𝑘𝑢^∗₂^𝑘‖𝑣‖^𝑚𝑀^𝑘=𝑒𝑀‖𝑣‖𝑢^∗₁+𝑢^∗₂^𝑚×𝑃_𝑚𝑓(𝑥)_𝐵𝑋=12^𝑚𝑃_𝑚𝑓(𝑥)_{𝐵𝑋(0,2𝐾)}≤12^𝑚𝑓(𝑥)_{𝐵𝑋(0,𝑅+2𝐾)}, (75) where 𝐾=𝑒𝑀‖𝑣‖(‖𝑢^∗₁‖+‖𝑢^∗₂‖). As a consequence ^∞𝑚_𝑚=0𝑘=0𝑚𝑘sup𝑃_𝑚𝑓_{‖𝑥‖≤𝑅}^∨(𝑥)𝑢^∗₁(𝑧)^{(𝑚−𝑘)},𝑢^∗₂(𝑧)^(𝑘)𝜆^𝑘≤^∞_𝑚=012^𝑚𝑓(𝑥)_{𝐵𝑋(0,𝑅+2𝐾)}=2𝑓(𝑥)_{𝐵𝑋(0,𝑅+2𝐾)}. (76) Hence we have that the function 𝑔_𝑧,𝑘(𝑥)=∑^∞_𝑘=𝑚(^𝑚_𝑘)𝑃_𝑚𝑓^∨(𝑧)((𝑢^∗₁(𝑧))^{(𝑚−𝑘)},(𝑢^∗₂(𝑧))^(𝑘)) is well defined and actually belongs to ℋ_𝑏(𝑋) for every nonnegative integer 𝑘 and every 𝑧 in 𝑌^∗∗. Moreover, the equality 𝑓𝐽_𝑋𝑥+𝑢^∗₁(𝑧)+𝜆𝑢^∗₂(𝑧)=^∞_𝑘=0𝑔_𝑧,𝑘(𝑥)𝜆^𝑘 (77) holds for every 𝑥 in 𝑋, 𝑧 in 𝑌^∗∗ and 𝜆 in ℂ. Actually, (76) implies also that the series ∑^∞_𝑘=0𝑔_𝑧,𝑘(⋅)𝜆^𝑘 converges in ℋ_𝑏(𝑋) for each 𝑧 in 𝑌^∗∗. Hence, Φ𝑥⟼𝑓𝐽_𝑋𝑥+𝑢^∗₁(𝑧)+𝜆𝑢^∗₂(𝑧)=^∞_𝑘=0Φ𝑔_𝑧,𝑘𝜆^𝑘, (78) with Φ(𝑔_𝑧,𝑘) in ℋ_∞(𝐵_𝑌) for every 𝑧 and 𝑘 and the series converges in ℋ_∞(𝐵_𝑌) for each 𝜆 in ℂ. Now, if we take 𝑧=𝑦∈𝐵_𝑌, we have 𝑇𝑢₁+𝜆𝑢₂𝑦=^∞_𝑘=0Φ𝑔_𝑦,𝑘𝑦𝜆^𝑘. (79) To finish we just have to observe that the function 𝑦↦Φ(𝑔_𝑦,𝑘)(𝑦) belongs to ℋ_∞(𝐵_𝑌) for every 𝑘. This is an immediate consequence of the fact that Φ(𝑔_𝑦,𝑘)(𝑦)=𝐺(𝑦,𝑦), where 𝐺 is the function 𝐺∶𝑌^∗∗×𝐵_𝑌𝑔⟶ℂ,𝑧,𝑦⟼𝐺𝑧,𝑦=Φ_𝑧,𝑘𝑦, (80) which clearly is separately holomorphic. Thus, by Hartogs’ theorem, 𝐺 is holomorphic.

