K-theory for the tame C^*-algebra of a separated graph

© 2015 Elsevier Inc.. A separated graph is a pair (E, C) consisting of a directed graph E and a set C=v∈E0Cv, where each Cv is a partition of the set of edges whose terminal vertex is v. Given a separated graph (E, C), such that all the sets X∈C are finite, the K-theory of the graph C*-algebra C*(E, C) is known to be determined by the kernel and the cokernel of a certain map, denoted by 1C-A(E,C), from Z(C) to Z(E0). In this paper, we compute the K-theory of the tame graph C*-algebra O(E,C) associated to (E, C), which has been recently introduced by the authors. Letting π denote the natural surjective homomorphism from C*(E, C) onto O(E,C), we show that K1(π) is a group isomorphism, and that K0(π) is a split monomorphism, whose cokernel is a torsion-free abelian group. We also prove that this cokernel is a free abelian group when the graph E is finite, and determine its generators in terms of a sequence of separated graphs {(En,Cn)}n=1∞ naturally attached to (E, C). On the way to showing our main results, we obtain an explicit description of a connecting map arising in a six-term exact sequence computing the K-theory of an amalgamated free product, and we also exhibit an explicit isomorphism between ker(1C-A(E,C)) and K1(C*(E, C)).