Boundary maps and covariant representations

TL;DR

This paper extends Furstenberg boundary theory to analyze $C^*$-algebras from minimal group actions, introducing boundary maps and characterizing simplicity and intersection properties of these algebras.

Contribution

It introduces boundary maps on $( ext{Gamma},X)$-$C^*$-algebras and characterizes simplicity conditions for $C^*$-algebras generated by covariant representations.

Findings

Determines when $C^*$-algebras from stabilizer subgroups are simple.

Characterizes intersection property of $ ext{Gamma}$-spaces.

Links simplicity of crossed products to space properties.

Abstract

We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $\Gamma\!\curvearrowright\! X$ of discrete groups $\Gamma$ on locally compact spaces $X$. We introduce boundary maps on $(\Gamma,X)$-$C^*$-algebras and investigate their applications in this context. Among other results, we completely determine when $C^*$-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.