Cartan semigroups and twisted groupoid C*-algebras

TL;DR

This paper characterizes twisted groupoid C*-algebras using Cartan semigroups, extending classical theory to more general and non-effective groupoids, thus broadening the understanding of their structure.

Contribution

It introduces Cartan semigroups as a new tool to classify twisted groupoid C*-algebras, generalizing existing theories to non-effective and non-reduced cases.

Findings

Twisted groupoid C*-algebras are characterized by Cartan semigroups.

Extension of Kumjian-Renault theory to non-effective groupoids.

Provides a classification framework for a broader class of C*-algebras.

Abstract

We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted \'etale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.