$\mathcal{Z}$-stable Graph Algebras

Gregory Faurot

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arXiv · 2511.02760

$\mathcal{Z}$-stable Graph Algebras

Gregory Faurot

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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.

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BibTeX RIS

Gregory Faurot. 2025-11-04. $\mathcal{Z}$-stable Graph Algebras. https://arxiv.org/abs/2511.02760

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