Group Algebra Modules. I

Abstract

Some time ago, J. G. Wendel proved that the operators on the group algebra L1(G) which commute with convolution correspond in a natural way to the measure algebra M(G) (13). One might ask if Wendel's theorem can be restated in a more general setting. It is this question that is the point of departure for our present paper. Let K be a Banach module over L1(G). Our interest is in operators from L1(G) into K, and from K into L∞(G), which commute with the module composition (where L∞(G) is thought of as a module over L1(G) also). Such operators we call (L1(G), K)- and (K, L∞(G))-homomorphisms, respectively. Investigations of various other kinds of module homomorphisms occur in A. Figà-Talamanca (6) and B. E. Johnson (9; 10).