Uniformly recurrent subgroups and the ideal structure of reduced crossed products

Abstract

We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $\Gamma$, we consider the absence of a non-zero ideals in the reduced crossed product $C(X){⋊}_r\Gamma$ which has a zero intersection with $C(X)$. We characterize this condition by a property for amenable subgroups of the stabilizer subgroups of $X$ in terms of the Chabauty space of $\Gamma$. This generalizes Kennedy's algebraic characterization of the simplicity for a reduced group ${\mathrm{C}}^{\ast }$-algebra of a countable discrete group.

arXiv comment16 pages

References in corpus (2)

• The ideal structure of reduced crossed products
• Subgroup dynamics and $C^{\ast }$-simplicity of groups of homeomorphisms

Cited by in corpus (9)

• Noncommutative boundaries and the ideal structure of reduced crossed products
• Crossed products of operator systems
• A generalized Powers averaging property for commutative crossed products
• Amenable uniformly recurrent subgroups and lattice embeddings
• The Furstenberg Boundary of a Groupoid
• On uniformly recurrent subgroups of finitely generated groups
• The ideal intersection property for essential groupoid C*-algebras
• On minimal actions of countable groups
• Uniformly recurrent subgroups and simple $C^{\ast }$-algebras