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

Abstract

A separated graph is a pair ( E , C ) consisting of a directed graph E and a set C = ⨆ v ∈ E 0 C v , where each C v 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 1 C − A ( E , C ) , from Z ( C ) to Z ( E 0 ) . 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 K 1 ( π ) is a group isomorphism, and that K 0 ( 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 { ( E n , C n ) } 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 ⁡ ( 1 C − A ( E , C ) ) and K 1 ( C ⁎ ( E , C ) ) .