2006/04/05 by Daniel Lenz, Lenz, Daniel
Mathematics · #18B35 #20M18 #22A22 #52C23 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric and Algebraic Topology #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.math/0604105
openalex publication_date 2006/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a construction, which assigns two groupoids, \Gugamma and \Gmgamma, to an inverse semigroup Γ. By definition, \Gmgamma is a subgroupoid (even a reduction) of \Gugamma. The construction unifies known constructions for groupoids. More precisely, the groupoid \Gugamma is shown to be isomorphic to the universal groupoid of Γ introduced by Paterson. For Γ arising from graphs resp. tilings, the groupoid \Gmgamma is the graph groupoid introduced by Kumjian et al. resp. the tiling groupoid introduced by Kellendonk. We obtain a characterisation of open invariant sets in \Gmgamma(0) in terms of certain order ideals of \Gammanull for a large class of Γ (including those arising from graphs and from tilings). If \Gmgamma is essentially principal this gives a characterization of the ideal structure of \Cred(\Gmgamma) by a theory of Renault. In particular, we then obtain necessary and sufficient conditions on Γ for simplicity of \Cred(\Gmgamma). Our approach relies on a detailed analysis of the order structure of Γ.