In this paper we show that for a locally compact group G, the group algebra-Li(G) has nontrivial center if and only if G possesses a compact neighborhood of 1, invariant under inner automorphisms. Moreover, G has a basis of such neighborhoods at 1 if and only if Z-i(G) has an approximate identity consisting of functions in the center of L\. This constitutes part of a program of finding conditions on the group algebra which characterize groups satisfying various compactness conditions (see e.g., [3]).
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 3Cb of 5-characters to consist of the nonzero extreme points of the set of continuous positive-definite 5-invariant functions 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.