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

TL;DR

This paper investigates the structure of intermediate $C^*$-algebras in dynamical systems, revealing existence conditions for non-trivial subalgebras in non-$C^*$-simple groups and providing a complete classification for irrational rotations of the circle.

Contribution

It characterizes when non-crossed-product intermediate $C^*$-algebras exist for non-$C^*$-simple groups and fully describes these subalgebras for irrational circle rotations.

Findings

Existence of intermediate subalgebras depends on invariant measures for non-$C^*$-simple groups.

Complete classification of subalgebras for irrational circle rotations.

Non-crossed-product subalgebras are characterized in specific dynamical settings.

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 $C^*$-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.