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Journal of Operator Theory

Volume 53, Issue 2, Spring 2005  pp. 273-302.

Ideal Structure In Free Semigroupoid Algebras From Directed Graphs

Authors Michael T. Jury (1) and David W. Kribs (2)
Author institution: (1) Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada

Summary:  A {\it 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 $\wot$-closed ideal structure for these algebras. We prove a distance formula to ideals, and this gives an appropriate version of the Carath\'{e}odory interpolation theorem. Our analysis rests on an investigation of predual properties, specifically the $\bbA_n$ 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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