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The classification problem for operator algebraic varieties and their multiplier algebras

Hartz, Michael and Lupini, Martino (2018) The classification problem for operator algebraic varieties and their multiplier algebras. Transactions of the American Mathematical Society, 370 (3). pp. 2161-2180. ISSN 0002-9947.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2017 American Mathematical Society. Received by the editors September 7, 2015 and, in revised form, December 3, 2016. Published electronically: November 1, 2017. 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. The authors would like to thank the anonymous referee for carefully reviewing the paper and providing a large number of useful comments.
Funding AgencyGrant Number
Ontario Trillium ScholarshipUNSPECIFIED
York UniversityUNSPECIFIED
European Research Council (ERC)259527
Subject Keywords:Non-selfadjoint operator algebra, reproducing kernel Hilbert space, multiplier algebra, Nevanlinna-Pick kernel, Borel complexity, turbulence, Polish groupoid, Blaschke sequence
Classification Code:MSC (2010): Primary 47L30, 03E15; Secondary 46E22, 47A13
Record Number:CaltechAUTHORS:20180111-134221146
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:84271
Deposited By: George Porter
Deposited On:12 Jan 2018 00:08
Last Modified:12 Jan 2018 00:08

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