Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$C^*$-correspondences for ordinal graphs

Benjamin Jones

Abstract

We introduce a family of $C^*$-correspondences $X_α$ naturally associated to every ordinal graph $Λ$. When $Λ$ is a directed graph, $X_0$ is isomorphic to the usual $C^*$-correspondence associated to a graph. We show that ordinal graphs satisfying a weak assumption have the property that the $C^*$-algebra of $Λ_{α+ 1}$ is isomorphic to the Cuntz-Pimsner algebra of $X_α$. As a consequence, the $C^*$-algebra of $Λ$ may be constructed starting from $c_0(Λ_0)$ by iteratively applying the Cuntz-Pimsner construction and inductive limits. We apply this result to strengthen the author's previous Cuntz-Krieger uniqueness theorem.

$C^*$-correspondences for ordinal graphs

Abstract

We introduce a family of -correspondences naturally associated to every ordinal graph . When is a directed graph, is isomorphic to the usual -correspondence associated to a graph. We show that ordinal graphs satisfying a weak assumption have the property that the -algebra of is isomorphic to the Cuntz-Pimsner algebra of . As a consequence, the -algebra of may be constructed starting from by iteratively applying the Cuntz-Pimsner construction and inductive limits. We apply this result to strengthen the author's previous Cuntz-Krieger uniqueness theorem.
Paper Structure (6 sections, 44 theorems, 163 equations, 2 figures)

This paper contains 6 sections, 44 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ordinal Graphs
  4. $C^{*}$-correspondences
  5. Main Results
  6. Inductive Step

Key Result

Theorem 2.1

Every $\alpha\in\mathrm{Ord}$ may be expressed uniquely as for $n,\gamma_{k}\in[0,\omega)$ and $\beta_{k}\in\mathrm{Ord}$ satisfying $\beta_{1}\geq\beta_{2}\geq\dots\geq\beta_{n}$.

Figures (2)

  • Figure 4.1: A category generated by two objects $v,w$, two morphisms $e,f$, and a morphism $g$ such that $g=efg$
  • Figure 4.2: A directed graph $F$ with vertices $v'$, $w'$ and edges $f'$, $g'$

Theorems & Definitions (108)

  • Theorem 2.1: Cantor Normal Form CARDINALORDINAL
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: KATIDEAL2
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2: ORDGRAPH
  • Definition 3.3: ORDGRAPH
  • ...and 98 more