Proceeding now as in Section 4 we are going to see how we can insert big sets into the fibers of ℳ_𝑏,∞(𝑋,𝐵_𝑌). Indeed, given 𝑔∈ℋ_∞(𝐵_𝑌,𝑋^∗∗) we consider Φ_𝑔∈ℳ_𝑏,∞(𝑋,𝐵_𝑌) given by Φ_𝑔(𝑓)=𝑓∘𝑔. This is well defined, since 𝑔(𝐵_𝑌) is bounded in 𝑋^∗∗ and 𝑓∈ℋ_𝑏(𝑋^∗∗). This again gives an inclusion 𝑗∶ℋ_∞𝐵_𝑌,𝑋^∗∗⟶ℳ_𝑏,∞𝑋,𝐵_𝑌,𝑔⟼Φ_𝑔 (81) that for each 𝑢∈ℒ(𝑋^∗,𝑌^∗) maps the set 𝑔∈ℋ_∞𝐵_𝑌,𝑋^∗∗∶𝑑𝑔(0)=𝑢^∗∘𝐽_𝑌 (82) into the fiber of 𝑢. Also in this case we have a projection 𝜁∶ℳ_𝑏,∞𝑋,𝐵_𝑌⟶ℋ_∞𝐵_𝑌,𝑋^∗∗,Φ⟼𝑦⟼𝑥^∗⟼Φ𝑥^∗𝑦. (83) Again like in Section 4 the map [𝑦∈𝐵_𝑌↦(𝑥^∗↦Φ(𝑥^∗)(𝑦))] is holomorphic. The fact that it is also bounded follows immediately from the fact that 𝐵_𝑋^∗ is a bounded subset in ℋ_𝑏(𝑋) and then {Φ(𝑥^∗)∶𝑥^∗∈𝐵_𝑋^∗} is bounded in ℋ_∞(𝐵_𝑌).

Like in the case of ℳ_𝑏(𝑋,𝑌) we have that if finite type polynomials are dense in ℋ_𝑏(𝑋) then for each Φ∈ℳ_𝑏,∞(𝑋,𝐵_𝑌) there exists 𝑔∈ℋ_∞(𝐵_𝑌,𝑋^∗∗) such that Φ=Φ_𝑔.

On the other hand for every nontrivial Banach space 𝑌, there exists an element in ℳ_𝑏,∞(ℓ₂,𝐵_𝑌) that does not belong to {Φ_𝑔∶𝑔∈ℋ_∞(𝐵_𝑌,ℓ₂)}. Indeed, in [18] it is shown that there exists a continuous homomorphism 𝜑∶ℋ_𝑏(ℓ₂)→ℂ such that 𝜑(𝑄)=1, where 𝑄=∑^∞_𝑛=1𝑥²_𝑛, and vanishes on all continuous homogeneous polynomial on ℓ₂ of odd degree. We define 𝜑⊗𝟏_𝑌 as 𝜑⊗𝟏_𝑌(𝑓)=𝜙(𝑓)𝟏_𝑌 for 𝑓∈ℋ_𝑏(ℓ₂). Clearly 𝜑⊗𝟏_𝑌 is in ℳ_𝑏,∞(ℓ₂,𝐵_𝑌). Observe that if 𝜑⊗𝟏_𝑌=Φ_𝑔 for a certain 𝑔∈ℋ_∞(𝐵_𝑌,ℓ₂), then 0=𝜑𝑥^∗=𝜑⊗𝟏_𝑌𝑦𝑥^∗=𝑔𝑦𝑥^∗, (84) for every 𝑦∈𝐵_𝑌 and every 𝑥^∗∈(ℓ₂)^∗=ℓ₂, and we have obtained that 𝑔=0. Hence Φ_𝑔(𝑓)(𝑦)=𝑓∘𝑔(𝑦)=𝑓∘0(𝑦)=𝑓(0) for every 𝑦∈𝐵_𝑌 and every 𝑓∈ℋ_𝑏(ℓ₂). In particular, 1=𝜑(𝑄)=𝑄(0)=0, a contradiction.

6. Homomorphisms between Algebras of Bounded Functions

In this last section we introduce, for two given 𝑋 and 𝑌 complex Banach spaces, ℳ_∞𝐵_𝑋,𝐵_𝑌=ℳℋ_∞𝐵_𝑋,ℋ_∞𝐵_𝑌=Φ∶ℋ_∞𝐵_𝑋⟶ℋ_∞𝐵_𝑌algebrahomomorphisms⧵{0}. (85)

In this case, no Riemann manifold structure is known, but on the other hand we are going to show that some results very close to the Gelfand transform of the elements of a uniform algebra can be obtained.

The space ℋ_∞(𝐵_𝑌) is a dual space. That was proved by Mujica in [19], where he found a topological predual 𝒢_∞(𝐵_𝑌) that is a subspace of ℋ_∞(𝐵_𝑌)^∗ such that 𝜑∈𝒢_∞(𝐵_𝑌) if and only if the restriction of 𝜑 to 𝐵_𝑌 is continuous when endowed with the compact open topology 𝜏₀. Let us observe that {𝛿_𝑦∶𝑦∈𝐵_𝑌} is a subset of 𝒢_∞(𝐵_𝑌). We denote by 𝑤^∗ the weak-star topology 𝑤(ℋ_∞(𝐵_𝑌),𝒢_∞(𝐵_𝑌)).

