On the Dales-Zelazko conjecture for Beurling algebras on discrete groups

TL;DR

This paper investigates the structure of maximal left ideals in Beurling algebras on certain infinite groups, confirming a conjecture for these cases and providing examples where the property fails, highlighting the complexity of the conjecture.

Contribution

It proves the Dales-Zelazko conjecture for Beurling algebras on virtually soluble or free groups and constructs examples where the conjecture does not hold.

Findings

Maximal left ideals of finite codimension may not be finitely generated.

The conjecture holds for groups that are virtually soluble or free.

Counterexamples show the property can fail strongly in certain weighted groups.

Abstract

Let $G$ be a group which is either virtually soluble or virtually free, and let $\omega$ be a weight on $G$. We prove that, if $G$ is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$ which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Zelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance we describe a Beurling algebra on an infinite group in which every left ideal of finite codimension is finitely generated, and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Zelazko's conjecture.