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

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