Reconstructing a totally disconnected groupoid from its ample semigroup

TL;DR

This paper demonstrates that a second countable, etale, totally disconnected groupoid can be reconstructed from its ample semigroup, and explores the universal property of its C*-algebra related to tight representations.

Contribution

It establishes a method to reconstruct such groupoids solely from their ample semigroup and analyzes the universal property of their C*-algebras.

Findings

Reconstruction of groupoids from ample semigroups

Universal property of C*(G) related to tight representations

Applicable to non-Hausdorff, second countable groupoids

Abstract

We show that a (not necessarily Hausdorff) etale, second countable groupoid G with totally disconnected unit space may be reconstructed solely from the algebraic structure of its ample semigroup S. We also show that C*(G) possesses a universal property related to tight representations of S.