A C'-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal structure of the crossed product algebra is related to that of the original algebra. One consequence is that a minimal topologically free discrete system has a simple reduced crossed product. Sharper results are obtained when the algebra is abelian.1991 Mathematics subject classification: Primary 46L55.Let (A, G,a) be a C*-dynamical system with A abelian and G discrete. In [4] it was shown that the reduced crossed-product A x ^G is simple if G acts minimally and if each automorphism a,, t^e, is properly outer (see below for the definition). As a minimal action can be properly outer without being free, this was a significant advance from earlier results of Zeller-Meier and Effros-Hahn (see [9, Corollary 4.5] for naturally occurring examples). Elliott's result applies to actions of discrete groups on a larger class of C*-algebras than abelian ones. In [8] a general version is proved using a condition (apparently) stronger than proper outerness. In [7] the discrete abelian case is studied. A condition on the ideals of the reduced crossed product is presented which corresponds to the dynamical condition of proper outerness. This is used to give another proof of simplicity of reduced crossed-products.In this paper we present a somewhat simpler proof than that of [7] of the equivalence of these two conditions. In one direction, that the dynamical condition implies the condition on ideals, the argument is valid for arbitrary discrete systems. In fact, this implication is contained in the proof of [4] and [8] (using the apparently stronger condition of [8]); however, in the general case our argument is much simpler. We remark that no assumptions are made regarding separability, countability, or possession of an identity element. All ideal are assumed to be closed and two-sided.Much of the work described in this article was done while the second author was an SERC fellow at the University of Aberdeen. He wishes to thank SERC for its support, and the members of the Mathematics Department at the University of Aberdeen for their warm hospitality.

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AbstactA necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.

Let G be a second countable group, A be a separable C * -algebra with bounded trace and α a strongly continuous action of G on A. Suppose that the action of G on A induced by α is free and the G-orbits are locally closed. We show that the crossed product A × α G has bounded trace if and only if G acts integrably (in the sense of Rieffel and an Huef) on A. In the course of this, we show that the extent of non-properness of an integrable action gives rise to a lower bound for the size of the (finite) upper multiplicities of the irreducible representations of the crossed product. Mathematics Subject Classification (1991): 46L55