Structure of free semigroupoid algebras

TL;DR

This paper develops a comprehensive structure theory for free semigroupoid algebras, including decompositions, reflexivity, and classification, extending concepts from free semigroup algebra theory to graph-based operator algebras.

Contribution

It introduces a structure theory for free semigroupoid algebras, including Lebesgue-von Neumann-Wold decomposition and classification results, advancing the understanding of these operator algebras.

Findings

Established a Lebesgue-von Neumann-Wold decomposition for TCK families

Proved reflexivity and a Kaplansky density theorem for free semigroupoid algebras

Classified atomic free semigroupoid algebras up to unitary equivalence

Abstract

A free semigroupoid algebra is the closure of the algebra generated by a TCK family of a graph in the weak operator topology. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clarify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence.