C*-algebras associated with topological group quivers II: K-groups

Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^{\ast }$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^{\ast }(Q)\mathrm{.}$ Given $\mathrm{\Gamma }$ a locally compact group and $\alpha$ and $\beta$ endomorphisms on $\mathrm{\Gamma },$ one may construct a topological quiver $Q_{\alpha ,\beta }(\mathrm{\Gamma })$ with vertex set $\mathrm{\Gamma },$ and edge set $\Omega_{\alpha,\beta}(\Gamma)= {(x,y)\in\Gamma\times\Gamma\st \alpha(y)=\beta(x)}.$ In \cite{Mc1}, the author examined the Cuntz-Pimsner algebra ${\text{cO}}_{\alpha ,\beta }(\mathrm{\Gamma }):=C^{\ast }(Q_{\alpha ,\beta }(\mathrm{\Gamma }))$ and found generators (and their relations) of ${\text{cO}}_{\alpha ,\beta }(\mathrm{\Gamma })\mathrm{.}$ In this paper, the author uses this information to create a six term exact sequence in order to calculate the $K$-groups of ${\text{cO}}_{\alpha ,\beta }(\mathrm{\Gamma })\mathrm{.}$