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

TL;DR

This paper computes the K-groups of C*-algebras associated with topological group quivers, extending the understanding of their structure through exact sequences derived from the quivers' properties.

Contribution

It introduces a method to calculate K-groups of C*-algebras from topological group quivers using a six-term exact sequence, building on previous generator and relation analysis.

Findings

Derived a six-term exact sequence for K-group calculations.

Calculated K-groups for specific topological group quiver C*-algebras.

Extended the classification framework for C*-algebras associated with topological quivers.

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 $\cO_{\alpha,\beta}(\Gamma).$