The coarse geometry of Tsirelson’s space and applications
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- by F. Baudier, G. Lancien and Th. Schlumprecht
- J. Amer. Math. Soc. 31 (2018), 699-717
- DOI: https://doi.org/10.1090/jams/899
- Published electronically: February 28, 2018
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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
- Israel Aharoni, Every separable metric space is Lipschitz equivalent to a subset of $c^{+}_{0}$, Israel J. Math. 19 (1974), 284–291. MR 511661, DOI 10.1007/BF02757727
- Dale Alspach, Robert Judd, and Edward Odell, The Szlenk index and local $l_1$-indices, Positivity 9 (2005), no. 1, 1–44. MR 2139115, DOI 10.1007/s11117-002-9781-0
- Florent Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429. MR 2363693, DOI 10.1007/s00013-007-2108-4
- Florent P. Baudier, Quantitative nonlinear embeddings into Lebesgue sequence spaces, J. Topol. Anal. 8 (2016), no. 1, 117–150. MR 3463248, DOI 10.1142/S1793525316500011
- F. Baudier and G. Lancien, Tight embeddability of proper and stable metric spaces, Anal. Geom. Metr. Spaces 3 (2015), no. 1, 140–156. MR 3365754, DOI 10.1515/agms-2015-0010
- F. Baudier, N. J. Kalton, and G. Lancien, A new metric invariant for Banach spaces, Studia Math. 199 (2010), no. 1, 73–94. MR 2652598, DOI 10.4064/sm199-1-5
- Bernard Beauzamy, Banach-Saks properties and spreading models, Math. Scand. 44 (1979), no. 2, 357–384. MR 555227, DOI 10.7146/math.scand.a-11818
- B. Beauzamy and J.-T. Lapresté, Modèles étalés des espaces de Banach, Travaux en Cours. [Works in Progress], Hermann, Paris, 1984 (French). MR 770062
- B. M. Braga, Asymptotic structure and coarse Lipschitz geometry of Banach spaces, Studia Math. 237 (2017), no. 1, 71–97. MR 3612891, DOI 10.4064/sm8604-11-2016
- Antoine Brunel and Louis Sucheston, On $B$-convex Banach spaces, Math. Systems Theory 7 (1974), no. 4, 294–299. MR 438085, DOI 10.1007/BF01795947
- Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
- V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225. MR 1489106, DOI 10.1007/BF02773800
- Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg, A history and survey of the Novikov conjecture, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 7–66. MR 1388295, DOI 10.1017/CBO9780511662676.003
- T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no $l_{p}$, Compositio Math. 29 (1974), 179–190. MR 355537
- G. Godefroy, G. Lancien, and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529–1583. MR 3295641, DOI 10.1216/RMJ-2014-44-5-1529
- Robert C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR 39915, DOI 10.2307/1969430
- Robert C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N.J., 1972, pp. 159–175. MR 0454600
- William B. Johnson and N. Lovasoa Randrianarivony, $l_p\ (p>2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1045–1050. MR 2196037, DOI 10.1090/S0002-9939-05-08415-7
- N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58 (2007), no. 3, 393–414. MR 2354924, DOI 10.1093/qmath/ham018
- N. J. Kalton, Lipschitz and uniform embeddings into $\ell _\infty$, Fund. Math. 212 (2011), no. 1, 53–69. MR 2771588, DOI 10.4064/fm212-1-4
- Nigel J. Kalton and N. Lovasoa Randrianarivony, The coarse Lipschitz geometry of $l_p\oplus l_q$, Math. Ann. 341 (2008), no. 1, 223–237. MR 2377476, DOI 10.1007/s00208-007-0190-3
- Nigel J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. MR 1324212, DOI 10.1515/crll.1995.461.137
- Gennadi Kasparov and Guoliang Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), no. 1, 1–56. MR 2261750, DOI 10.1016/j.aim.2005.08.004
- Manor Mendel and Assaf Naor, Metric cotype, Ann. of Math. (2) 168 (2008), no. 1, 247–298. MR 2415403, DOI 10.4007/annals.2008.168.247
- A. Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012), no. 2, 167–233., DOI 10.1007/s11537-012-1222-7
- Assaf Naor, Discrete Riesz transforms and sharp metric $X_p$ inequalities, Ann. of Math. (2) 184 (2016), no. 3, 991–1016. MR 3549628, DOI 10.4007/annals.2016.184.3.9
- Assaf Naor and Gideon Schechtman, Pythagorean powers of hypercubes, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 1093–1116 (English, with English and French summaries). MR 3494167, DOI 10.5802/aif.3032
- Assaf Naor and Gideon Schechtman, Metric $X_p$ inequalities, Forum Math. Pi 4 (2016), e3, 81. MR 3456183, DOI 10.1017/fmp.2016.1
- Piotr W. Nowak, On coarse embeddability into $l_p$-spaces and a conjecture of Dranishnikov, Fund. Math. 189 (2006), no. 2, 111–116. MR 2214573, DOI 10.4064/fm189-2-2
- Piotr W. Nowak and Guoliang Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR 2986138, DOI 10.4171/112
- E. Odell and T. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281., DOI 10.1007/BF02398436
- E. Odell, Th. Schlumprecht, and A. Zsák, Banach spaces of bounded Szlenk index, Studia Math. 183 (2007), no. 1, 63–97. MR 2360257, DOI 10.4064/sm183-1-4
- M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009), 163–183. MR 2471569
- Mikhail I. Ostrovskii, Metric embeddings, De Gruyter Studies in Mathematics, vol. 49, De Gruyter, Berlin, 2013. Bilipschitz and coarse embeddings into Banach spaces. MR 3114782, DOI 10.1515/9783110264012
- M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), no. 2-3, 139–147. MR 770696, DOI 10.1007/BF02761159
- Jonathan Rosenberg, Novikov’s conjecture, Open problems in mathematics, Springer, [Cham], 2016, pp. 377–402. MR 3526942
- B. S. Tsirel$^\prime$son, It is impossible to imbed $l_{p}$ of $c_{0}$ into an arbitrary Banach space, Funkcional. Anal. i Priložen. 8 (1974), 57–60 (Russian).
- A. Valette, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. From notes taken by Indira Chatterji, With an appendix by Guido Mislin., DOI 10.1007/978-3-0348-8187-6
- Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI 10.1007/s002229900032
Bibliographic 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