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

Abstract

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