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

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 T, and then using duality and the Pimsner-Voiculescu exact sequence we compute the K-theory of C ^*(E) ⊗K ≅ (C^*(E) × T) × ℤ. 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 assumptions 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.