Boundary maps and covariant representations

Abstract

We extend applications of Furstenberg boundary theory to the study of C ∗ $C^*$ -algebras associated to minimal actions Γ ↷ X $\Gamma \!\curvearrowright \!X$ of discrete groups Γ $\Gamma$ on locally compact spaces X $X$ . We introduce boundary maps on ( Γ , X ) $(\Gamma ,X)$ - C ∗ $C^*$ -algebras and investigate their applications in this context. Among other results, we completely determine when C ∗ $C^*$ -algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact Γ $\Gamma$ -spaces and simplicity of their associated crossed products.