The dual spaces of Banach algebras

Abstract

Introduction. A great deal of work has been done in the last fifteen years on the theory and classification of irreducible unitary representations of locally compact groups. It is also generally recognized (see for example [12, ? 8]) that, at least for some classes of groups, it is artificial to confine oneself to unitary representations; one ought to study the irreducible Banach space representations also. One cogent reason for this is that continuation of representations inevitably leads us outside the unitary domain. Thus one ought to try to extend to the case as much as possible of the theory developed for the unitary situation. The topology of the dual of C*-algebras and locally compact groups (that is, the of their irreducible unitary representations) has been studied in several papers (see [2], [3], [4], [6]). In the present paper we begin the study of the topology of the nonunitary dual spaces of algebras and groups. Two closely related motives suggest this: First, one would like to generalize the now classical theory of commutative Banach algebras to the noncommutative case even in the absence of an involution in particular to obtain a noncommutative version of the Banach-algebraic approach to the theory of analytic functions (see [ 14]). Secondly, one would like to use this to develop an abstract theory of continuation of group representations. Unfortunately, the space of all irreducible Banach space representations of an arbitrary Banach algebra or locally compact group seems too vast and unstructured to submit to a thorough analysis at present. We do not even know what we ought to mean by the equivalence of two such representations. Some happy of our subject-matter is called for. The work of Harish-Chandra and Godement [8] suggests what this restriction ought to be. Among Banach algebras A, it suggests that we should look first at those whose irreducible representations are all of bounded finite dimension, that is, whose dual are of bounded degree. Among locally compact groups, it suggests that we first examine groups G having a compact subgroup. The group algebra of such a group will have a large stock of subalgebras whose duals are of bounded degree; and the study of the irreducible Banach space representations of G reduces to that of the (finite-dimensional) irreducible representations of these subalgebras. The class of groups having compact subgroups includes all Euclidean