The coarse geometry of Tsirelson’s space and applications

Baudier, F.¹ ; Lancien, G.² ; Schlumprecht, Th.³

¹ Department of Mathematics, Texas A&M University, College Station, Texas 77843
² Laboratoire de Mathématiques de Besançon, CNRS UMR-6623, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cédex, France
³ 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

Journal of the American Mathematical Society, Tome 31 (2018) no. 3, pp. 699-717

Résumé

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.

Détail

DOI : 10.1090/jams/899

Bibliographie

[1] Aharoni, Israel Every separable metric space is Lipschitz equivalent to a subset of 𝑐⁺₀ Israel J. Math. 1974 284 291

[2] Alspach, Dale, Judd, Robert, Odell, Edward The Szlenk index and local 𝑙₁-indices Positivity 2005 1 44

[3] Baudier, Florent Metrical characterization of super-reflexivity and linear type of Banach spaces Arch. Math. (Basel) 2007 419 429

[4] Baudier, Florent P. Quantitative nonlinear embeddings into Lebesgue sequence spaces J. Topol. Anal. 2016 117 150

[5] Baudier, F., Lancien, G. Tight embeddability of proper and stable metric spaces Anal. Geom. Metr. Spaces 2015 140 156

[6] Baudier, F., Kalton, N. J., Lancien, G. A new metric invariant for Banach spaces Studia Math. 2010 73 94

[7] Beauzamy, Bernard Banach-Saks properties and spreading models Math. Scand. 1979 357 384

[8] Beauzamy, B., Laprestã©, J.-T. Modèles étalés des espaces de Banach 1984

[9] Braga, B. M. Asymptotic structure and coarse Lipschitz geometry of Banach spaces Studia Math. 2017 71 97

[10] Brunel, Antoine, Sucheston, Louis On 𝐵-convex Banach spaces Math. Systems Theory 1974 294 299

[11] Dvoretzky, Aryeh Some results on convex bodies and Banach spaces 1961 123 160

[12] Ferenczi, V. A uniformly convex hereditarily indecomposable Banach space Israel J. Math. 1997 199 225

[13] Ferry, Steven C., Ranicki, Andrew, Rosenberg, Jonathan A history and survey of the Novikov conjecture 1995 7 66

[14] Figiel, T., Johnson, W. B. A uniformly convex Banach space which contains no 𝑙_{𝑝} Compositio Math. 1974 179 190

[15] Godefroy, G., Lancien, G., Zizler, V. The non-linear geometry of Banach spaces after Nigel Kalton Rocky Mountain J. Math. 2014 1529 1583

[16] James, Robert C. Bases and reflexivity of Banach spaces Ann. of Math. (2) 1950 518 527

[17] James, Robert C. Some self-dual properties of normed linear spaces 1972 159 175

[18] Johnson, William B., Randrianarivony, N. Lovasoa 𝑙_{𝑝}(𝑝>2) does not coarsely embed into a Hilbert space Proc. Amer. Math. Soc. 2006 1045 1050

[19] Kalton, N. J. Coarse and uniform embeddings into reflexive spaces Q. J. Math. 2007 393 414

[20] Kalton, N. J. Lipschitz and uniform embeddings into ℓ_{∞} Fund. Math. 2011 53 69

[21] Kalton, Nigel J., Randrianarivony, N. Lovasoa The coarse Lipschitz geometry of 𝑙_{𝑝}⊕𝑙_{𝑞} Math. Ann. 2008 223 237

[22] Kalton, Nigel J., Werner, Dirk Property (𝑀), 𝑀-ideals, and almost isometric structure of Banach spaces J. Reine Angew. Math. 1995 137 178

[23] Kasparov, Gennadi, Yu, Guoliang The coarse geometric Novikov conjecture and uniform convexity Adv. Math. 2006 1 56

[24] Mendel, Manor, Naor, Assaf Metric cotype Ann. of Math. (2) 2008 247 298

[25] Naor, A. An introduction to the Ribe program Jpn. J. Math. 2012 167 233

[26] Naor, Assaf Discrete Riesz transforms and sharp metric 𝑋_{𝑝} inequalities Ann. of Math. (2) 2016 991 1016

[27] Naor, Assaf, Schechtman, Gideon Pythagorean powers of hypercubes Ann. Inst. Fourier (Grenoble) 2016 1093 1116

[28] Naor, Assaf, Schechtman, Gideon Metric 𝑋_{𝑝} inequalities Forum Math. Pi 2016

[29] Nowak, Piotr W. On coarse embeddability into 𝑙_{𝑝}-spaces and a conjecture of Dranishnikov Fund. Math. 2006 111 116

[30] Nowak, Piotr W., Yu, Guoliang Large scale geometry 2012

[31] Odell, E., Schlumprecht, T. The distortion problem Acta Math. 1994

[32] Odell, E., Schlumprecht, Th., Zsã¡K, A. Banach spaces of bounded Szlenk index Studia Math. 2007 63 97

[33] Ostrovskii, M. I. Coarse embeddability into Banach spaces Topology Proc. 2009 163 183

[34] Ostrovskii, Mikhail I. Metric embeddings 2013

[35] Ribe, M. Existence of separable uniformly homeomorphic nonisomorphic Banach spaces Israel J. Math. 1984 139 147

[36] Rosenberg, Jonathan Novikov’s conjecture 2016 377 402

[37] Tsirel′Son, B. S. It is impossible to imbed 𝑙_{𝑝} of 𝑐₀ into an arbitrary Banach space Funkcional. Anal. i Priložen. 1974

[38] Valette, A. Introduction to the Baum-Connes conjecture 2002

[39] Yu, Guoliang The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space Invent. Math. 2000 201 240