On the Amenability of Compact and Discrete Hypergroup Algebras

TL;DR

This paper characterizes the amenability of hypergroup algebras $L^1(K)$ for compact and discrete hypergroups, linking it to the boundedness of the Plancherel weight and properties of the dual space.

Contribution

It provides necessary and sufficient conditions for the amenability of $L^1(K)$ in terms of the Plancherel weight and dual space properties, extending understanding of hypergroup algebra structures.

Findings

$L^1(K)$ is amenable iff the Plancherel weight $ ext{pi}_K$ is bounded.

If $K$ is an infinite discrete hypergroup with a dual element vanishing at infinity, then $L^1(K)$ is not amenable.

$L^1(K)$ fails to be $ ext{alpha}$-left amenable if $ ext{pi}_K( ext{alpha})=0$.

Abstract

Let $K$ be a commutative compact hypergroup and $L^1(K)$ the hypergroup algebra. We show that $L^1(K)$ is amenable if and only if $\pi_K$, the Plancherel weight on the dual space $\widehat{K}$, is bounded. Furthermore, we show that if $K$ is an infinite discrete hypergroup and there exists $\alpha\in \widehat{K}$ which vanishes at infinity, then $L^1(K)$ is not amenable. In particular, $L^1(K)$ fails to be even $\alpha$-left amenable if $\pi_K(\{\alpha\})=0$.