Convolution operators on the dual of hypergroup algebras

Let X be a hypergroup. In this paper, we define a locally convex topology β on L(X) such that (L(X), β) 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 LC(X) ∗∗ = cl{F ∈ L(X);F has compact carrier}. Moreover, we study the operators on L(X) and L 0 (X) which commute with translations and convolutions. We prove, among other things, that if 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.