Narrow Results

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra . In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of can be determined from . We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

We show that the method to construct C*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C*-algebras constructed from them. We also give a characterization of our C*-algebras in terms of their representation theory.

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type $(2,0)$. We say that a graph $G$ satisfies a term equation $s\approx t$ if the corresponding graph algebra $\underline{A(G)}$ satisfies $s\approx t$. The set of all term equations $s\approx t$, which the graph $G$ satisfies, is denoted by $\mathrm{\text{Id}}(\{G\})$. The class of all graph algebras satisfy all term equations in $\mathrm{\text{Id}}(\{G\})$ is called the graph variety generated by $G$ denoted by ${{𝒱}}_g(\{G\})$. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation $s\approx t$ is called a linear term equation if $s$ and $t$ are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.