Nest representations of directed graph algebras

TL;DR

This paper thoroughly investigates nest representations of graph algebras, establishing conditions for their separation of points, faithfulness, and irreducibility based on graph properties like transitivity and cycles.

Contribution

It characterizes when finite dimensional, irreducible, and faithful nest representations exist for graph algebras, linking these to specific graph transitivity and cycle conditions.

Findings

Finite dimensional nest representations separate points in the algebra.

Irreducible finite dimensional representations separate points iff the graph is transitive in components.

Existence of faithful irreducible representations is equivalent to the graph being strongly transitive.

Abstract

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.