2013/11/04 by Nicolai Stammeier, Stammeier, Nicolai · 1 citation
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #math.DS #math.OA #msc:46L55
paper · pdf · doi:10.48550/arxiv.1311.0793
42 pages, replacing "C*-Algebras associated to certain semigroups of local homeomorphisms"
openalex publication_date 2013/11/04 · arxiv created 2014/11/17 · arxiv updated 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A countable family of *-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space X gives rise to an action θ of a countably generated, free abelian monoid P. For such a triple (X,P,θ), which we call an irreversible *-commutative dynamical system, we construct a universal C^*-algebra O[X,P,θ]. Within this setting we show that the following four conditions are equivalent: (X,P,θ) is topologically free, C(X) ⊂ O[X,P,θ] has the ideal intersection property, the natural representation of O[X,P,θ] on ℓ2(X) is faithful, and C(X) is a masa in O[X,P,θ]. As an application, we characterise simplicity of O[X,P,θ] by minimality of (X,P,θ). We also show that O[X,P,θ] is isomorphic to the Cuntz-Nica-Pimsner algebra of a product system of Hilbert bimodules naturally associated to (X,P,θ). 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,θ), the dual model (G,P,θ) is an irreversible *-commutative dynamical system and O[G,P,θ] is canonically isomorphic to O[G,P,θ]. This allows us to conclude that minimality of (G,P,θ) is not only sufficient, but also necessary for simplicity of O[G,P,θ] if (G,P,θ) is commutative and of finite type.