Summary: We prove simplicity of all intermediate \(C^{\ast}\)-algebras \(C_r^{\ast}(\Gamma)\subseteq\mathcal{B}\subseteq\Gamma\ltimes_rC(X)\) in the case of minimal actions of \(C^{\ast}\)-simple groups \(\Gamma\) on compact spaces \(X\). For this, we use the notion of stationary states, recently introduced by Y. Hartman and M. Kalantar [“Stationary \(C^{\ast}\)-dynamical systems”, Preprint, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product \(\Gamma\ltimes_r\mathcal{A}\) for any action \(\Gamma\curvearrowright\mathcal{A}\) of a \(C^{\ast}\)-simple group \(\Gamma\) on a unital \(C^{\ast}\)-algebra \(\mathcal{A}\), and use it to prove a one-to-one correspondence between stationary states on \(\mathcal{A}\) and those on \(\Gamma\ltimes_r\mathcal{A}\).