Hypergroups with Unique Alpha-Means

Let $K$ be a commutative hypergroup and $\alpha \in \widehat{K}$. 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 $\widehat{K}$. In contrast to the case of amenable noncompact locally compact groups, examples of polynomial hypergroups with unique $\alpha$-means ($\alpha ≠1$) are given. Further examples emphasize that the $\alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.