Alling's conjecture on closed prime ideals in \(H^ \infty\)
type: Item
instance of: scholarly article
title: Alling's conjecture on closed prime ideals in \(H^ \infty\)
zbMATH Open document ID: 0887.46032
author: Pamela Gorkin
author: Raymond Mortini
published in: Journal of Functional Analysis
publication date: 1998-05-25T00:00:00Z
review text: Let \(H^\infty\) be the algebra of all bounded analytic functions in the open unit disk. At a conference in Oberwolfach around 1970 Norman Alling asked for a characterization of the closed prime ideals in \(H^\infty\). In the paper under review, it is shown that every such ideal is an intersection of maximal ideals, hence uniquely determined by its hull. The proof is based on \textit{L. Carleson's} contour construction developed for the proof of the Corona theorem [Ann. Math., II. Ser. 76, 547-559 (1962; Zbl 0112.29702)], on some recent results of \textit{F. D. Suarez} on separation properties of \(H^\infty\) [J. Funct. Anal. 123, No. 2, 233-263 (1994; Zbl 0808.46076)] and on some factorization theorems of K. Hoffman, K. Izuchi and D. Suarez. Preliminary results on closed prime ideals in \(H^\infty\) have been obtained by the authors in [\textit{R. Mortini}, Bull. Austral. Math. Soc. 35, 213-229 (1987; Zbl 0601.46052)]. Some sufficient conditions for a closed prime ideal to be maximal are given. Alling's conjecture whether the hull of every nonmaximal, closed prime ideal in \(H^\infty\) is the closure of a single Gleason part remains unsolved.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 1054128
zbMATH Keywords: Carleson's contour construction
zbMATH Keywords: Corona theorem
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: closed prime ideals
zbMATH Keywords: intersection of maximal ideals
zbMATH Keywords: factorization theorems
zbMATH Keywords: Gleason part
MaRDI profile type: Publication
full work available at URL: jfan.1996.3066
OpenAlex ID: W2035424986
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DOI: 10.1006/JFAN.1996.3066
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description: scientific article; zbMATH DE number 1054128