Radicals of Biduals of Beurling Algebras Can Be Different for the Two Arens Products

TL;DR

This paper provides an example of a Beurling algebra where the Jacobson radical of its bidual differs depending on the Arens product used, addressing a question in Banach algebra theory.

Contribution

It constructs a specific Beurling algebra with a radical that varies between the two Arens products, showing these radicals can be different.

Findings

The Jacobson radical of the bidual can differ for the two Arens products.

An explicit example using the free group on three generators.

Answers an open question by Dales and Lau.

Abstract

Let $\operatorname{rad}$ denote the Jacobson radical of a Banach algebra, and let $\Box$ and $\Diamond$ denote the two Arens products on its bidual. We give an example of a Beurling algebra $\mathcal{A}$ for which $\operatorname{rad}(\mathcal{A}^{**}, \Box) \neq \operatorname{rad}(\mathcal{A}^{**}, \Diamond)$, answering a question of Dales and Lau. The underlying group in our example is the free group on three generators.