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

Abstract

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

21 pages