On Intermediate Subalgebras of C*-simple Group Actions

We show that for a large class of actions $\Gamma \curvearrowright \mathcal{A}$ of $C^*$-simple groups $\Gamma $ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-subalgebra $\mathcal{B}$, $C_{\lambda }^*(\Gamma )\subseteq \mathcal{B} \subseteq \mathcal{A}\rtimes _{r}\Gamma $, is of the form $\mathcal{A}_1\rtimes _{r}\Gamma $, where $\mathcal{A}_1$ is a unital $\Gamma $-$C^*$-subalgebra of $\mathcal{A}$.