Lebesgue-Fourier algebra of a hypergroup

TL;DR

This paper investigates the structure and properties of the Lebesgue-Fourier algebra associated with hypergroups, focusing on its Banach algebra structure and amenability characteristics.

Contribution

It introduces the Lebesgue-Fourier algebra for hypergroups and analyzes its Banach algebra properties and amenability in the context of regular Fourier hypergroups.

Findings

${ m extbf{L}A(H)}$ is a Banach algebra with inherited multiplication from $L^1(H)$.

${ m extbf{L}A(H)}$ is a Banach algebra with pointwise multiplication for regular Fourier hypergroups.

The paper studies the conditions for amenability and character amenability of these algebras.

Abstract

Let ${\mathcal L} A(H)$ be the Lebesgue-Fourier space of a hypergroup $H$ considered as a Banach space on $H$. In addition ${\mathcal L}A(H)$ is a Banach algebra with the multiplication inherited from $L^1(H)$. Moreover If $H$ is a regular Fourier hypergroup, ${\mathcal L}A(H)$ is a Banach algebra with pointwise multiplication. We study the amenability and character amenability of these two Banach algebras.