Theorem 11. ℳ_∞(𝐵_𝑋,𝐵_𝑌) is a 𝜏_𝑝-compact subset of  ℒ(ℋ_∞(𝐵_𝑋),(ℋ_∞(𝐵_𝑌),𝑤^∗)).

Proof. An application of the Banach Steinhaus theorem yields that ℒ(ℋ_∞(𝐵_𝑋),ℋ_∞(𝐵_𝑌)) and ℒ(ℋ_∞(𝐵_𝑋),(ℋ_∞(𝐵_𝑌),𝑤^∗)) coincide as sets.
By Alaoglu-Bourbaki theorem any bounded subset of (ℋ_∞(𝐵_𝑌),𝑤^∗) is weak-star relatively compact. Thus, by [20, § 39.4(5)] the space ℒ(ℋ_∞(𝐵_𝑋),(ℋ_∞(𝐵_𝑌),𝑤^∗)) has the property that every equicontinuous subset of it is relatively 𝜏_𝑝-compact. As the spectrum ℳ_∞(𝐵_𝑋,𝐵_𝑌) is equicontinuous, we obtain that it is relatively 𝜏_𝑝-compact. Now we check that the spectrum is 𝜏_𝑝-closed. Take 𝑇 in the 𝜏_𝑝 closure of ℳ_∞(𝐵_𝑋,𝐵_𝑌) and let (Φ_𝛼) be a net in the spectrum 𝜏_𝑝 convergent to 𝑇. Then, given 𝑓,𝑔 in ℋ_∞(𝐵_𝑋) we have that Φ_𝛼(𝑓), Φ_𝛼(𝑔), and Φ_𝛼(𝑓𝑔)=Φ_𝛼(𝑓)Φ_𝛼(𝑔)𝑤^∗-converge to 𝑇(𝑓), 𝑇(𝑔), and 𝑇(𝑓𝑔), respectively. But the 𝑤^∗ topology in ℋ_∞(𝐵_𝑌) coincides with the topology of uniform convergence on the compact subsets of 𝒢_∞(𝐵_𝑌). On the other hand in [19, 2.1 Theorem] it is proved that the mapping 𝑔_𝐵𝑌∶𝐵_𝑌⟶𝒢_∞𝐵_𝑌 (86) defined by 𝑔_𝐵𝑌(𝑦)=𝛿_𝑦 is holomorphic. Hence if 𝐾 is a compact subset of 𝐵_𝑌, then {𝛿_𝑦∶𝑦∈𝐾} is a compact subset of 𝒢_∞(𝐵_𝑌) and we obtain that Φ_𝛼(𝑓) converges uniformly to 𝑇(𝑓) on 𝐾. Also Φ_𝛼(𝑔) converges uniformly to 𝑇(𝑔) on 𝐾. As a consequence the net Φ_𝛼(𝑓)Φ_𝛼(𝑔) converges to 𝑇(𝑓)𝑇(𝑔) on the compact open topology of 𝐵_𝑌. This implies, by the definition of 𝒢_∞(𝐵_𝑌), that 𝜑(Φ_𝛼(𝑓)Φ_𝛼(𝑔)) converges to 𝜑(𝑇(𝑓)𝑇(𝑔)) for every 𝜑 in 𝒢_∞(𝐵_𝑌). In other words the net Φ_𝛼(𝑓)Φ_𝛼(𝑔)𝑤^∗-converges to 𝑇(𝑓)𝑇(𝑔) and we have 𝑇𝑓𝑔=𝑇𝑓𝑇𝑔, (87) for every 𝑓,𝑔 in ℋ_∞(𝐵_𝑋). Thus 𝑇 belongs to ℳ_∞(𝐵_𝑋,𝐵_𝑌) and the conclusion follows.

In our setting, taking ℳ_∞(𝐵_𝑋,𝐵_𝑌) as a 𝜏_𝑝-compact set, we are going to extend the concept of Gelfand transform of an element of a Banach algebra in the following way. Given 𝑓∈ℋ_∞(𝐵_𝑋), define 𝑓∶ℳ_∞𝐵_𝑋,𝐵_𝑌⟶ℋ_∞𝐵_𝑌 (88) by 𝑓(Φ)=Φ(𝑓).

