Acta Mathematica Sinica, English Series April, 2004, Vol.20, No.2, pp. 201–208 Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions Ali Ghaffari Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran Email: ghaffari 1380@yahoo.com Alireza Medghalchi Department of Mathematics, Teacher Training University, Tehran, Iran. 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. Keywords Hypergroup algebras, Group algebras, Operators, Translations, Convolutions, Invariant MR(2000) Subject Classification 43A10, 43A62 1 Introduction For a locally compact group G, L ∞ (G), the dual of L 1 (G), is a Banach algebra which is exactly characterized by essentially bounded measureable functions on G. The bounded linear operators on L ∞ (G) into L ∞ (G) which commute with convolutions and translations have been studied by Lau in [1] and by Pym and Lau in [2]. They also went further, and for several subspaces H of L ∞ (G), they have obtained a number of interesting and nice results. The aim of this paper is to go further and generalize the above results to very general hypergroup algebras which include not only group algebras but also most semigroup algebras as well. For a locally compact Hausdorff space K let M (K) be the Banach space of all bounded complex regular measures on K, so that M (K)= C 0 (K) ∗ which is a Banach algebra with convolution defined by μ ∗ ν (ψ)=  ψ(x ∗ y)dμ(x)dν (y)(ψ ∈ C 0 (K)) ([3–6]). Let L(K)= {μ|μ ∈ M (K),x -→ |μ|∗ δ k is continuous} which is defined ([7–8]) and studied by the second Received July 2, 2002, Accepted December 13, 2002