2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103811We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph whose each weak connected component is a rooted (possibly infinite) directed tree, and the labelling map is almost proper, which is proved to be a reasonable generalization of the earlier case, then the crossed product can be embedded to an $AF$ algebra.Operator Algebras46L05;19K33text