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 infinit
e discrete hypergroup and there exists $alphain widehat{K}$ which vanishes at infinity, then $L^1(K)$ is not amenable. In particular, $L^1(K)$ fails to be even $alpha$-left amenable if $pi_K({alpha})=0$.
Let $K$ denote a locally compact commutative hypergroup, $L^1(K)$ the hypergroup algebra, and $alpha$ a real-valued hermitian character of $K$. We show that $K$ is $alpha$-amenable if and only if $L^1(K)$ is $alpha$-left amenable. We also consider
the $alpha$-amenability of hypergroup joins and polynomial hypergroups in several variables as well as a single variable.
