Nest Representations of Directed Graph Algebras

This paper is a comprehensive study of the nest representationsfor the free semigroupoid algebra L_G of a countable directedgraph G as well as its norm-closed counterpart, the tensor algebraT⁺(G). We prove that the finite-dimensional nest representations separatethe points in L_G, and a fortiori, in T⁺(G). The irreduciblefinite-dimensional representations separate the points in L_Gif and only if G is transitive in components (which is equivalentto being semisimple). Also the upper triangular nest representationsseparate points if and only if for every vertex x T(G) supportinga cycle, x also supports at least one loop edge. We also study faithful nest representations. We prove that L_G(or T⁺(G) admits a faithful irreducible representation if andonly if G is strongly transitive as a directed graph. More generally,we obtain a condition on G which is equivalent to the existenceof a faithful nest representation. We also give a conditionthat determines the existence of a faithful nest representationfor a maximal type N nest. 2000 Mathematics Subject Classification47L80, 47L55, 47L40.