  We investigate the tracial and ideal structures of $C^*$-algebras of
quasi-regular representations of stabilizers of boundary actions. Our main tool
is the notion of boundary maps, namely $\Gamma$-equivariant unital completely
positive maps from $\Gamma$-$C^*$-algebras to $C(\partial_F\Gamma)$, where
$\partial_F\Gamma$ denotes the Furstenberg boundary of a group $\Gamma$. For a
unitary representation $\pi$ coming from the groupoid of germs of a boundary
action, we show that there is a unique boundary map on $C^*_\pi(\Gamma)$.
Consequently, we completely describe the tracial structure of the
$C^*$-algebras $C^*_\pi(\Gamma)$, and for any $\Gamma$-boundary $X$, we
completely characterize the simplicity of the $C^*$-algebras generated by the
quasi-regular representations $\lambda_{\Gamma/\Gamma_x}$ associated to
stabilizer subgroups $\Gamma_x$ for any $x\in X$. As an application, we show
that the $C^*$-algebra generated by the quasi-regular representation
$\lambda_{T/F}$ associated to Thompson's groups $F\leq T$ does not admit traces
and is simple.
