Boundary maps and covariant representations

We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $Γ\!\curvearrowright\! X$ of discrete groups $Γ$ on locally compact spaces $X$. We introduce boundary maps on $(Γ,X)$-$C^*$-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 $Γ$-spaces and simplicity of their associated crossed products.

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