$\mathcal{Z}$-stable Graph Algebras

We introduce a divisibility-type condition for directed graphs that is necessary for $\mathcal{Z}$-stability of the corresponding graph $C^{\ast }$-algebra. We prove that this condition is sufficient if either the graph $E$ has no cycles or the algebra $C^{\ast }(E)$ has finitely many ideals. Under the further assumption that $E$ is a finite graph, we provide a complete characterization of $\mathcal{Z}$-stability of $C^{\ast }(E)$. We conjecture that our divisibility condition and Condition (K) are equivalent to $\mathcal{Z}$-stability of the graph algebra. We prove that it is equivalent to $C^{\ast }(E)$ being pure, verifying the Generalized Toms–Winter Conjecture for graph algebras with finitely many ideals.