Finite codimensional ideals in Banach algebras with one generator
type: Item
instance of: scholarly article
title: Finite codimensional ideals in Banach algebras with one generator
zbMATH Open document ID: 0806.46055
author: Krzysztof Jarosz
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-01-17T00:00:00Z
review text: It has been shown by Gleason and, independently, by Kahane and Zelazko, that, if \(M\) is a closed subspace of codimension 1 in a complex Banach algebra \(A\), then \(M\) is an ideal if, and only if, \(M\) contains only non- invertible elements of \(A\). In the article under review, the author addresses the problem, still largely open, of generating this result to the case where \(M\) is a closed subspace of \(A\) of codimension \(n>1\) and offers the following principal theorem as a partial solution. Theorem. Let \(A\) be a Banach algebra with one generator and let \(M\) be a closed subspace of \(A\) of codimension \(n<\infty\). Assume that each element of \(M\) is contained in at least \(n\) distinct maximal ideals of \(A\). Then \(M\) is an ideal, namely, an intersection of \(n\) maximal ideals. Following the proof of this theorem, the author closes with a question (to which he conjectures an affirmative answer): If \(A\) is a singly generated Banach algebra and \(M\) is a closed subspace of finite codimension \(n\geq 2\) consisting only of non-invertible elements, then is \(M\) contained in a non-trivial ideal of \(A\)?
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 97431
zbMATH Keywords: Banach algebra
zbMATH Keywords: intersection of maximal ideals
reviewed by: Timothy G. Feeman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072754
OpenAlex ID: W2070005009
cites work: A characterization of maximal ideals
cites work: Finite Codimensional Ideals in Function Algebras
cites work: Generalizations of the Gleason-Kahane-Żelazko theorem
cites work: A characterization of maximal ideals in commutative Banach algebras
cites work: A characterization of regular maximal ideals
cites work: Ideals of Finite Codimension in C[ 0, 1 ] and L 1 (R)
DOI: 10.1216/RMJM/1181072754
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description: scientific article; zbMATH DE number 97431