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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 $\cO_{\alpha,\beta}(\Gamma).$, Comment: 27 pages

Bibliography: p. 141-147, 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 r a locally compact group and a and (3 endomorphisms on r, one may construct a topological quiver Qae,,a(r) with vertex set r, and edge set n ,,8 (r) = { ( X ) y) E r X r I a(y) = (3( X)}. In this dissertation, the author examines the Cuntz-Pimsner algebra C* ( Q a,,B (r)). The investigative topics include generators of the C*-algebras, spatial structure (i.e., colimits, tensor products and crossed prod­ucts), K-groups, simplicity, and lattice properties.

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