Simplicity of crossed products by FC-hypercentral groups

Abstract

In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A{⋊}_{\lambda }G$ is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely-generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every element has a finite conjugacy class. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to characterize when the crossed product $A{⋊}_{\lambda }G$ is prime.

arXiv comment53 pages, 2 figures; To appear in Compositio Mathematica. This is the accepted manuscript, which includes all changes requested by the referee. Some redundant or elementary results and proofs were cut and many theorem and section numbers got shifted as a result

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