C*-algebras associated with topological group quivers I: generators, relations and spatial structure

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 this paper, the author examines the Cuntz-Pimsner algebra ${\text{cO}}_{\alpha ,\beta }(\mathrm{\Gamma }):=C^{\ast }(Q_{\alpha ,\beta }(\mathrm{\Gamma }))\mathrm{.}$ The investigative topics include a notion for topological quiver isomorphisms, generators (and their relations) of the $C^{\ast }$-algebras ${\text{cO}}_{\alpha ,\beta }(\mathrm{\Gamma })$, and its spatial structure (i.e., colimits, tensor products and crossed products) and a few properties of its $C^{\ast }$-subalgebras.