Bounded analytic functions and closed ideals
type: Item
instance of: scholarly article
title: Bounded analytic functions and closed ideals
zbMATH Open document ID: 0633.46055
DOI: 10.1007/BF02790326
published in: Journal d'Analyse Mathématique
publication date: 1987-01-01T00:00:00Z
review text: This paper is a revision of part of the author's thesis [Ideals and subalgebras of analytic functions (1985; Zbl 0596.46049)]. One of the main results in {\S} 2 is a complete characterization of the closed primary ideals in \(H^{\infty}\) contained in a maximal ideal whose Gleason part is nontrivial. Another result in {\S} 2 tells us that every closed ideal in \(H^{\infty}\) whose hull is contained in the fiber \(M_ 1(H^{\infty})\) of the maximal ideal space of \(H^{\infty}\) has the form \(I=k\cdot J\), where k is the singular inner function \(k(z)=\exp (\alpha (z+1)/(z-1))\) (\(\alpha\geq 0)\) and where J is a closed ideal in \(H^{\infty}\) containing all functions whose Gelfand transforms vanish on \(M_ 1(H^{\infty})\). Extensions of this result may be found in [\textit{P. Gorkin}, \textit{H. Hedenmalm} and \textit{R. Mortini}, Ill. J. Math. 31, 629-644 (1987; Zbl 0615.46049)]. {\S}{\S} 3 and 4 contain a lot of interesting results relating the structure of closed ideals of Banach algebras B of analytic functions on planar domains W to that of subalgebras A of B which are characterized by the condition that their elements are holomorphically extendable to some of the components of \({\mathbb{C}}\setminus W.\) The author's techniques consist in the use of the holomorphic functional calculus. Also, for the first time, the powerful tool of the analytic Carleman transform is successfully applied in the case of nonseparable Banach algebras like \(H^{\infty}(W)\).
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46H30
zbMATH DE Number: 4030262
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: closed primary ideals
zbMATH Keywords: maximal ideal
zbMATH Keywords: Gleason part
zbMATH Keywords: singular inner function
zbMATH Keywords: Gelfand transforms
zbMATH Keywords: structure of closed ideals of Banach algebras
zbMATH Keywords: holomorphic functional calculus
zbMATH Keywords: analytic Carleman transform
author: Håkan Hedenmalm
reviewed by: Raymond Mortini
Wikidata QID: Q59288730
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02790326
OpenAlex ID: W2012178140
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description: scientific article; zbMATH DE number 4030262