Generators of maximal left ideals in Banach algebras

Dales, H.G. and Zelazko, W. (2012) Generators of maximal left ideals in Banach algebras. Studia Mathematica, 212 (2). pp. 173-193. ISSN 0039-3223

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In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples. We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.

Item Type:
Journal Article
Journal or Publication Title:
Studia Mathematica
Uncontrolled Keywords:
?? banach algebramaximal left idealmathematics(all) ??
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Deposited On:
13 May 2013 08:22
Last Modified:
09 Jan 2024 00:15