Nest Representations of Directed Graph Algebras

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra LG 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 LG, and a fortiori, in T (G). The irreducible finite dimensional representations separate the points in LG 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 ∈ V(G) supporting a cycle, x also supports at least one loop edge. We also study faithful nest representations. We prove that LG (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 N nest.