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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
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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  • [1] Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111-124. MR 1774853,
  • [2] Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259
  • [3] Daniel Alpay, Mihai Putinar, and Victor Vinnikov, A Hilbert space approach to bounded analytic extension in the ball, Commun. Pure Appl. Anal. 2 (2003), no. 2, 139-145. MR 1975056,
  • [4] N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107-1180. MR 2419381,
  • [5] William Arveson, Subalgebras of $ C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159-228. MR 1668582,
  • [6] Bruce Blackadar, $ K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
  • [7] Françoise Dal'Bo, Geodesic and horocyclic trajectories, translated from the 2007 French original, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. MR 2766419
  • [8] Kenneth R. Davidson, Christopher Ramsey, and Orr Moshe Shalit, Operator algebras for analytic varieties, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1121-1150. MR 3280039,
  • [9] Kenneth R. Davidson, Michael Hartz, and Orr Moshe Shalit, Multipliers of embedded discs, Complex Anal. Oper. Theory 9 (2015), no. 2, 287-321; erratum, 9 (2015), no. 2, 323-327. MR 3311940,
  • [10] Kenneth R. Davidson, Christopher Ramsey, and Orr Moshe Shalit, The isomorphism problem for some universal operator algebras, Adv. Math. 228 (2011), no. 1, 167-218. MR 2822231,
  • [11] Harvey Friedman and Lee Stanley, A Borel reducibility theory for classes of countable structures, J. Symbolic Logic 54 (1989), no. 3, 894-914. MR 1011177,
  • [12] Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR 2455198
  • [13] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
  • [14] Michael Hartz, Topological isomorphisms for some universal operator algebras, J. Funct. Anal. 263 (2012), no. 11, 3564-3587. MR 2984075,
  • [15] Michael Hartz, On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, Canad. J. Math. 69 (2017), no. 1, 54-106. MR 3589854,
  • [16] Greg Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000. MR 1725642
  • [17] G. Hjorth and A. S. Kechris, The complexity of the classification of Riemann surfaces and complex manifolds, Illinois J. Math. 44 (2000), no. 1, 104-137. MR 1731384
  • [18] Alexander S. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 283-295. MR 1176624,
  • [19] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
  • [20] Matt Kerr, John E. McCarthy, and Orr Moshe Shalit, On the isomorphism question for complete pick multiplier algebras, Integral Equations Operator Theory 76 (2013), no. 1, 39-53. MR 3041720,
  • [21] Martino Lupini, Polish groupoids and functorial complexity, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6683-6723. MR 3660238,
  • [22] John E. McCarthy and Orr Moshe Shalit, Spaces of Dirichlet series with the complete Pick property, Israel J. Math. 220 (2017), no. 2, 509-530. MR 3666434,
  • [23] H. A. Priestley, Introduction to complex analysis, 2nd ed., Oxford University Press, Oxford, 2003. MR 2014542
  • [24] Arlan Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), no. 2, 358-374. MR 1081649,
  • [25] Arlan B. Ramsay, Polish groupoids, Descriptive set theory and dynamical systems (Marseille-Luminy, 1996) London Math. Soc. Lecture Note Ser., vol. 277, Cambridge Univ. Press, Cambridge, 2000, pp. 259-271. MR 1774429
  • [26] Pedro Resende, Lectures on étale groupoids, inverse semigroups and quantales, 2006.
  • [27] Walter Rudin, Function theory in the unit ball of $ \mathbb{C}^n$, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1980 edition. MR 2446682
  • [28] Guy Salomon and Orr Moshe Shalit, The isomorphism problem for complete Pick algebras: a survey, Operator theory, function spaces, and applications, Oper. Theory Adv. Appl., vol. 255, Birkhäuser/Springer, Cham, 2016, pp. 167-198. MR 3617006
  • [29] Orr Shalit, Operator theory and function theory in Drury-Arveson space and its quotients, Operator Theory (Daniel Alpay, ed.), Springer, 2015, pp. 1125-1180.
  • [30] M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898

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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

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

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.
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