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

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.