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

TL;DR

This paper explores the structure and properties of C*-algebras derived from topological group quivers, focusing on generators, relations, and spatial structure, extending the theory of Cuntz-Pimsner algebras in a topological group context.

Contribution

It introduces a framework for analyzing C*-algebras associated with topological group quivers, including isomorphisms, generators, relations, and spatial structures, advancing the understanding of their algebraic and topological properties.

Findings

Characterization of topological quiver isomorphisms

Explicit generators and relations for the C*-algebras

Analysis of spatial structures like colimits and tensor products

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