Ideal Structure in Free Semigroupoid Algebras from Directed Graphs

Abstract

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.

32 pages, J. Operator Theory, to appear

References in corpus (2)

• Free Semigroupoid Algebras
• Factoring in Non-commutative Analytic Toeplitz Algebras

Cited by in corpus (22)

• Free Semigroupoid Algebras
• Classifying Higher Rank Analytic Toeplitz Algebras
• Applications of Automata and Graphs: Labeling Operators in Hilbert Space II
• Graph W*-probability Theory
• Finitely Correlated Representations of Product Systems of $C^{\ast }$-Correspondences over ${\mathbb{N}}^k$
• Random Variables in Graph W*-Probability Spaces
• Nest representations of directed graph algebras
• The $H^{\infty }$ algebras of higher rank graphs
• Interpolation in Semigroupoid Algebras
• Noncommutative point derivations for matrix function algebras
• Isomorphisms of algebras associated with directed graphs
• Compressed Random Variables in a Graph W*-Probability Space
• The C*-envelope of the tensor algebra of a directed graph
• Classification of Graph Fractaloids
• Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs
• Free Product Structure of a Graph W*-probability space
• Structure of free semigroupoid algebras
• Agler interpolation families of kernels
• Applications of the Wold decomposition to the study of row contractions associated with directed graphs
• Diagonal Compressed Random Variables in a Graph W*-probability Spaces
• The Moments of the Block Operators in a Group von Neumann Algebra
• Finite dimensional point derivations for graph algebras