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

作　　者：Ali Ghaffari Alireza Medghalchi 

机构地区：[1]Department of Mathematics,Semnan University,P.O.Box 35195-363,Semnan,Iran [2]Department of Mathematics,Teacher Training University,Tehran,Iran.

出　　处：《Acta Mathematica Sinica,English Series》2004年第2期201-208,共8页数学学报（英文版）

摘　　要：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.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.

关 键 词：Hypergroup algebras Group algebras OPERATORS TRANSLATIONS CONVOLUTIONS INVARIANT 

分 类 号：O152[理学—数学]