Results from a few years ago of Kennedy and Schafhauser characterize simplicity of reduced crossed products $A \rtimes_\lambda G$, where $A$ is a unital C*-algebra and $G$ is a discrete group, under an assumption which they call vanishing obstruction. However, this is a strong condition that often fails, even in cases of $A$ being finite-dimensional and $G$ being finite.

In our work, we find the correct two-way characterization of when the crossed product is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups which have polynomial growth. With some additional effort, we also characterize the intersection property of $A \rtimes_\lambda G$ for the slightly less general class of FC-groups. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to generalize a result by Hamana for finite groups, and characterize when the crossed product $A \rtimes_\lambda G$ is prime.

All of our characterizations are initially given in terms of the dynamics of $G$ on $I(A)$, the injective envelope of $A$. This is a somewhat mysterious object that is not that easy to get a handle on in practice. If $A$ is separable, our characterization is shown to be equivalent to an intrinsic condition on the dynamics of $G$ on $A$ itself.

This is joint work with Shirly Geffen. The project was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 427320536 - SFB 1442, as well as under Germany's Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics-Geometry-Structure, and by the ERC Advanced Grant 834267 - AMAREC.