TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 163, January 1972 THE L1- AND C*-ALGEBRAS OF [FIA]ä GROUPS, AND THEIR REPRESENTATIONS BY RICHARD D. MOSAKO 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]ä 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 3Cbof 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 identifica- tion 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 funda- mental 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 repre- sentations. Introduction. The now classical theory of abstract harmonic analysis on com- pact groups and locally compact abelian groups furnishes detailed information on the structure of such groups, on their representations, and on their group algebras. The attempt to extend this theory to other classes of groups leads quickly to the study of groups with certain more general compactness conditions. The class of [Z] groups (locally compact groups G such that G/Z is compact, where Z is the center of G) was studied extensively by S. Grosser and M. Moskowitz in [11], [12], and [13]; they have shown that in this class all the essential characteristics of the Received by the editors July 6, 1970. AMS 1969 subject classifications. Primary 2260, 4680; Secondary 2265. Key words and phrases. Group and C*-algebras, structure space, #-operator, unitary representation, 5-character, spherical function, [FIA]£ group, [SIN]B group, central group, type I group. (*) This paper was written while the author held a National Science Foundation Graduate Fellowship. It constitutes part of the author's doctoral dissertation at Columbia University, written under Professor M. Moskowitz. Copyright © 1972, American Mathematical Society 277 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use