$\mathbb{Z}$-graded rings as Cuntz-Pimsner rings

TL;DR

This paper investigates conditions under which $Z$-graded rings, including Steinberg algebras of graded groupoids, can be represented as Cuntz-Pimsner rings, linking graded ring theory with operator algebra constructions.

Contribution

It provides sufficient conditions for realizing $Z$-graded rings and Steinberg algebras as Cuntz-Pimsner rings of certain $R$-systems, expanding the understanding of their structure.

Findings

Derived conditions for $Z$-graded rings to be Cuntz-Pimsner rings.

Applied these conditions to Steinberg algebras of graded groupoids.

Established a link between graded ring structures and Cuntz-Pimsner constructions.

Abstract

Given a $\mathbb{Z}$-graded ring $A$ and a subring $R\subseteq A$, it is natural to ask whether $A$ can be realised as the Cuntz-Pimsner ring of some $R$-system. In this paper, we derive sufficient conditions on $A$ and $R$ for this to be the case. As an application, we give conditions under which the Steinberg algebra $A_K(\mathcal{G})$ associated to a $\mathbb{Z}$-graded groupoid $\mathcal{G}=\sqcup_{n\in \mathbb{Z}} \mathcal{G}_n$ can be realised as the Cuntz-Pimsner ring of an $A_K(\mathcal{G}_0)$-system.