OBSTRUCTIONS TO A GENERAL CHARACTERIZATION OF GRAPH CORRESPONDENCES

Abstract

AbstractFor a countable discrete space $V$, every nondegenerate separable ${C}^{\ast } $-correspondence over ${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set $V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of ${C}^{\ast } $-correspondences) and for topological graphs (where $V$ is locally compact Hausdorff), and we discuss the obstructions that arise.