Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

Details Ideal Structure in Free Semigroupoid Algebras from Directed Graphs Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

2003-09-24

arxiv.org/abs/math/0309397v1

J. Operator Theory, 53 (2005), 273-302.

Mathematics

Operator Algebras

32 pages, J. Operator Theory, to appear

Scientific paper

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.

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