Research Article
Homomorphisms between Algebras of Holomorphic Functions
Verónica Dimant,
1
Domingo García,
2
Manuel Maestre,
2
and Pablo Sevilla-Peris
3
1
Departamento de Matem´ atica, Universidad de San Andr´ es, Vito Dumas 284, Victoria, B1644BID Buenos Aires, Argentina
2
Departamento de An´ alisis Matem´ atico, Universidad de Valencia, Doctor Moliner 50, Burjasot, 46100 Valencia, Spain
3
Instituto Universitario de Matem´ atica Pura y Aplicada, Universitat Polit` ecnica de Val` encia, 46022 Valencia, Spain
Correspondence should be addressed to Pablo Sevilla-Peris; psevilla@mat.upv.es
Received 26 December 2013; Accepted 13 March 2014; Published 12 May 2014
Academic Editor: Alfredo Peris
Copyright © 2014 Ver´ onica Dimant et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
For two complex Banach spaces and , in this paper, we study the generalized spectrum M
(,) of all nonzero algebra
homomorphisms from H
(), the algebra of all bounded type entire functions on , into H
(). We endow M
(,) with a
structure of Riemann domain over L(
∗
,
∗
) whenever is symmetrically regular. Te size of the fbers is also studied. Following
the philosophy of (Aron et al., 1991), this is a step to study the set M
,∞
(,
) of all nonzero algebra homomorphisms from
H
() into H
∞
(
) of bounded holomorphic functions on the open unit ball of and M
∞
(
,
) of all nonzero algebra
homomorphisms from H
∞
(
) into H
∞
(
).
1. Introduction
Te 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 infnite dimensional spaces (see
Section 2 for precise defnitions): H
(), the holomorphic
functions of bounded type (which is a Fr´ echet algebra), and
H
∞
(
), the bounded holomorphic functions on the open
unit ball (which is a Banach algebra). Here, as a frst step in
the study of the set of homomorphisms between H
∞
(
)
spaces, we mainly focus on algebras of holomorphic functions
of bounded type and homomorphisms between them; :
H
() → H
() (i.e., continuous, linear, and multiplicative
mappings). Tese were already considered in [1]. Tere the
focus was to study the homomorphisms as “individuals,”
seeking properties of single ones. We have here a diferent
interest: we treat them as a whole, considering the set
M
(,) = M (H
(), H
())
={Φ: H
() → H
()
algebra homomorphisms}\{0}.
(1)
We will call this set the generalized spectrum or simply the
spectrum. Our main aim is to study M
(,) and to defne
on it a topological and a diferential structure.
Tis problem has the same favor as considering M
(),
the spectrum of the algebra H
() (i.e., the set of nonzero
continuous, linear, and multiplicative Φ: H
() → C).
Tis was studied in [2, 3], where a structure of Riemannian
manifold over the bidual
∗∗
was defned 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,
defning on M
(,) a Riemann structure over L(
∗
,
∗
)
(note that
∗∗
= L(
∗
, C)). We will also be interested in the
fbers of elements of L(
∗
,
∗
).
Te outline of the paper is the following. In Section 3,
for two complex Banach spaces and , we study the
generalized spectrum M
(,) of all nonzero algebra homo-
morphisms from H
() to H
(). We endow it with a
structure of Riemann domain over L(
∗
,
∗
) whenever
is symmetrically regular. In Section 4, we focus on the sets
(fbers) of elements in M
(,) that are projected on the
same element of L(
∗
,
∗
). Te size of these fbers is
studied and we prove that they are big by showing that
they contain big sets. Following the philosophy of [2], all
about M
(,) is a step to study in Section 5 the spectrum
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 612304, 12 pages
http://dx.doi.org/10.1155/2014/612304