AF‐embeddable labeled graph C∗‐algebras

AF‐embeddability, quasidiagonality and stable finiteness of a C∗‐algebra have been studied by many authors and shown to be equivalent for certain classes of C∗‐algebras. The crossed products C(X)⋊σZ (by Pimsner) and AF⋊αZ (by Brown) are such classes, and recently Schfhauser proves the equivalence for C∗‐algebras of compact topological graphs. Clark, an Huef and Sims prove similar results for k‐graph algebras. In this paper, we show that this is the case for labeled graph C∗‐algebras C∗(E,L). Motivated by Schfhauser's result, we also provide another equivalent condition which is easy to check in terms of labeled paths when (E,L) is a labeled graph over a finite alphabet. As a corollary, we have that if C∗(E,L) is simple, then it is AF‐embeddable if and only if labeled edges have disjoint ranges.