On the structure of hypergroups with respect to the induced topology

Abstract

Let $H$ be a hypergroup with left Haar measure and let $L^1(H)$ be the complex Lebesgue space associated with it. Let $L^{\infty }(H)$ 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 $L^{\infty }(H)$ into $\mathcal{ℬ}(L^1(H),L^{\infty }(H))$. We denote the strong operator topology and the weak operator topology on $L^{\infty }(H)$ by ${\tau }_{\text{so}}$ and ${\tau }_{\text{wo}}$. Unlike the uniform norm and weak topologies on $L^{\infty }(H)$, they depend essentially on the hypergroup structure of $H$. We derive that the ${\tau }_{\text{so}}$-topology is always different from the weak${}^{\ast }$-topology whenever $H$ is infinite. We can conclude that for a compact hypergroup $H$, $L^1(H)$ is the dual of $(L^{\infty }(H),{\tau }_{\text{so}})$. The properties of ${\tau }_{\text{so}}$ and ${\tau }_{\text{wo}}$ are then studied further and we pay attention to the ${\tau }_{\text{wo}}$-almost periodic elements of $L^{\infty }(H)$. Finally we give some further results about bounded linear operators which are ${\tau }_{\text{so}}$-${\tau }_{\text{so}}$-continuous.