Convolution operators on the dual of hypergroup algebras. (English) Zbl 1098.43001

Summary: Let \(X\) be a hypergroup. 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 Haar measure, then the dual to this subspace is \(L_C(X)^{**}=\text{cl}\{F\in L(X)^{**}; F\text{ has compact carrier}\}\). 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 \(\text{wap}(L(X))\) 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.


43A10 Measure algebras on groups, semigroups, etc.
43A62 Harmonic analysis on hypergroups
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