The Ideal Structure of Reduced Crossed Products

Let (A, G) be a C*-dynamical system with G discrete. In this paper we investigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient – and in some cases also necessary – conditions for A to separate the ideals in A⋊r G. When A separates the ideals in A ⋊r G, then there is a one-to-one correspondence between the ideals in A⋊r G and the invariant ideals in A. We extend the concept of topological freeness and present a generalization of the Rokhlin property. Exactness properties of (A, G) turns out to be crucial in these investigations. In this paper we examine conditions on a C*-dynamical system (A,G) with G discrete assuring that A separates the ideals in the reduced crossed product A⋊r G, i.e. when the (surjective) map J 7→ J ∩A, from the ideals in A⋊r G into the invariant ideals in A, is injective. Simplicity of A⋊rG obviously implies that A separates the ideas in A⋊rG. Some of the first results about simplicity of crossed products go back to works of Effros-Hahn [7] and Zeller-Meier [22]. For results restricted to abelian or Powers groups, we refer to [14] and [6]. Elliott showed in [8] that A ⋊r G is simple, provided that A is an AF-algebra and the action is minimal and properly outer. Kishimoto showed in [13] that the reduced crossed product of a C*-algebra A by a discrete group G is simple if the action is minimal and fulfills the strong Connes spectrum condition. Archbold and Spielberg generalized the result of Elliott by introducing topological freeness. The action of G on A is called topologically free if ⋂ t∈F {x ∈  : t.x 6= x} is dense in the spectrum  of A ( ) for any finite subset F ⊆ G \ {e}. They show in [1] that if the action is minimal and topologically free, then the reduced crossed product A ⋊r G is simple. For (A,G) with A abelian, the notion of topological freeness and proper outerness coincide, cf. [1]. We say that an action of G on A has the intersection property if every nontrivial ideal in A⋊rG has a non-trivial intersection with A. The intersection property is a necessary condition to ensure that A separates the ideals in A ⋊r G. Kawamura and Tomiyama showed in [11] that if A is abelian and unital and G is amenable, then topological freeness of G acting on A is equivalent to the intersection property. There is also some recent work of Svensson and Tomiyama on a related ideal intersection property in the case where A = C(X) and G = Z, cf. [19]. Archbold and Spielberg have made a 1 Here b A denotes the unitary equivalence classes of irreducible representations equipped with the not necessarily separated topology induced be the natural surjection onto the T0 space Prim(A). 1