Cartan subalgebras of self-similar graph C*-algebras

Abstract

Self-similar graph C*-algebras, introduced by Exel and Pardo, generalize graph C*-algebras by encoding self-similar group actions on directed graphs. This class of C*-algebras admits natural groupoid models and is broad, covering Nekrashevych algebras and Katsura algebras (and hence UCT Kirchberg algebras). In joint work (WOA III) with Archey, Duwenig, McCormick, Norton, and Yang, we study Cartan subalgebras in self-similar graph C*-algebras beyond the “locally faithful” setting.

For finite source-free graphs, associated graph C*-algebras have Cartan subalgebras described either via the interior of isotropy of the path groupoid, or combinatorially through the so-called "cycline pairs" encoding the dynamics. We obtain analogous results for a large class of self-similar graph C*-algebras, producing Cartan subalgebras through the understanding of “cycline triples” based on dynamical data.