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

Benjamin Jones

TL;DR

The paper introduces ordinal graphs, a generalization of directed graphs built from left cancellative categories endowed with a length functor to the ordinal monoid. It develops a CK-type framework using C*-correspondences, culminating in a Cuntz-Krieger uniqueness theorem for the associated algebras $\mathcal{O}(\Lambda)$. By constructing a family of correspondences $X_{\alpha}$ and leveraging the Katsura ideals, the authors relate $\mathcal{O}(\Lambda)$ to a sequence of CK-like algebras and prove injectivity results under condition (S) and the absence of $1$-regular vertices. The results extend the graph/K-graph operator algebra theory to ordinal graphs, enabling analysis via infinite path spaces and exhaustive-sets techniques, and provide a path to CK-type uniqueness in this broad setting.

Abstract

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

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

TL;DR

The paper introduces ordinal graphs, a generalization of directed graphs built from left cancellative categories endowed with a length functor to the ordinal monoid. It develops a CK-type framework using C*-correspondences, culminating in a Cuntz-Krieger uniqueness theorem for the associated algebras . By constructing a family of correspondences and leveraging the Katsura ideals, the authors relate to a sequence of CK-like algebras and prove injectivity results under condition (S) and the absence of -regular vertices. The results extend the graph/K-graph operator algebra theory to ordinal graphs, enabling analysis via infinite path spaces and exhaustive-sets techniques, and provide a path to CK-type uniqueness in this broad setting.

Abstract

We introduce a class of left cancellative categories we call ordinal graphs for which there is a functor by which morphisms of factor. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. In particular, we construct a -correspondence for each in order to apply Eryüzlü and Tomforde's condition (S) and prove a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Paper Structure

This paper contains 8 sections, 43 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ordinal Graphs
  4. Exhaustive Sets
  5. Infinite Paths
  6. Correspondences
  7. Cuntz-Krieger Uniqueness
  8. Condition (S)

Key Result

Theorem 2.1

For every $\alpha>0$ and $\beta>1$ there exists unique $k\in\mathbb{N}$ such that $\alpha$ may be represented uniquely as where $\alpha_{n}\geq\alpha_{n+1}$ and $0<\gamma_{n}<\beta$.

Figures (5)

  • Figure 3.1: $\Lambda=\left[0,\omega^{2}\right)$ as an ordinal graph, where $v=0$, $e=1$, and $f=\omega$
  • Figure 3.2: The ordinal graph $\Lambda$ in \ref{['exa:long-path']}
  • Figure 4.1: $\Lambda$ as in \ref{['exa:two-loop-example']}
  • Figure 8.1: The ordinal graph $\Lambda$ in \ref{['exa:two-loops-two-omega']}
  • Figure 8.2: A connected component of $\mathcal{F}_{\alpha}$ for $\Lambda$ defined in \ref{['exa:complicated']}

Theorems & Definitions (86)

  • Theorem 2.1: CARDINALORDINAL
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • Definition 3.7
  • Proposition 3.8
  • Definition 3.9
  • ...and 76 more