When universal edge-colored directed graph $C^{\ast }$-algebras are exact

Abstract

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

14 pages, a complete revision which corrects errors in Propositions 1, 3, and 5 and throughout subsection 5.1 of the previous version. The errors in Propositions 3 and 5 were pointed out by an anonymous referee whose comments also significantly improved the exposition. Comments welcome

References in corpus (1)

• A direct approach to co-universal algebras associated to directed graphs

Cited by in corpus (1)

• Exchange rings and real rank zero C*-algebras associated with finitely separated graphs