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 $Γ$ a locally compact group and $α$ and $β$ endomorphisms on $Γ,$ one may construct a topological quiver $Q_{α,β}(Γ)$ with vertex set $Γ,$ and edge set $Ω_{α,β}(Γ)= \{(x,y)\inΓ\timesΓ\st α(y)=β(x)\}.$ In \cite{Mc1}, the author examined the Cuntz-Pimsner algebra $\cO_{α,β}(Γ):=C^*(Q_{α,β}(Γ))$ and found generators (and their relations) of $\cO_{α,β}(Γ).$ In this paper, the author uses this information to create a six term exact sequence in order to calculate the $K$-groups of $\cO_{α,β}(Γ).$