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 â 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 â 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.