C${}^{\ast }$-algebras of twisted étale groupoids and Cartan pairs

Abstract

This contribution provides an introduction to the theory of C${}^{\ast }$-algebras arising from twisted étale groupoids, with a particular emphasis on the role of Cartan subalgebras and their groupoid models. We present the construction of groupoid C${}^{\ast }$-algebras, the theory of twists and their reinterpretation as Fell line bundles, and the structure of Fell bundles over étale groupoids.

A central theme is the reconstruction of a twisted groupoid from a Cartan inclusion $A\subseteq B$, following the framework developed by Kumjian and Renault. Throughout the notes, we highlight the importance of effectiveness—rather than topological principality—as the key condition ensuring the existence of a Cartan subalgebra and the associated groupoid model.

We illustrate the theory with a range of examples, including graph C${}^{\ast }$-algebras, AF-algebras and their realization via Bratteli diagrams, coarse groupoids and uniform Roe algebras, and twisted crossed products by (partial) group actions. Motivational background from the Feldman–Moore theorem is also discussed.