Harmonic analysis on central topological groups

Abstract

Introduction.In the present paper we continue and, in a sense, complete the investigations concerning [Z]-groups carried out in [8], [9], [10], and [11]; our notation and terminology is for the most part that employed in these papers.In particular, we utilize the following: ^"(A') and ^¡,(X) denote, resp., the continuous functions vanishing at co and the bounded functions on the locally compact Hausdorff space X, <ps is the characteristic function of Ss X. G is a locally compact group; M(G) and X(G) denote, resp., the measure algebra and set of (this time) normalized characters of the elements p of 0t,,4ß) ; here, 01(G) denotes continuous irreducible unitary representations (or their equivalence classes) and ^fln(G) those elements of 0t(G) which are finite-dimensional; in the case of a [Z]-group, they coincide.Mn(C) is the matrix algebra of order « over C. The symbol ~ denotes both topological closure and complex conjugation.The results used here as well as the methods of proof depend primarily on those of [9], and only familiarity with [9] is actually presupposed.However, an overall perspective of the subject is provided in the introduction to [10].As for harmonic analysis, another motivation for, and historical antecedent of, the present work is the program initiated by R. Godement in [3], [4], and [5], and carried out, in the more general case of [S77V]-groups, in [6].The great detail and explicitness of the present work is of course due to the special position of [Z]-groups within the class of [577V]-groups and, in fact, within all the classes of groups, satisfying compactness conditions, that have arisen to date.(For a full discussion of these classes see [10].)An excellent survey of the literature concerning harmonic analysis on groups, until rather recently, is provided in E. Hewitt [12], G. W. Mackey [17], [18], and I. E. Segal [24].The paper is organized as follows.§1 contains approximation theorems (Theorem (1.2) and Corollary (1.4)) related to and dependent on those of [9]; however, in the