The radical of the bidual of a Beurling algebra

Abstract We prove that the bidual of a Beurling algebra on Z, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that rad(ℓ1(⊕i=1∞Z)′′) contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight ω on Z such that the bidual of ℓ1(Z,ω) contains a radical element which is not nilpotent.