Free Semigroupoid Algebras

By David W. Kribs, S. Power. 2003

83 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f340c99f612785fddd47f0a82f26dcc874d231b4

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In Algebra and Number Theory

1. K. Davidson
2. Elias G. Katsoulis
3. B. Solel
4. P. Muhly
5. David W. Kribs
6. M. Jury
7. Gelu Popescu
8. Orr Shalit

In Applied Mathematics

1. K. Davidson
2. Elias G. Katsoulis
3. P. Muhly
4. B. Solel
5. M. Jury
6. David W. Kribs
7. Gelu Popescu
8. Orr Shalit

In Mathematical Physics

1. K. Davidson
2. Elias G. Katsoulis
3. B. Solel
4. David W. Kribs
5. P. Muhly
6. M. Jury
7. Gelu Popescu
8. Orr Shalit

Citations in the corpus

• Operator Algebras for Multivariable Dynamics (2007, background): “A first effort to address the periodic case was the study of isomorphisms between graph algebras [22, 28, 50].”
• IDEAL STRUCTURE IN FREE SEMIGROUPOID ALGEBRAS FROM DIRECTED GRAPHS (2003, methodology): “An interesting subclass of the algebras LG discovered in [18, 19] are characterized by commutant LG = RG containing a pair of isometries with mutually orthogonal ranges.”
• Isomorphisms of algebras associated with directed graphs (2003, methodology): “(This construction originates from [1, 4] and was reiterated in [14]).”
• Hardy algebras, W*-correspondences and interpolation theory (2003, background): “d that mi = 1 for all i. In this case, B(Cmj ,H ji ⊗Cmi)} is naturally isomorphic to Hji and we see that Eσ = E(Ct). From this, Proposition 5.4 of [26] is immediate. We note, too, that Theorem 5.8 of [22], which extends Proposition 5.4 of [26] beyond the multiplicity one case, also follows from Theo”