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

A {\it separated graph} is a pair $(E,C)$ consisting of a directed graph $E$ and a set $C={⨆}_{v\in E^0}{C}_v$, 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 $X\in C$ are finite, the K-theory of the graph C*-algebra $C^{\ast }(E,C)$ is known to be determined by the kernel and the cokernel of a certain map, denoted by $1_C-A_{(E,C)}$, from ${\mathbb{Z}}^{(C)}$ to ${\mathbb{Z}}^{(E^0)}$. In this paper, we compute the K-theory of the {\it tame} graph C*-algebra $\mathcal{O}(E,C)$ associated to $(E,C)$, which has been recently introduced by the authors. Letting $\pi$ denote the natural surjective homomorphism from $C^{\ast }(E,C)$ onto $\mathcal{O}(E,C)$, we show that $K_1(\pi )$ is a group isomorphism, and that $K_0(\pi )$ 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 $\{(E_n,C^n){\}}_{n=1}^{\mathrm{\infty }}$ 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 $\mathrm{k}\mathrm{e}\mathrm{r}(1_C-A_{(E,C)})$ and $K_1(C^{\ast }(E,C))$.