Nest representations of directed graph algebras

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra $\flgee$ of countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra $\T^{+}(G)$. We prove that the finite dimensional nest representations separate the points in $\flgee$, and a fortiori, in $\T^{+}(G)$. The irreducible finite dimensional representations separate the points in $\flgee$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in \V(G)$ supporting a cycle, $x$ also supports at least one loop edge. We also study \textit{faithful} nest representations. We prove that $\flgee$ (or $\T^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence a faithful nest representation for a maximal type $\bN$ nest.