Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

 
 

 

The coarse geometry of Tsirelson's space and applications


Authors: F. Baudier, G. Lancien and Th. Schlumprecht
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
Published electronically: February 28, 2018
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 46B20, 46B85, 46T99, 05C63, 20F65

Retrieve articles in all journals with MSC (2010): 46B20, 46B85, 46T99, 05C63, 20F65


Additional Information

F. Baudier
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
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
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
Email: schlump@math.tamu.edu

DOI: https://doi.org/10.1090/jams/899
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.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society