AF-embeddable labeled graph $C^{\ast }$-algebras

Finiteness conditions for $C^{\ast }$-algebras like AF-embeddability, quasidiagonality, stable finiteness have been studied by many authors and shown to be equivalent for certain classes of $C^{\ast }$-algebras. For example, Schfhauser proves that these conditions are all equivalent for $C^{\ast }$-algebras of compact topological graphs, and similar results were established by Clark, an Huef, and Sims for $k$-graph algebras. If $C^{\ast }(E,\mathcal{L})$ is a labeled graph $C^{\ast }$-algebra over finite alphabet, it can be viewed as a $C^{\ast }$-algebra of a compact topological graph. For these labeled graph $C^{\ast }$-algebras, we provide conditions on labeled paths and show that they are equivalent to AF-embeddability of $C^{\ast }(E,\mathcal{L})$.