$\mathcal{Z}$-stable Graph Algebras

TL;DR

This paper establishes a divisibility condition for directed graphs that characterizes when the associated graph $C^*$-algebra is $ ext{Z}$-stable, providing a complete characterization for finite graphs and connecting to the Toms--Winter conjecture.

Contribution

It introduces a new divisibility condition for graphs that determines $ ext{Z}$-stability of the corresponding graph algebra, with complete characterization for finite graphs.

Findings

Divisibility condition is necessary for $ ext{Z}$-stability.

For graphs with no cycles or finitely many ideals, the condition is sufficient.

For finite graphs, the condition fully characterizes $ ext{Z}$-stability.

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.