On amenability of hypergroup algebras

In this paper, among the other things, we prove a variety of characterizations of amenable hypergroups. We give sufficient conditions and some necessary conditions for $H$ to have a left invariant mean. We know that every topologically left invariant mean on $L^\infty(H)$ is also left invariant mean. Sufficient conditions on a left invariant mean to be a topologically left invariant mean are given. One of the main results of this paper implies that, if $\varphi\in P^1(H)$ is left invariant mean, then $H$ is compact.