Abstract

Let $E$ be a rowfinite directed graph. We prove that there exists a $C^*$algebra $\Cr{E}$ with the following couniversal property: given any $C^*$algebra $B$ generated by a ToeplitzCuntzKrieger $E$family in which all the vertex projections are nonzero, there is a canonical homomorphism from $B$ onto $\Cr{E}$. We also identify when a homomorphism from $B$ to $\Cr{E}$ obtained from the couniversal property is injective. When every loop in $E$ has an entrance, $\Cr{E}$ coincides with the graph $C^*$algebra $C^*(E)$, but in general, $\Cr{E}$ is a quotient of $C^*(E)$. We investigate the
properties of $\Cr{E}$ with emphasis on the utility of couniversality as the defining property of the algebra.