The radical of the bidual of a Beurling algebra

TL;DR

This paper investigates the structure of the biduals of Beurling algebras on integers, proving they are never semisimple and revealing the presence of nilpotent and radical elements, thus answering longstanding questions in the field.

Contribution

It establishes that biduals of Beurling algebras are never semisimple and characterizes the radical elements, including nilpotent and non-nilpotent examples, settling questions by Dales and Lau.

Findings

Biduals of Beurling algebras are never semisimple.

The radical of certain ℓ¹ spaces contains nilpotent elements of all indices.

Existence of weights with biduals containing non-nilpotent radical elements.

Abstract

We prove that the bidual of a Beurling algebra on $\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that ${\rm rad\,}(\ell^{\, 1}(\oplus_{i=1}^\infty \mathbb{Z})")$ 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 $\omega$ on $\mathbb{Z}$ such that the bidual of $\ell^{\, 1}(\mathbb{Z}, \omega)$ contains a radical element which is not nilpotent.