On the spectrum G of a locally compact group G

Abstract

Separation properties of the Fell topology, on the spectrum G of a locally compact group G, characterize important properties of G. We will develop three equivalent ways to describe the Fell topology on the spectrum  of any C* algebra A. Specifically, we show that both the relative weak*-topology on P(A), the set of pure states of A, and the Jacobson topology on Prim(A), the set of all primative ideals on A, can be mapped onto  so that both topologies agree with the Fell topology. We will also study the correspondences, both between the set of strongly continuous unitary representations of G and the irreducible representations of the group C*-algebra G*(G), and between the continuous functions of positive type on G and the set of pure states of G*(G). As well, we give a survey of results outlining the characterization of G by simple separation properties of the Fell topology on G.