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

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.