Arens products for some convolution algebras of measures

TL;DR

This paper investigates the Arens products on the second dual of measure algebras associated with Chébli-Trimèche hypergroups, revealing differences from classical group cases through analysis of hypergroup properties.

Contribution

It provides new insights into the structure of measure algebras on hypergroups and their Arens products, extending previous group-based results to hypergroup settings.

Findings

Distinct behavior of Arens products compared to group cases

Analysis of multiplication and translation properties in hypergroups

Asymptotic behavior of hypergroup measure algebras

Abstract

We consider the measure algebra of a Ch\'ebli-\!Trim\`eche hypergroup (in particular, double coset spaces of classical Lie groups) and study the corresponding Arens products on its second dual. The behaviour turns out to be different to the group case investigated in [LNPS]. For this, we study more closely properties of the multiplication and generalized translation in a Ch\'ebli-\!Trim\`eche hypergroup and the asymptotic behaviour.