Cluster values for algebras of analytic functions

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet alge...

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Autores principales: Carando, D., Galicer, D., Muro, S., Sevilla-Peris, P.
Formato: JOUR
Materias:
Analytic functions of bounded type
Ball algebra
Cluster value problem
Corona Theorem
Fiber
Spectrum
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00018708_v329_n_p157_Carando
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00018708_v329_n_p157_Carando2023-10-03T13:52:23Z Cluster values for algebras of analytic functions Carando, D. Galicer, D. Muro, S. Sevilla-Peris, P. Analytic functions of bounded type Ball algebra Cluster value problem Corona Theorem Fiber Spectrum The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet algebra Hb(X) of holomorphic functions of bounded type on X (more generally, for Hb(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U⊂X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras. © 2017 Elsevier Inc. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00018708_v329_n_p157_Carando
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Analytic functions of bounded type
Ball algebra
Cluster value problem
Corona Theorem
Fiber
Spectrum
spellingShingle Analytic functions of bounded type
Ball algebra
Cluster value problem
Corona Theorem
Fiber
Spectrum
Carando, D.
Galicer, D.
Muro, S.
Sevilla-Peris, P.
Cluster values for algebras of analytic functions
topic_facet Analytic functions of bounded type
Ball algebra
Cluster value problem
Corona Theorem
Fiber
Spectrum
description The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet algebra Hb(X) of holomorphic functions of bounded type on X (more generally, for Hb(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U⊂X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras. © 2017 Elsevier Inc.
format JOUR
author Carando, D.
Galicer, D.
Muro, S.
Sevilla-Peris, P.
author_facet Carando, D.
Galicer, D.
Muro, S.
Sevilla-Peris, P.
author_sort Carando, D.
title Cluster values for algebras of analytic functions
title_short Cluster values for algebras of analytic functions
title_full Cluster values for algebras of analytic functions
title_fullStr Cluster values for algebras of analytic functions
title_full_unstemmed Cluster values for algebras of analytic functions
title_sort cluster values for algebras of analytic functions
url http://hdl.handle.net/20.500.12110/paper_00018708_v329_n_p157_Carando
work_keys_str_mv AT carandod clustervaluesforalgebrasofanalyticfunctions
AT galicerd clustervaluesforalgebrasofanalyticfunctions
AT muros clustervaluesforalgebrasofanalyticfunctions
AT sevillaperisp clustervaluesforalgebrasofanalyticfunctions
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