Hypergroups with Unique Alpha-Means

Let $K$ be a commutative hypergroup and $alphain hat{K}$. We show that $K$ is $alpha$-amenable with the unique $alpha$-mean $m_alpha$ if and only if $m_alphain L^1(K)cap L^2(K)$ 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 ($alpha ot=1$) are given. Further examples emphasize that the $alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.