Finite codimensional ideals in Banach algebras with one generator
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 \(A\)?
scientific article; zbMATH DE number 97431
scientific article