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arXiv · 1209.3530

Generators of maximal left ideals in Banach algebras

H. G. Dales·W. Żelazko
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Abstract

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 \v{S}ilov boundary of $A$'. We shall give a shorter proof of this latter result, together with some extensions and related examples.

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BibTeXRIS

H. G. Dales, W. Żelazko. 2012-09-17. Generators of maximal left ideals in Banach algebras. https://arxiv.org/abs/1209.3530

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