Characterization of operators on the dual of hypergroups which commute with translations and convolutions (Q1887413)

In the context of locally compact groups the bounded linear operators on \( L^{\infty }(G)\) (identified with \(L^{1}(G)^{\ast }\)) into a subspace of \( L^{\infty }(G)\) (and, particularly into \(L^{\infty }(G)\)) which commute with convolutions and translations have been studied by \textit{A. To-Ming Lau} [Colloq. Math. 39, 351--359 (1978; Zbl 0411.47025)] and \textit{A. To-Ming Lau} and \textit{J. S. Pym} [J. Lond. Math. Soc. 41, 445--460 (1990; Zbl 0667.43004)]. The aim of the present paper is to generalize the study of such kind of operators to \(L(K)^{\ast }\), the dual of a hypergroup algebra. We notice that the hypergroup \(K\) is not supposed to have a Haar measure. Among other things, it is shown that a necessary and sufficient condition that \(T\in \text{Conv} (L(K)^{\ast })=\{T| T:L(K)^{\ast }\rightarrow L(K)^{\ast }\), \(T\) a bounded linear map and \(T(f\mu )=T(f)\mu \) for \(\mu \in L(K)\), \(f\in L(K)^{\ast }\}\) be weak\(^{\ast }\)-weak\(^{\ast }\) continuous is that \(K\) is compact and a structure result is obtained regarding the operators in \(\Hom (L(K)^{\ast })=\{T| T:L(K)^{\ast }\rightarrow L(K)^{\ast }\), \(T\) a bounded linear map and \(T(f\delta _{x})=T(f)\delta _{x}\) for \(x\in K\), \(f\in L(K)^{\ast }\}\) which are weak\(^{\ast }\)-weak\(^{\ast }\) continuous.