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∗-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 Qα,β(Γ) with ver-tex set Γ, and edge set Ωα,β(Γ) = {(x, y) ∈ Γ × Γ ∣∣α(y) = β(x)}. In this paper, the author examines the Cuntz-Pimsner algebra Oα,β(Γ): = C∗(Qα,β(Γ)). The investigative topics include a notion for topological quiver isomorphisms, gener-ators (and their relations) of the C∗-algebras Oα,β(Γ), and its spatial structure (i.e., colimits, tensor products and crossed products) and a few properties of its C∗-subalgebras