Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups

Let $\Gamma$ be a countable group and $(X, \Gamma)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(\Gamma)<\mathcal{A}<C(X)\rtimes_r\Gamma,$$ which is not of the form $\mathcal{A} = C(Y) \rtimes_r \Gamma$, corresponding to a factor map $(X,\Gamma) \to (Y,\Gamma)$. Here $C^{*}_{r} (\Gamma)$ and $C(X) \rtimes_r \Gamma$ are the reduced $C^*$-algebras of $\Gamma$ and $(X,\Gamma)$ respectively. Our main results are (1) For $\Gamma$, which is not $C^*$-simple, if $(X,\Gamma)$ admits a $\Gamma$-invariant probability measure, then such a sub-algebra always exists. (2) For $\Gamma = \mathbb{Z}$ and $(X, \Gamma)$ an irrational rotation of the circle $X = S^1$, we give a full description of all these non-crossed-product subalgebras.