Arens regularity of weighted convolution algebras that arise from totally ordered semilattices
M. Eugenia Celorrio
TL;DR
This work analyzes Arens regularity for weighted semigroup convolution algebras arising from totally ordered semilattices, extending prior unweighted results to a weighted setting. By introducing the $\Omega$-function and $0$-cluster criteria, it characterizes when $\mathcal{A}_\omega=\ell^1(S,\omega)$ is Arens regular (equivalently $\Lim_{s\to\infty}\omega(s)=\infty$) and when bidual products vanish on $E_\omega^\perp$, while also addressing duality questions via Banach-algebra preduals. The paper develops general existence results for approximate identities and provides conditions for the (non)existence and (non)uniqueness of preduals, including both cancellative and non-cancellative totally ordered semilattices. In the concrete case $(\mathbb{N},\wedge)$, it establishes strong Arens irregularity through DTC sets, identifies minimal two-point DTC sets, and analyzes duality structure and preduals, offering a detailed view of how weight growth shapes regularity properties and dual behavior.
Abstract
We study Arens regularity of weighted semigroup convolution algebras for the specific case of totally ordered semilattices. This paper is a natural continuation of that of Dales and Strauss (2022) [DS22], where they studied the unweighted case. We also study some other properties of these algebras, such as the existence of Banach-algebra preduals.