On Intermediate $C^{\ast }$-subalgebras of $C^{\ast }$-simple Group Actions

We show that for a large class of actions $\Gamma \curvearrowright $\mathcal{A}$$of$C^$-simplegroups$\Gamma$onunital$C^$-algebras$$\mathcal{A}$$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^{\ast }$-subalgebra $\mathcal{B}$, $C_{\lambda}^*(\Gamma)\subseteq $\mathcal{B}$ \subseteq $\mathcal{A}$\rtimes_{r}\Gamma$,isoftheform$$\mathcal{A}$1\rtimes{r}\Gamma$, where ${\mathcal{A}}_1$ is a unital $\mathrm{\Gamma }$-$C^{\ast }$-subalgebra of $\mathcal{A}$.