On the Amenability of Compact and Discrete Hypergroup Algebras

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 $\pi_K(\{\alpha\})=0$.