A free semigroupoid algebra is the WOT-closure of the algebra generated by a TCK family of a graph. 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.
Bibliographical noteFunding Information:
The first author was partially supported by a grant 2018-03973 from NSERC.The second author was partially supported by an Ontario Trillium Scholarship.
© 2019 Elsevier Inc.
- Free semigroupoid algebras
- Graph algebras
- Road colouring
- Wandering vectors and absolute continuity
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