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

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 $\mathcal{G}={\bigsqcup }_{n\in \mathbb{Z}}{\mathcal{G}}_n$ can be realised as the Cuntz-Pimsner ring of an $A_K({\mathcal{G}}_0)$-system.

16 pages, fixed a small number of typographical errors. arXiv admin note: text overlap with arXiv:0810.3254 by other authors