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

TL;DR

This paper introduces ordinal graphs, a new class of left cancellative categories, and studies their associated Cuntz-Krieger $ ext{C}^*$-algebras using generators, relations, and a constructed $ ext{C}^*$-correspondence, proving a uniqueness theorem.

Contribution

It defines ordinal graphs and develops a framework for their Cuntz-Krieger algebras, including a new $ ext{C}^*$-correspondence and a uniqueness theorem.

Findings

Constructed a $ ext{C}^*$-correspondence $X_eta$ for each ordinal $eta$

Proved a Cuntz-Krieger uniqueness theorem for ordinal graphs

Applied Ery"uzl"u and Tomforde's condition (S) to these algebras

Abstract

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