Representation theory of central topological groups

Abstract

By a central topological group we mean a group G such that G/Z is compact, where Z (or Z(G)) denotes the center of G. A locally compact group satisfying this condition will be called a [Z]-group, and [Z] denotes the class of these groups. In [7] we developed the structure theory of [Z]-groups; in particular, we showed that every [Z]-group G obeys the following Structure Theorem: G = V x H (direct product), where V is a vector group and H contains a compact open normal subgroup K(3). Both the structure theory and, as will be shown in this paper, the representation theory of [Z]-groups generalize and unify in a natural manner the corresponding theories for compact groups on the one hand and for locally compact abelian groups on the other. In fact, many of the features common to these two classes appear in their natural setting only when viewed as being characteristic of [Z]-groups. In addition, there are strong indications that [Z] marks the utmost degree of generality in which all these features are still present; not the least of these is the fact that the representation theory of [Z]-groups is essentially finite-dimensional. By contrast, that of the slightly larger class of [FIA]--groups is not; an [FIA]-group being a locally compact group G such that Z(G), the group of inner automorphisms, has compact closure in %(G), the group of all topological group automorphisms of G, in the natural topology. The class [FIA] was introduced by R. Godement [4]. (For a full discussion of the relation between [Z] and [FIA] -, see [7].) The finite-dimensionality of representations referred to above in combination with the compactness condition which defines [Z] allows us to obtain less general but considerably sharper results than those laid down by Godement in [6]. The present paper gives complete proofs of the results announced in Bull. Amer. Math. Soc. 72 (1966), 836-841, under the same title. The paper is organized as follows. After establishing the basic definitions and a number of technical results in ?1 we proceed to the proof of the fundamental fact that continuous irreducible unitary Hilbert space representations of [Z]-groups are finite-dimensional (Theorem 2.1) and we obtain an orthogonality relation for the