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Hypergroups with Unique Alpha-Means · arXivDesk

0706.3620Jun 25, 20078 pages, keywords: Hypergroups: orthogonal polynomial, of Nevaei classes. $α$-Amenable Hypergroups

Hypergroups with Unique Alpha-Means

Ahmadreza Azimifard

Abstract

Let $K$ be a commutative hypergroup and $α\in \hat{K}$ . We show that $K$ is $α$ -amenable with the unique $α$

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-mean

m_α

if and only if

m_α\in L^1(K)\cap L^2(K)

and

α

is isolated in

\hat{K}

. In contrast to the case of amenable noncompact locally compact groups, examples of polynomial hypergroups with unique

α

-means (

α\not=1

) are given. Further examples emphasize that the

α

-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.

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