Let $X$ be a compact Hausdorff $\Gamma$-space. We say that the corresponding crossed-product $C^*$-algebra $C(X)\rtimes_r\Gamma$ is reflecting when every intermediate $C^*$-algebra $C_r^*(\Gamma)\subset\mathcal{A}\subset C(X)\rtimes_r\Gamma$, which we consider a" non-commutative factor," is of the form $\mathcal{A}=C(Y)\rtimes_r\Gamma$, corresponding to a dynamical factor $X\to Y$. We establish (dynamical) sufficient conditions on $(X, \Gamma)$ under which $C(X)$ is reflecting and provide several examples where these sufficient conditions apply. It turns out that when our conditions are satisfied, the acting group \Gamma is necessarily $C^*$-simple. In the second part of the talk, we will examine the opposite aspect of the story when $\Gamma$ is not C^*-simple.

This talk is based on two recent joint works with Eli Glasner and Yair Glasner.