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

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 $C^{\ast }$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^{\ast }(Q).$ 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 $Ω_{α,β}(Γ)= \{(x,y)\inΓ\timesΓ\st α(y)=β(x)\}.$ In this paper, the author examines the Cuntz-Pimsner algebra $\cO_{α,β}(Γ):=C^*(Q_{α,β}(Γ)).$ The investigative topics include a notion for topological quiver isomorphisms, generators (and their relations) of the $C^{\ast }$-algebras $\cO_{α,β}(Γ)$, and its spatial structure (i.e., colimits, tensor products and crossed products) and a few properties of its $C^{\ast }$-subalgebras.

40 pages

References in corpus (2)

• Adding tails to C*-correspondences
• Graph C*-algebras and Z/2Z-quotients of quantum spheres

Cited by in corpus (2)

• C*-algebras associated with topological group quivers II: K-groups
• Simplicity criterion for $C^{\ast }$-algebras associated with topological group quivers