Simplicity of crossed products by FC-hypercentral groups
Shirly Geffen, Dan Ursu
TL;DR
This work provides a complete, two-way characterization of when reduced crossed products $A\rtimes_\lambda G$ are simple for FC-hypercentral $G$, using dynamics on the injective envelope $I(A)$ (and on $A$ itself in separable cases). The authors develop a unified framework based on pseudo-expectations and monotone-complete injective envelopes to translate between intrinsic $A$-dynamics and $I(A)$-dynamics, yielding precise conditions for simplicity, the intersection property, and primality across FC-groups, FC-hypercentral groups, and minimal actions. They further connect these high-level criteria to concrete, intrinsic descriptions in $A$ and demonstrate the transfer of prime/simplicity properties through $I(A)$ and $I_G(A)$. The paper also extends the theory to non-unital algebras and provides illustrative examples, including simple algebras, a finite-dimensional case, and cyclic-group actions, clarifying when standard outerness conditions suffice. Altogether, this advances the understanding of the ideal structure of noncommutative crossed products and supplies practical, dynamics-based criteria for key dynamical properties in broad dynamical settings.
Abstract
In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_λG$ 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 $A \rtimes_λG$ is prime.