\'Etale groupoids and their $C^*$-algebras

TL;DR

This paper provides an overview of $C^*$-algebras associated with étale groupoids, focusing on the representation and measure-theoretic analysis within a simplified setting to facilitate understanding.

Contribution

It offers a concise introduction to the theory of $C^*$-algebras from étale groupoids, emphasizing the case with reduced complexity for easier comprehension.

Findings

Summarizes key concepts in étale groupoid $C^*$-algebras

Highlights representation-theoretic techniques in a simplified setting

Prepares for further research in operator algebras and dynamics

Abstract

These notes were written as supplementary material for a five-hour lecture series presented at the Centre de Recerca Mathem\`atica at the Universitat Aut\`onoma de Barcelona from the 13th to the 17th of March 2017. The intention of these notes is to give a brief overview of some key topics in the area of $C^*$-algebras associated to \'etale groupoids. The scope has been deliberately contained to the case of \'etale groupoids with the intention that much of the representation-theoretic technology and measure-theoretic analysis required to handle general groupoids can be suppressed in this simpler setting. A published version of these notes will appear in the volume tentatively titled "Operator algebras and dynamics: groupoids, crossed products and Rokhlin dimension" by Gabor Szabo, Dana P. Williams and myself, and edited by Francesc Perera, in the series "Advanced Courses in Mathematics. CRM Barcelona." The pagination of this arXiv version is not identical to Birkh\"auser's style, but I have tried to make it close. The theorem numbering should be correct. I'm grateful to the CRM and Birkh\"auser for allowing me to post a version on arXiv.