2025/11/04 by Gregory Faurot, Faurot, Gregory
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Spectral Theory in Mathematical Physics #math.OA #msc:46L05
paper · pdf · doi:10.48550/arxiv.2511.02760
18 pages, corrected a small error in the proof of Lemma 2.13 and added Remark 2.4. To appear in J. Oper. Theory
openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · arxiv created 2026/07/28 · arxiv updated 2026/07/30 · openalex updated_date 2026/08/01
We introduce a divisibility-type condition for directed graphs that is necessary for 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 Z-stability of C^*(E). We conjecture that our divisibility condition and Condition (K) are equivalent to 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.