On the structure of hypergroups with respect to the induced topology

Abstract

Let H be a hypergroup with left Haar measure and let L1(H) be the complex Lebesgue space associated with it. Let L∞(H) be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed L∞(H) into ℬ(L1(H),L∞(H)). We denote the strong operator topology and the weak operator topology on L∞(H) by τso and τwo. Unlike the uniform norm and weak topologies on L∞(H), they depend essentially on the hypergroup structure of H. We derive that the τso-topology is always different from the weak∗-topology whenever H is infinite. We can conclude that for a compact hypergroup H, L1(H) is the dual of (L∞(H),τso). The properties of τso and τwo are then studied further and we pay attention to the τwo-almost periodic elements of L∞(H). Finally we give some further results about bounded linear operators which are τso-τso-continuous.