Finitely-generated left ideals in Banach algebras on groups and semigroups

Jared T. White¹

¹ Department of Mathematics and Statistics University of Lancaster Lancaster LA1 4YF, United Kingdom

Studia Mathematica, Tome 239 (2017) no. 1, pp. 67-99

Résumé

Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason.

Détail

DOI : 10.4064/sm8743-1-2017

Keywords: locally compact group prove augmentation ideal algebraically finitely generated ideal only finite investigate weighted versions result version semigroup algebras weighted measure algebras considered motivated recent conjecture dales elazko which states unital banach algebra which every maximal ideal finitely generated necessarily finite dimensional prove conjecture holds many algebras considered finally theory have developed construct examples commutative banach algebras relate theorem gleason