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

We introduce a family of $C^{\ast }$-correspondences $X_{\alpha }$ naturally associated to every ordinal graph $\mathrm{\Lambda }$. When $\mathrm{\Lambda }$ 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 ${\mathrm{\Lambda }}_{\alpha +1}$ is isomorphic to the Cuntz-Pimsner algebra of $X_{\alpha }$. As a consequence, the $C^{\ast }$-algebra of $\mathrm{\Lambda }$ may be constructed starting from $c_0({\mathrm{\Lambda }}_0)$ by iteratively applying the Cuntz-Pimsner construction and inductive limits. We apply this result to strengthen the author’s previous Cuntz-Krieger uniqueness theorem.