Free Semigroupoid Algebras from Categories of Paths

Given a directed graph G, we can define a Hilbert space HG with basis indexed by the path space of the graph, then represent the vertices of the graph as projections on HG and the edges of the graph as partial isometries on HG. The weak operator topology closed algebra generated by these projections and partial isometries is called the free semigroupoid algebra for G. Kribs and Power showed that these algebras are reflexive, and that they are semisimple if and only if each path in the graph lies on a cycle. We extend the free semigroupoid algebra construction to categories of paths, which are a generalization of graphs, and provide examples of free semigroupoid algebras from categories of paths that cannot arise from graphs (or higher rank graphs). We then describe conditions under which these algebras are semisimple, and we prove reflexivity for a class of examples