Arens Products and Asymptotic Structures on Ch\'{e}bli-Trim\`eche Hypergroups under Low Regularity Conditions

TL;DR

This paper extends the analysis of Arens products and asymptotic structures on hypergroup algebras under low regularity conditions, broadening the class of hypergroups where these properties are understood.

Contribution

It relaxes classical smoothness assumptions on the Sturm--Liouville function and develops new asymptotic tools, extending results to a larger class of hypergroups and providing new criteria for Arens irregularity.

Findings

Extended the class of hypergroups with well-understood asymptotic measures

Provided new criteria for strong Arens irregularity based on spectral behavior

Compared topological centres for non-classical hypergroups

Abstract

We investigate the Arens products on the second duals of convolution algebras associated with Ch\'{e}bli--Trim\`{e}che hypergroups, particularly focusing on the left and right topological centres of $L^{1}(H)^{\prime\prime}$ and $M(H)^{\prime\prime}$. Building on the recent framework established by Losert, we relax the classical smoothness assumptions on the underlying Sturm--Liouville function $A$ and develop new asymptotic analysis tools for measure-valued and low-regularity perturbations. This allows us to extend the existence and continuity of the asymptotic measures $\nu_{x}$ and the limit measure $\nu_{\infty}$ to a strictly larger class of hypergroups. We further provide new necessary and sufficient conditions for strong Arens irregularity of $L^{1}(H)$ in terms of the spectral behaviour of $\nu_{\infty}$, explore weighted (Beurling-type) hypergroup algebras, and obtain the first detailed comparison between the left and right topological centres for a wide class of non-classical examples. Several concrete applications to Jacobi, Naimark, and Bessel--Kingman hypergroups are presented.