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.