Topological freeness for $*$-commuting covering maps

Abstract

A countable family of $*$-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space $X$ gives rise to an action $\theta$ of a countably generated, free abelian monoid $P$. For such a triple $(X,P,\theta)$, which we call an irreversible $*$-commutative dynamical system, we construct a universal $C^*$-algebra $\mathcal{O}[X,P,\theta]$. Within this setting we show that the following four conditions are equivalent: $(X,P,\theta)$ is topologically free, $C(X) \subset \mathcal{O}[X,P,\theta]$ has the ideal intersection property, the natural representation of $\mathcal{O}[X,P,\theta]$ on $\ell^2(X)$ is faithful, and $C(X)$ is a masa in $\mathcal{O}[X,P,\theta]$. As an application, we characterise simplicity of $\mathcal{O}[X,P,\theta]$ by minimality of $(X,P,\theta)$. We also show that $\mathcal{O}[X,P,\theta]$ is isomorphic to the Cuntz-Nica-Pimsner algebra of a product system of Hilbert bimodules naturally associated to $(X,P,\theta)$. Moreover, we find a close connection between $*$-commutativity and independence of group endomorphisms, a notion introduced by Cuntz and Vershik. This leads to the observation that, for commutative irreversible algebraic dynamical systems of finite type $(G,P,\theta)$, the dual model $(\hat{G},P,\hat{\theta})$ is an irreversible $*$-commutative dynamical system and $\mathcal{O}[\hat{G},P,\hat{\theta}]$ is canonically isomorphic to $\mathcal{O}[G,P,\theta]$. This allows us to conclude that minimality of $(G,P,\theta)$ is not only sufficient, but also necessary for simplicity of $\mathcal{O}[G,P,\theta]$ if $(G,P,\theta)$ is commutative and of finite type.