Representations of twisted group algebras

Abstract

We construct a general class of Banach algebras which include as special cases the group algebra of a locally compact group, the group algebra of a group extension (in terms of the subgroup and quotient group), and some other examples, special cases of which have been studied under the name of covariance algebras.We develop the general representation theory and generalize Mackey's theory of induced representations.1. Introduction.In this paper we begin the investigation of a very general class of algebras of functions on locally compact groups taking values in a Banach algebra.A number of authors have investigated special cases of such algebras, notably G. P. Johnson [12], J. Glimm [8], E. Effros and F. Hahn [6], Turumaru [23], Zeller-Meier [26], I. Segal [20], and Doplicher, Kastler and Robinson [5].Of these, Zeller-Meier's algebras come closest in generality to those we discuss.He limits himself to discrete groups and C*-algebras, and in this context he formulates many of the important concepts.While a number of our results generalize his, our techniques are almost always different.Horst Leptin in [14] (see also [13] and [15]) defines an essentially equivalent class of algebras, however he confines himself to functions taking values in commutative algebras.Our construction differs from his in a number of technical details (we have tried to use concepts and notations more in agreement with previous work on special cases, and usual conventions), and there is very little overlap in subject matter.His representation theory and ours are distinct.Several authors have studied what are essentially representations of the algebras we discuss, but without defining the algebras.This is true of G. Mackey [16], [17], [18], M. Takesaki [22], Doplicher, Kastler and Robinson [5], and a number of other authors.We restrict our interest in the present paper to the representation theory of the algebras, leaving functorial considerations and their connection with the ideas centered around induced representations, as well as the detailed study of special cases and the structure theory to a later occasion.In §2, we introduce a few needed concepts and cite some results for later use.