Free Semigroupoid Algebras

Abstract

Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras. We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined. We provide a diverse collection of examples including; algebras with free behaviour, and examples which can be represented as matrix function algebras. We show how these algebras can be presented and decomposed in terms of amalgamated free products. We determine the commutant, consider invariant subspaces, obtain a Beurling theorem for them, conduct an eigenvalue analysis, give an elementary proof of reflexivity, and discuss hyper-reflexivity. Our main theorem shows the graph to be a complete unitary invariant for the algebra. This classification theorem makes use of an analysis of unitarily implemented automorphisms. We give a graph-theoretic description of when these algebras are partly free, in the sense that they contain a copy of a free semigroup algebra.

45 pages, J. Ramanujan Math. Soc., to appear

References in corpus (3)

• Ideal Structure in Free Semigroupoid Algebras from Directed Graphs
• Factoring in Non-commutative Analytic Toeplitz Algebras
• Universal operator algebras of directed graphs

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• Graph W*-probability Theory
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• You Can See the Arrows in a Quiver Operator Algebra
• The $H^{\infty }$ algebras of higher rank graphs
• Compressed Random Variables in a Graph W*-Probability Space
• The C*-envelope of the tensor algebra of a directed graph
• Duality of $W^{\ast }$-correspondences and applications
• On Bilateral Weighted Shifts in Noncommutative Multivariable Operator Theory
• Hardy Algebras, W*-Correspondences and Interpolation Theory
• The Moments of the Block Operators in a Group von Neumann Algebra
• Diagonal Compressed Random Variables in a Graph W*-probability Spaces
• Explicit construction and uniqueness for universal operator algebras of directed graphs
• Applications of the Wold decomposition to the study of row contractions associated with directed graphs
• Free Product Structure of a Graph W*-probability space