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Generators of maximal left ideals in Banach algebras

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<mark>Journal publication date</mark>2012
<mark>Journal</mark>Studia Mathematica
Issue number2
Number of pages21
Pages (from-to)173-193
Publication StatusPublished
<mark>Original language</mark>English


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.