Uniqueness Theorems for Twisted Steinberg Algebras

TL;DR

This paper proves generalized uniqueness theorems for twisted Steinberg algebras associated with ample Hausdorff groupoids, extending known results and deriving a Cuntz-Krieger theorem as a special case.

Contribution

It establishes a unified framework of uniqueness theorems for twisted Steinberg algebras, including graded and Cuntz-Krieger types, for effective groupoids.

Findings

Established a generalized uniqueness theorem for twisted Steinberg algebras.

Derived a Cuntz-Krieger uniqueness theorem as a corollary for effective groupoids.

Proved a generalized graded uniqueness theorem for these algebras.

Abstract

Given an ample Hausdorff groupoid $G$, a unital commutative ring $R$, and a discrete twist $(\Sigma,i,q)$, we establish a generalised uniqueness theorem for the twisted Steinberg algebra $A_R(G;\Sigma)$. By applying this theorem when $G$ is effective, we establish a Cuntz-Krieger uniqueness theorem as a corollary. We also prove a generalised graded uniqueness theorem for $A_R(G;\Sigma)$.