$\mathcal{Z}$-stable Graph Algebras

Abstract

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.

18 pages, corrected a small error in the proof of Lemma 2.13 and added Remark 2.4. To appear in J. Oper. Theory