Hypergroups with Unique Alpha-Means

TL;DR

This paper characterizes when a commutative hypergroup has a unique alpha-mean, linking it to measure integrability and character isolation, and provides examples contrasting group cases.

Contribution

It establishes necessary and sufficient conditions for alpha-amenability with unique alpha-means in commutative hypergroups, including new examples and insights.

Findings

Unique alpha-mean exists iff measure is in L^1 and L^2 and alpha is isolated

Examples of polynomial hypergroups with unique alpha-means for nontrivial alpha

Alpha-amenability depends on Haar measure and character asymptotics

Abstract

Let $K$ be a commutative hypergroup and $\alpha\in \hat{K}$. We show that $K$ is $\alpha$-amenable with the unique $\alpha$-mean $m_\alpha$ if and only if $m_\alpha\in L^1(K)\cap L^2(K)$ and $\alpha$ is isolated in $\hat{K}$. In contrast to the case of amenable noncompact locally compact groups, examples of polynomial hypergroups with unique $\alpha$-means ($\alpha\not=1$) are given. Further examples emphasize that the $\alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.