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 $\textit{free semigroupoid algebra}$ is the weak operator topology closed algebra generated by the left regular representation of a directed graph. $\newcommand{\wot}{\texttt{wot}}$ 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 Carathéodory 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.

Keywords: Hilbert space, Fock space, directed graph, partial isometry, nonselfadjoint operator algebra, partly free algebra, Wold Decomposition, distance formula, Carathéodory Theorem.

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