Ordinal graphs and their ${\mathrm{C}}^{\ast }$-algebras

Abstract

We introduce a class of left cancellative categories we call ordinal graphs for which there is a functor $d:\hat{I}›\to \mathrm{Ord}$ by which morphisms of $\hat{I}›$ factor. We use generators and relations to study the Cuntz-Krieger algebra $\mathcal{O}\left(\hat{I}›\right)$ defined by Spielberg. In particular, we construct a ${\mathrm{C}}^{\ast }$-correspondence $X_{\hat{I}}\pm$ for each $\hat{I}\pm \in \mathrm{Ord}$ in order to apply Eryüzlü and Tomforde's condition (S) and prove a Cuntz-Krieger uniqueness theorem for ordinal graphs.

31 pages, 5 figures. v2 - Fixed typos, generalized condition (S), clarified abstract