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arXiv · 1708.09635

The radical of the bidual of a Beurling algebra

Jared T. White
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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 $ω$ on $\mathbb{Z}$ such that the bidual of $\ell^{\, 1}(\mathbb{Z}, ω)$ contains a radical element which is not nilpotent.

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Jared T. White. 2018-01-17. The radical of the bidual of a Beurling algebra. https://arxiv.org/abs/1708.09635

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