On the Amenability of Compact and Discrete Hypergroup Algebras

Details On the Amenability of Compact and Discrete Hypergroup Algebras On the Amenability of Compact and Discrete Hypergroup Algebras

2009-08-11

arxiv.org/abs/0908.1590v2

Mathematics

Functional Analysis

Scientific paper

Let $K$ be a commutative compact hypergroup and $L^1(K)$ the hypergroup algebra. We show that $L^1(K)$ is amenable if and only if $\pi_K$, the Plancherel weight on the dual space $\widehat{K}$, is bounded. Furthermore, we show that if $K$ is an infinite discrete hypergroup and there exists $\alpha\in \widehat{K}$ which vanishes at infinity, then $L^1(K)$ is not amenable. In particular, $L^1(K)$ fails to be even $\alpha$-left amenable if $\pi_K(\{\alpha\})=0$.

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