Noncommutative boundaries and the ideal structure of reduced crossed products

Abstract

A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenberg's notion of topological boundary for a group.

43 pages; revised

References in corpus (5)

• The ideal structure of reduced crossed products
• Uniformly recurrent subgroups and the ideal structure of reduced crossed products
• An intrinsic characterization of C*-simplicity
• Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles
• Unique expectations for discrete crossed products

Cited by in corpus (8)

• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Characterizing traces on crossed products of noncommutative C*-algebras
• A generalized Powers averaging property for commutative crossed products
• Partial generalized crossed products and a seven-term exact sequence
• The ideal intersection property for essential groupoid C*-algebras
• Partial Galois cohomology and related homomorphisms (expanded version)
• The third partial cohomology group and existence of extensions of semilattices of groups by groups
• Simplicity of crossed products by FC-hypercentral groups