Boundary maps and covariant representations

Abstract

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

17 pages. Minor changes. Accepted in the Bulletin of the LMS