Finitely-Generated Left Ideals in Banach Algebras on Groups and Semigroups

TL;DR

This paper characterizes when augmentation ideals in group Banach algebras are finitely generated, showing it only occurs for finite groups, and explores related properties in weighted and semigroup algebras, addressing a conjecture by Dales and Zelazko.

Contribution

It proves the finite-generation of augmentation ideals characterizes finite groups and extends the analysis to weighted and semigroup algebras, supporting a conjecture on maximal left ideals.

Findings

Augmentation ideal in L^1(G) is finitely generated iff G is finite.

The conjecture holds for many Banach algebras considered.

Constructs examples of commutative Banach algebras related to Gleason's theorem.

Abstract

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 \.Zelazko, 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.