Reconstruction of twisted Steinberg algebras

TL;DR

This paper establishes a method to reconstruct discrete twists over ample Hausdorff groupoids from algebraic data, characterizing which twists can be recovered and demonstrating a bijective correspondence between algebraic pairs and twists.

Contribution

It introduces a novel reconstruction technique for twists from algebraic pairs and characterizes the class of twists satisfying the local bisection hypothesis.

Findings

Reconstruction of twists from algebraic pairs is possible for those satisfying the local bisection hypothesis.

The correspondence between twisted Steinberg algebras and twists is bijective.

Identification of algebraic pairs corresponding to effective and principal groupoids.

Abstract

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.