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

Abstract

Let $\Gamma$ be a countable group and $(X,\Gamma )$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^{\ast }$-subalgebra $\mathcal{A}$ $$C_r^{\ast }(\Gamma )<\mathcal{A}<C(X){⋊}_r\Gamma ,$$ which is not of the form $\mathcal{A}=C(Y){⋊}_r\Gamma$, corresponding to a factor map $(X,\Gamma )\to (Y,\Gamma )$. Here $C_r^{\ast }(\Gamma )$ and $C(X){⋊}_r\Gamma$ are the reduced $C^{\ast }$-algebras of $\Gamma$ and $(X,\Gamma )$ respectively. Our main results are (1) For $\Gamma$, which is not $C^{\ast }$-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.

arXiv commentAll the changes suggested by the referee have been implemented. There is a change in the title, which aptly describes the paper's contents. The paper has been accepted into the Journal of Functional Analysis