K-theory of C^*-algebras of directed graphs

2009/06/21 by Menassie Ephrem, Ephrem, Menassie, Jack Spielberg +1
Mathematics · Physics and Astronomy · #46L05 #46L35 #46L55 #Advanced Operator Algebra Research #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0906.3901

openalex publication_date 2009/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a directed graph E, we compute the K-theory of the C^*-algebra C^*(E) from the Cuntz-Krieger generators and relations. First we compute the K-theory of the crossed product C^*(E)×_γ\IT, and then using duality and the Pimsner-Voiculescu exact sequence we compute the K-theory of C^*(E)⊗\CK ≅ (C^*(E)×\IT)×\IZ. The method relies on the decomposition of C^*(E) as an inductive limit of Toeplitz graph C^*-algebras, indexed by the finite subgraphs of E. The proof and result require no special asssumptions about the graph, and is given in graph-theoretic terms. This can be helpful if the graph is described by pictures rather than by a matrix.

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