The $L\sp{1}$- and $C\sp{\ast} $-algebras of $[FIA]\sp{-}\sb{B}$\ groups, and their representations

Abstract

Let G be a locally compact group, and B a subgroup of the (topologized) group Aut (G) of topological automorphisms of G; G is an [FIA] group if B has compact closure in Aut (G). Abelian and compact groups are [FIA] groups, with B=I(G); the purpose of this paper is to generalize certain theorems about the group algebras and representations of these familiar groups to the case of general [FIA] groups. One defines the set 3Cb of 5-characters to consist of the nonzero extreme points of the set of continuous positive-definite 5-invariant functions <f> on G with <^(1)^1. 3cb is naturally identified with the set of pure states on the subalgebra of .B-invariant elements of C*(G). When this subalgebra is commutative, this identification yields generalizations of known duality results connecting the topology of G with that of G. When B=I(G), Xb can be identified with the structure spaces of C*(G) and LHG), and one obtains thereby information about representations of G and ideals in Z-HG). When G is an [FIA]b group, one has under favorable conditions a simple integral formula and a functional equation for the 5-characters. LHG) and C*(G) are "semisimple" in a certain sense (in the two cases B = (l) and B = I(G) this "semisimplicity" reduces to weak and strong semisimplicity, respectively). Finally, the ^-characters have certain separation properties, on the level of the group and the group algebras, which extend to [S1N]B groups (groups which contain a fundamental system of compact fl-invariant neighborhoods of the identity). When B -I(G) these properties generalize known results about separation of conjugacy classes by characters in compact groups; for example, when B={\) they reduce to a form of the Gelfand-Raikov theorem about "sufficiently many" irreducible unitary representations.