Topological freeness for $\ast$-commuting covering maps

Abstract

A countable family of $\ast$-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 $\ast$-commutative dynamical system, we construct a universal $C^{\ast }$-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 $\ast$-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 $\ast$-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.

42 pages, replacing "C*-Algebras associated to certain semigroups of local homeomorphisms"

References in corpus (3)

• The ideal structure of reduced crossed products
• C*-algebras associated to product systems of Hilbert bimodules
• On C*-algebras of irreversible algebraic dynamical systems

Cited by in corpus (7)

• The boundary quotient for algebraic dynamical systems
• Topological aperiodicity for product systems over semigroups of Ore type
• On C*-algebras of irreversible algebraic dynamical systems
• The inner structure of boundary quotients of right LCM semigroups
• Algebraic actions I. C*-algebras and groupoids
• KMS states on $C^{\ast }$-algebras associated to a family of $\ast$-commuting local homeomorphisms
• On Cohomology for Product Systems