The local bisection hypothesis for twisted groupoid C*-algebras

TL;DR

This paper introduces a criterion called the local bisection hypothesis, linking the effectiveness of a groupoid to properties of normalisers in its twisted C*-algebra, with implications for understanding their structure.

Contribution

It establishes the equivalence between effectiveness of a groupoid and the local bisection hypothesis involving normalisers in the C*-algebra.

Findings

Normalisers in the reduced twisted groupoid C*-algebra are supported on open bisections.

The semigroup of normalisers is key to characterizing effectiveness.

The paper connects normalisers in cyclic group C*-algebras to the local bisection hypothesis.

Abstract

In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid $G$ being effective. One of these conditions is that $G$ satisfies the "C*-algebraic local bisection hypothesis"; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.