Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

Abstract

A free semigroupoid algebra is the weak operator topology closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a complete description of the weak operator topology closed ideal structure for these algebras. We prove a distance formula to ideals, and this gives an appropriate version of the Caratheodory interpolation theorem. Our analysis rests on an investigation of predual properties, specifically the $A_n$ properties for linear functionals, together with a general Wold Decomposition for $n$-tuples of partial isometries. A number of our proofs unify proofs for subclasses appearing in the literature.