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On groupoids and $C^*$-algebras from self-similar actions (2012.14519v1)
Published 28 Dec 2020 in math.OA
Abstract: Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo \'etale groupoid ${\mathcal G}(G,E)$ and its $C*$-algebra $C*(G,E)$. We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the $K$-theory of $C*(G,E)$ and the homology of ${\mathcal G}(G,E)$ in certain cases and illustrate with an example.