$C^{\ast }$-correspondences for ordinal graphs

Abstract

We introduce a family of $C^{\ast }$-correspondences $X_{\hat{I}}\pm$ naturally associated to every ordinal graph $\hat{I}›$. When $\hat{I}›$ is a directed graph, $X_0$ is isomorphic to the usual $C^{\ast }$-correspondence associated to a graph. We show that ordinal graphs satisfying a weak assumption have the property that the $C^{\ast }$-algebra of $\hat{I}{›}_{\hat{I}\pm +1}$ is isomorphic to the Cuntz-Pimsner algebra of $X_{\hat{I}}\pm$. As a consequence, the $C^{\ast }$-algebra of $\hat{I}›$ may be constructed starting from $c_0(\hat{I}{›}_0)$ by iteratively applying the Cuntz-Pimsner construction and inductive limits. We apply this result to strengthen the author's previous Cuntz-Krieger uniqueness theorem.

37 pages, 2 figures