Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

TL;DR

This paper characterizes the ideal structure of free semigroupoid algebras generated by directed graphs, establishing lattice isomorphisms and a distance formula, thus advancing understanding of their algebraic and functional properties.

Contribution

It provides a complete description of the weak operator topology closed ideals in free semigroupoid algebras from directed graphs, including a new distance formula and interpolation results.

Findings

Lattice isomorphisms between ideals and invariant subspaces

A distance formula to ideals for these algebras

A version of the Caratheodory interpolation theorem

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.