Primality and the ideal intersection property for reduced crossed products
Matthew Kennedy, Larissa Kroell, Camila F. Sehnem
TL;DR
This work addresses primality and the ideal intersection property for reduced crossed products arising from C*-dynamical systems over discrete groups. It develops an intrinsic framework based on induction and imprimitivity at the level of injective envelopes, linking dynamical properties to ideal structures via boundary theory. The authors provide complete intrinsic characterizations for primality and the regular-ideal-intersection property, prove their equivalence for FC-hypercentral groups, and derive practical criteria using concepts like meandering projections and derivations. The results extend and unify prior work, yield explicit simplifications for groups with restricted subgroup structure such as PSL$_2(\mathbb{Z})$, SL$_2(\mathbb{Z})$, and free products of cyclics, and offer a robust toolkit for analyzing reduced crossed products in noncommutative dynamics.
Abstract
We consider the ideal structure of reduced crossed products over discrete groups. First, we completely characterize primality for reduced crossed products. Second, we characterize the ideal intersection property for reduced crossed products over FC-hypercentral groups. Both of these characterizations are intrinsic, in terms of conditions on the underlying dynamics. A key intermediate result is a complete characterization of the regular ideal intersection property for reduced crossed products. For C*-dynamical systems over groups with restrictive subgroup structure, these characterizations simplify even further, which we demonstrate with a number of examples.