On groupoids and $C^*$-algebras from self-similar actions

TL;DR

This paper explores the structure of $C^*$-algebras arising from self-similar groupoid actions on graph path spaces, generalizing Renault's similarity results and providing methods for computing their $K$-theory and homology.

Contribution

It introduces a generalized similarity result for groupoids and a strategy to compute $K$-theory and homology for associated $C^*$-algebras from self-similar actions.

Findings

Generalized Renault's similarity result for groupoids.

Proposed a method to compute $K$-theory of $C^*(G,E)$.

Illustrated the approach with a concrete example.

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.