The co-universal C*-algebra of a row-finite graph

TL;DR

This paper introduces a new co-universal C*-algebra for row-finite graphs, generalizing graph C*-algebras and exploring its properties and conditions for injectivity of related homomorphisms.

Contribution

It constructs the co-universal C*-algebra C*_{min}(E) for row-finite graphs and characterizes when it coincides with the standard graph C*-algebra, expanding understanding of co-universality.

Findings

C*_{min}(E) exists with a co-universal property

C*_{min}(E) equals C*(E) when every loop has an entrance

Homomorphisms to C*_{min}(E) are injective under certain conditions

Abstract

Let E be a row-finite directed graph. We prove that there exists a C*-algebra C*_{min}(E) with the following co-universal property: given any C*-algebra B generated by a Toeplitz-Cuntz-Krieger E-family in which all the vertex projections are nonzero, there is a canonical homomorphism from B onto C*_{min}(E). We also identify when a homomorphism from B to C*_{min}(E) obtained from the co-universal property is injective. When every loop in E has an entrance, C*_{min}(E) coincides with the graph C*-algebra C*(E), but in general, C*_{min}(E) is a quotient of C*(E). We investigate the properties of C*_{min}(E) with emphasis on the utility of co-universality as the defining property of the algebra.