  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.
