Hypergroups with Unique Alpha-Means

Details Hypergroups with Unique Alpha-Means Hypergroups with Unique Alpha-Means

2007-06-25

arxiv.org/abs/0706.3620v3

Mathematics

Group Theory

8 pages, keywords: Hypergroups: orthogonal polynomial, of Nevaei classes. $\alpha$-Amenable Hypergroups

Scientific paper

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.

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