$\mathcal{Z}$-stable Graph Algebras
Gregory Faurot
TL;DR
The paper introduces a divisibility-type combinatorial condition called distinct detours on directed graphs and proves it is necessary for $\mathcal{Z}$-stability of the associated graph C*-algebra $C^*(E)$; under either the absence of cycles (AF case) or the condition of finitely many ideals, this condition is also sufficient. It then extends the analysis to graphs with finitely many ideals and to finite graphs, providing a complete characterization of $\mathcal{Z}$-stability in those regimes and connecting it to purity. The results use corner and central sequence techniques, along with in-tree constructions and Drinen--Tomforde desingularization, to build embeddings that certify $\mathcal{Z}$-stability via the generalized Toms--Winter framework. A central conjecture proposed is that Condition (K) together with distinct detours is equivalent to $\mathcal{Z}$-stability for graph algebras, which would align finite nuclear dimension with $\mathcal{Z}$-stability in these cases and validate the generalized Toms--Winter conjecture for graph algebras with finitely many ideals.
Abstract
We introduce a divisibility-type condition for directed graphs that is necessary for $\mathcal{Z}$-stability of the corresponding graph $C^*$-algebra. We prove that this condition is sufficient if either the graph $E$ has no cycles or the algebra $C^*(E)$ has finitely many ideals. Under the further assumption that $E$ is a finite graph, we provide a complete characterization of $\mathcal{Z}$-stability of $C^*(E)$. 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 $C^*(E)$ being pure, verifying the Generalized Toms--Winter Conjecture for graph algebras with finitely many ideals.