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

Abstract: 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 $C^*$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^*(Q).$ Given $\Gamma$ a locally compact group and $\alpha$ and $\beta$ endomorphisms on $\Gamma,$ one may construct a topological quiver $Q_{\alpha,\beta}(\Gamma)$ with vertex set $\Gamma,$ 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 $\cO_{\alpha,\beta}(\Gamma):=C^*(Q_{\alpha,\beta}(\Gamma))$ and found generators (and their relations) of $\cO_{\alpha,\beta}(\Gamma).$ In this paper, the author uses this information to create a six term exact sequence in order to calculate the $K$-groups of $\cO_{\alpha,\beta}(\Gamma).$