Generators of maximal left ideals in Banach algebras

• Autorzy: H. G. Dales , W. Żelazko
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46H10: Ideals and subalgebras
• 46J10: Banach algebras of continuous functions, function algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 212
• Numer: 2
• Strony: 173-193

Daty

wydano

2012

Twórcy

autor

H. G. Dales

• Department of Mathematics and Statistics, Fylde College, University of Lancaster, Lancaster LA1 4YF, United Kingdom

autor

W. Żelazko

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm212-2-5