The classification problem for operator algebraic varieties and their multiplier algebras
HTML articles powered by AMS MathViewer
- by Michael Hartz and Martino Lupini PDF
- Trans. Amer. Math. Soc. 370 (2018), 2161-2180 Request permission
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.References
- Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111–124. MR 1774853, DOI 10.1006/jfan.2000.3599
- 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, DOI 10.1090/gsm/044
- 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, DOI 10.3934/cpaa.2003.2.139
- 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, DOI 10.1016/j.aim.2008.03.001
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
- Françoise Dal’Bo, Geodesic and horocyclic trajectories, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Translated from the 2007 French original. MR 2766419, DOI 10.1007/978-0-85729-073-1
- 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, DOI 10.1090/S0002-9947-2014-05888-1
- Kenneth R. Davidson, Michael Hartz, and Orr Moshe Shalit, Multipliers of embedded discs, Complex Anal. Oper. Theory 9 (2015), no. 2, 287–321. MR 3311940, DOI 10.1007/s11785-014-0360-8
- 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, DOI 10.1016/j.aim.2011.05.015
- 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, DOI 10.2307/2274750
- Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR 2455198
- John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
- Michael Hartz, Topological isomorphisms for some universal operator algebras, J. Funct. Anal. 263 (2012), no. 11, 3564–3587. MR 2984075, DOI 10.1016/j.jfa.2012.08.028
- Michael Hartz, On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, Canad. J. Math. 69 (2017), no. 1, 54–106. MR 3589854, DOI 10.4153/CJM-2015-050-6
- Greg Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000. MR 1725642, DOI 10.1090/surv/075
- 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
- Alexander S. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 283–295. MR 1176624, DOI 10.1017/S0143385700006751
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- 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, DOI 10.1007/s00020-013-2048-2
- Martino Lupini, Polish groupoids and functorial complexity, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6683–6723. With an appendix by Anush Tserunyan. MR 3660238, DOI 10.1090/tran/7102
- 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, DOI 10.1007/s11856-017-1527-6
- H. A. Priestley, Introduction to complex analysis, 2nd ed., Oxford University Press, Oxford, 2003. MR 2014542
- Arlan Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), no. 2, 358–374. MR 1081649, DOI 10.1016/0022-1236(90)90018-G
- 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
- Pedro Resende, Lectures on étale groupoids, inverse semigroups and quantales, 2006.
- Walter Rudin, Function theory in the unit ball of $\Bbb C^n$, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1980 edition. MR 2446682
- 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
- Orr Shalit, Operator theory and function theory in Drury-Arveson space and its quotients, Operator Theory (Daniel Alpay, ed.), Springer, 2015, pp. 1125-1180.
- M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898
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
- MR Author ID: 997298
- 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
- MR Author ID: 1071243
- Email: lupini@caltech.edu
- 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.
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3739205