Arens products for some convolution algebras of measures
Viktor Losert
TL;DR
The paper analyzes Arens products on measure algebras associated with Chebli–Trimèche hypergroups, focusing on the left topological centre and its decomposition into explicit components that depend on asymptotic translation behavior. It proves that $Z_t(L^1(H))$ splits as $L^1(H) \oplus Z^\infty(H)$ with a potentially remote part $Z^\infty(H)$, while $Z_t(M(H))$ contains a nontrivial $M_0(H)_{LA}$, showing distinct irregularity patterns for $L^1(H)$ and $M(H)$. The work leverages detailed properties of hypergroup convolution and asymptotics, using results of Braaksma–de Snoo and Stein–Wainger to connect to real-line convolution, and applies the conclusions to double coset spaces from classical groups (e.g., $SL(2,\mathbb{C})$) and Euclidean motion groups. It also discusses implications for particular hypergroups (Jacobi, Naimark) and includes remarks correcting prior literature. Overall, the results illuminate how Arens regularity behaves in non-group hypergroup settings and provide a framework for understanding second-dual products in these algebras.
Abstract
We consider the measure algebra of a Chébli-\!Trimèche 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 Chébli-\!Trimèche hypergroup and the asymptotic behaviour.