Convolution operators on the dual of hypergroup algebras

Abstract

summary:Let $X$ be a hypergroup. In this paper, we define a locally convex topology $\beta$ on $L(X)$ such that $(L(X),\beta )^*$ with the strong topology can be identified with a Banach subspace of $L(X)^*$. We prove that if $X$ has a Haar measure, then the dual to this subspace is $L_C(X)^{**}= \operatorname{cl}\{F\in L(X)^{**}; F$ has compact carrier\}. Moreover, we study the operators on $L(X)^*$ and $L_0^\infty(X)$ which commute with translations and convolutions. We prove, among other things, that if $\operatorname{wap}(L(X))$ is left stationary, then there is a weakly compact operator $T$ on $L(X)^*$ which commutes with convolutions if and only if $L(X)^{**}$ has a topologically left invariant functional. For the most part, $X$ is a hypergroup not necessarily with an involution and Haar measure except when explicitly stated