Gelfand transforms and boundary representations of complete Nevanlinna–Pick quotients
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- by Raphaël Clouâtre and Edward J. Timko PDF
- Trans. Amer. Math. Soc. 374 (2021), 2107-2147 Request permission
Abstract:
The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna–Pick spaces, examples of which include the Drury–Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson’s notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson–Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.References
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Additional Information
- Raphaël Clouâtre
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
- MR Author ID: 841119
- ORCID: 0000-0002-9691-2906
- Email: raphael.clouatre@umanitoba.ca
- Edward J. Timko
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
- Email: edward.timko@umanitoba.ca
- Received by editor(s): December 21, 2019
- Received by editor(s) in revised form: August 4, 2020
- Published electronically: December 15, 2020
- Additional Notes: The first author was partially supported by an NSERC Discovery Grant.
The second author was partially supported by a PIMS postdoctoral fellowship. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 2107-2147
- MSC (2020): Primary 47L55, 46E22, 47A13
- DOI: https://doi.org/10.1090/tran/8279
- MathSciNet review: 4216734