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Commutants of weighted shift directed graph operator algebras


Authors: David W. Kribs, Rupert H. Levene and Stephen C. Power
Journal: Proc. Amer. Math. Soc. 145 (2017), 3465-3480
MSC (2010): Primary 47L75, 47L55, 47B37
DOI: https://doi.org/10.1090/proc/13477
Published electronically: February 21, 2017
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Abstract: We consider non-selfadjoint operator algebras $ \mathcal {L} (G,\lambda )$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $ G$. These algebras may be viewed as non-commutative generalizations of weighted Bergman space algebras or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for $ \mathfrak{L} (G,\lambda )$ in the case of the single vertex graph with two edges and a suitable choice of left weight function $ \lambda $.


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Additional Information

David W. Kribs
Affiliation: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

Rupert H. Levene
Affiliation: School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Stephen C. Power
Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom, LA1 4YF

DOI: https://doi.org/10.1090/proc/13477
Keywords: Directed graph, weighted shift, non-selfadjoint algebra, commutant, left regular representation, creation operators, Fock space.
Received by editor(s): May 24, 2016
Received by editor(s) in revised form: September 12, 2016, and September 14, 2016
Published electronically: February 21, 2017
Communicated by: Adrian Ioana
Article copyright: © Copyright 2017 American Mathematical Society