The ideal structure of reduced crossed products

By Adam Sierakowski. 2008

63 citations recorded by the source corpus.

Let (A, G) be a C*-dynamical system with G discrete. In this paper we in- vestigate 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 ⋊r G obviously implies that A separates the ideas in A ⋊r G. Some of the first results about simplicity of crossed products go back to works of Effros-Hahn (7) and Zeller-Meier (23). For results restricted to abelian or Powers groups, we refer to (15) 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 topo- logical freeness. The action of G on A is called topologically free if T t∈F {x ∈ b A : t.x 6 x} is dense in the spectrum b

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55d2fbebf7386a2c20ad1f07ead1aece2acfe3ba

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Purely infinite C*-algebras arising from crossed products (2010, background): “It was shown in [29] that A separates the ideals in A or G if G is a discrete group, the action of G on A is exact, and the action of G on Â is essentially free.”
Nuclear dimension and Z-stability of non-simple C*-algebras (2013, background): “[32] for example); since C(X) has the ideal property, it follows that A does as well.”