Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

Abstract For a locally compact group G, L ¹(G) is its group algebra and L ^∞(G) is the dual of L ¹(G). Lau has studied the bounded linear operators T : L ^∞(G) → L ^∞(G) which commute with convolutions and translations. For a subspace H of L ^∞(G), we know that M(L ^∞(G),H), the Banach algebra of all bounded linear operators on L ^∞(G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K)*, the dual of a hypergroup algebra L(K) in a very general setting, i. e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L ¹(G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results.