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The classification problem for operator algebraic varieties and their multiplier algebras


Authors: Michael Hartz and Martino Lupini
Journal: Trans. Amer. Math. Soc. 370 (2018), 2161-2180
MSC (2010): Primary 47L30, 03E15; Secondary 46E22, 47A13
DOI: https://doi.org/10.1090/tran/7146
Published electronically: November 1, 2017
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Abstract: We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier algebras associated with varieties in a finite-dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems. In particular, this shows that Blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc.


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Additional Information

Michael Hartz
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Address at time of publication: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130
Email: mphartz@wustl.edu

Martino Lupini
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Room 02.126, 1090 Wien, Austria
Address at time of publication: Department of Mathematics California Institute of Technology 1200 E. California Boulevard MC 253-37 Pasadena, California 91125
Email: lupini@caltech.edu

DOI: https://doi.org/10.1090/tran/7146
Keywords: Non-selfadjoint operator algebra, reproducing kernel Hilbert space, multiplier algebra, Nevanlinna-Pick kernel, Borel complexity, turbulence, Polish groupoid, Blaschke sequence
Received by editor(s): September 7, 2015
Received by editor(s) in revised form: December 3, 2016
Published electronically: November 1, 2017
Additional Notes: The first author was partially supported by an Ontario Trillium Scholarship. The second author was supported by the York University Susan Mann Dissertation Scholarship and by the ERC Starting Grant No. 259527 of Goulnara Arzhantseva. This work was initiated during a visit of the first-named author to the Fields Institute in March 2015. The hospitality of the Institute is gratefully acknowledged.
Article copyright: © Copyright 2017 American Mathematical Society

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