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.

This research was supported by the Australian Research Council.