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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The coarse geometry of Tsirelson’s space and applications
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by F. Baudier, G. Lancien and Th. Schlumprecht HTML | PDF
J. Amer. Math. Soc. 31 (2018), 699-717 Request permission

Abstract:

The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson’s original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive, and all of its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ coarsely contains neither $c_0$ nor $\ell _p$ for $p\in [1,\infty )$. We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.
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Additional Information
  • F. Baudier
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 825722
  • Email: florent@math.tamu.edu
  • G. Lancien
  • Affiliation: Laboratoire de Mathématiques de Besançon, CNRS UMR-6623, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cédex, France
  • MR Author ID: 324078
  • Email: gilles.lancien@univ-fcomte.fr
  • Th. Schlumprecht
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, and Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 16627, Prague, Czech Republic
  • MR Author ID: 260001
  • Email: schlump@math.tamu.edu
  • Received by editor(s): May 18, 2017
  • Received by editor(s) in revised form: November 15, 2017
  • Published electronically: February 28, 2018
  • Additional Notes: The first author was partially supported by the National Science Foundation under grant number DMS-1565826.
    The second author was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03) and as a participant of the “NSF Workshop in Analysis and Probability” at Texas A&M University.
    The third author was supported by the National Science Foundation under grant number DMS-1464713.
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 31 (2018), 699-717
  • MSC (2010): Primary 46B20, 46B85, 46T99, 05C63, 20F65
  • DOI: https://doi.org/10.1090/jams/899
  • MathSciNet review: 3787406