Proposition 12. The mapping ∶ℋ_∞𝐵_𝑋⟶𝐶ℳ_∞𝐵_𝑋,𝐵_𝑌,ℋ_∞𝐵_𝑌,𝑓⟼𝑓 (89) is an isometry of algebras.

Proof. Clearly it is an algebra homomorphism and 𝑓=sup_{Φ∈ℳ∞(𝐵𝑋,𝐵𝑌)}Φ𝑓≤𝑓, (90) for every 𝑓 in ℋ_∞(𝐵_𝑋). For a fixed 𝑓, given 𝜀>0 there exists 𝑥₀∈𝐵_𝑋 such that |𝑓(𝑥₀)|>‖𝑓‖−𝜀. Let 1<𝑅<1/‖𝑥₀‖ and consider any 𝑦^∗₀∈𝑋^∗ with ‖𝑦^∗₀‖=1 that attains its norm at a certain 𝑦₀∈𝑌 with ‖𝑦₀‖=1. Let 𝑔∶𝐵_𝑌→𝐵_𝑋 be defined by 𝑔(𝑦)=𝑦^∗₀(𝑦)𝑅𝑥₀. We have that Φ_𝑔 belongs to ℳ_∞(𝐵_𝑋,𝐵_𝑌) and 𝑓≥Φ_𝑔𝑓≥𝑦𝑓∘𝑔⁰𝑅=𝑓𝑦^∗₀𝑦₀𝑅𝑅𝑥₀=𝑓𝑥₀>𝑓−𝜀. (91) Hence the inequality ‖𝑓‖≥‖𝑓‖ also holds.

The scalar spectrum of ℋ_∞(𝐵_𝑋), denoted by ℳ_∞(𝐵_𝑋), can be considered as a subset of ℳ_∞(𝐵_𝑋,𝐵_𝑌) by associating to each 𝜑 in ℳ_∞(𝐵_𝑋) the homomorphism 𝜑⊗𝟏_𝑌 that belongs to ℳ_∞(𝐵_𝑋,𝐵_𝑌). In that way the new Gelfand transform 𝑓 of each 𝑓 in ℋ_∞(𝐵_𝑋) can be considered and extension of the classical 𝑓 by the equality 𝑓𝜑𝟏_𝑌=𝑓𝜑⊗𝟏_𝑌. (92)

It is natural to consider again ℳ_𝑏(𝑋,𝑌) and ℳ_𝑏,∞(𝑋,𝐵_𝑌) under the light of Proposition 12. We cannot give to any of these two spectra a topology that makes them compact sets. But in the case of ℳ_𝑏,∞(𝑋,𝐵_𝑌) it can be endowed with a topology in such a way that this set is a countable union of compact sets. Indeed, as above, an application of the Banach Steinhaus theorem, now for Fréchet spaces, yields that ℒ(ℋ_𝑏(𝑋),ℋ_∞(𝐵_𝑌)) and ℒ(ℋ_𝑏(𝑋),(ℋ_∞(𝐵_𝑌),𝑤^∗)) coincide as sets and, also as above, any equicontinuous subset of ℒ(ℋ_𝑏(𝑋),(ℋ_∞(𝐵_𝑌),𝑤^∗)) is relatively 𝜏_𝑝-compact. Given 𝑅>0, we denote ℳ_𝑏,∞𝑋,𝐵_𝑌𝑅=Φ∈ℳ_𝑏,∞𝑋,𝐵_𝑌∶Φ𝑓≤𝑓_{𝐵𝑋(0,𝑅)}∀𝑓∈ℋ_𝑏(𝑋). (93) Then, ℳ_𝑏,∞(𝑋,𝐵_𝑌)_𝑅 is an equicontinuous set, and a similar argument of one given in the proof of Proposition 12 implies that ℳ_𝑏,∞(𝑋,𝐵_𝑌)_𝑅 is a 𝜏_𝑝 compact set. Finally ℳ_𝑏,∞(𝑋,𝐵_𝑌)=⋃^∞_𝑛=1ℳ_𝑏,∞(𝑋,𝐵_𝑌)_𝑛.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was initiated in June 2012, when the first named author visited the Universidad de Valencia and the Universitat Politècnica de València. She wishes to thank all the people in and outside both universities who made that visit such a delightful time. The first named author was partially supported by research projects CONICET PIP 0624 and ANPCyT PICT 1456. The second, third, and fourth named authors were supported by research project MICINN MTM2011-22417. The second and third named authors also by research project Prometeo II/2013/013.