2012/09/17 by H. G. Dales, Dales, H. G., W. Żelazko +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1209.3530
arxiv created 2012/09/17 · openalex publication_date 2012/09/17 · arxiv updated 2012/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Šilov boundary of A'. We shall give a shorter proof of this latter result, together with some extensions and related examples.