When universal edge-colored directed graph $C^*$-algebras are exact

TL;DR

This paper investigates the conditions under which universal $C^*$-algebras from edge-colored directed graphs are exact, establishing a precise equivalence with isomorphism to graph $C^*$-algebras for countable graphs.

Contribution

It provides a characterization of exactness for universal $C^*$-algebras of edge-colored directed graphs, linking it to isomorphism with graph $C^*$-algebras.

Findings

Universal $C^*$-algebra is exact iff it is isomorphic to a graph $C^*$-algebra.

Exactness is equivalent to the universal and reduced $C^*$-algebras being isomorphic.

The result applies specifically to countable edge-colored directed graphs.

Abstract

We consider when the universal $C^*$-algebras associated to edge-colored directed graphs are exact. Specifically, for countable edge-colored directed graphs we show that the universal $C^*$-algebra is exact if and only if the $C^*$-algebra is isomorphic to a graph $C^*$-algebra which occurs precisely when the universal and reduced $C^*$-algebras of the edge-colored directed graph are isomorphic.