Obstructions to a general characterization of graph correspondences

TL;DR

This paper investigates why the straightforward characterization of graph correspondences for countable discrete spaces does not extend to higher-rank and topological graphs, highlighting the obstructions encountered.

Contribution

It identifies fundamental obstructions preventing the extension of graph correspondence characterizations to higher-rank and topological graphs.

Findings

Characterizations hold for countable discrete spaces but fail for higher-rank graphs.

Obstructions arise due to structural complexities in product systems and topological settings.

Highlights limitations in current frameworks for classifying C*-correspondences.

Abstract

For a countable discrete space V, every nondegenerate separable C*-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*-correspondences) and for topological graphs (where V is locally compact Hausdorff), and we discuss the obstructions that arise.