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Asymptotic geometry of Banach spaces and uniform quotient maps


Authors: S. J. Dilworth, Denka Kutzarova, G. Lancien and N. L. Randrianarivony
Journal: Proc. Amer. Math. Soc. 142 (2014), 2747-2762
MSC (2010): Primary 46B80; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-2014-12001-6
Published electronically: April 25, 2014
MathSciNet review: 3209329
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Abstract: Recently, Lima and Randrianarivony pointed out the role of the property $ (\beta )$ of Rolewicz in nonlinear quotient problems and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of $ (\beta )$ of the domain space. We also provide conditions under which this comparison can be improved.


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  • [1] G. Androulakis, C. D. Cazacu, and N. J. Kalton, Twisted sums, Fenchel-Orlicz spaces and property (M), Houston J. Math. 24 (1998), no. 1, 105-126. MR 1690211 (2000e:46020)
  • [2] J. M. Ayerbe, T. Domínguez Benavides, and S. Francisco Cutillas, Some noncompact convexity moduli for the property $ (\beta )$ of Rolewicz, Comm. Appl. Nonlinear Anal. 1 (1994), no. 1, 87-98. MR 1268081 (95a:46021)
  • [3] S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients, Geom. Funct. Anal. 9 (1999), no. 6, 1092-1127. MR 1736929 (2000m:46021), https://doi.org/10.1007/s000390050108
  • [4] 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 (2011d:46044), https://doi.org/10.4064/sm199-1-5
  • [5] Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673 (2001b:46001)
  • [6] Sudipta Dutta and Alexandre Godard, Banach spaces with property $ (M)$ and their Szlenk indices, Mediterr. J. Math. 5 (2008), no. 2, 211-220. MR 2427395 (2009c:46011), https://doi.org/10.1007/s00009-008-0145-2
  • [7] G. Godefroy, N. Kalton, and G. Lancien, Subspaces of $ c_0(\mathbf {N})$ and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000), no. 4, 798-820. MR 1791140 (2002k:46044), https://doi.org/10.1007/PL00001638
  • [8] G. Godefroy, N. J. Kalton, and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001), no. 10, 3895-3918 (electronic). MR 1837213 (2003c:46023), https://doi.org/10.1090/S0002-9947-01-02825-2
  • [9] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743-749. MR 595102 (82b:46016), https://doi.org/10.1216/RMJ-1980-10-4-743
  • [10] William B. Johnson, Joram Lindenstrauss, David Preiss, and Gideon Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), no. 3, 711-746. MR 1888429 (2003a:46064), https://doi.org/10.1112/S0024611502013400
  • [11] William B. Johnson, Joram Lindenstrauss, David Preiss, and Gideon Schechtman, Lipschitz quotients from metric trees and from Banach spaces containing $ l_1$, J. Funct. Anal. 194 (2002), no. 2, 332-346. MR 1934607 (2003h:46023), https://doi.org/10.1006/jfan.2002.3924
  • [12] N. J. Kalton, $ M$-ideals of compact operators, Illinois J. Math. 37 (1993), 147-169. MR 1193134
  • [13] H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), no. 2, 173-199. MR 1702641 (2001f:46011), https://doi.org/10.1023/A:1009786603119
  • [14] D. N. Kutzarova, On condition $ (\beta )$ and $ \Delta $-uniform convexity, C. R. Acad. Bulgare Sci. 42 (1989), no. 1, 15-18. MR 0991453
  • [15] Denka Kutzarova, An isomorphic characterization of property $ (\beta )$ of Rolewicz, Note Mat. 10 (1990), no. 2, 347-354. MR 1204212 (94a:46020)
  • [16] Denka Kutzarova, $ k$-$ \beta $ and $ k$-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162 (1991), no. 2, 322-338. MR 1137623 (93b:46018), https://doi.org/10.1016/0022-247X(91)90153-Q
  • [17] Gilles Lancien, On the Szlenk index and the weak$ ^*$-dentability index, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 59-71. MR 1380950 (97c:46021), https://doi.org/10.1093/qmath/47.1.59
  • [18] Vegard Lima and N. Lovasoa Randrianarivony, Property $ (\beta )$ and uniform quotient maps, Israel J. Math. 192 (2012), no. 1, 311-323. MR 3004085
  • [19] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer-Verlag, Berlin, 1977. MR 0500056 (58 #17766)
  • [20] V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73-149 (Russian). MR 0420226 (54 #8240)
  • [21] V. Montesinos and J. R. Torregrosa, A uniform geometric property of Banach spaces, Rocky Mountain J. Math. 22 (1992), no. 2, 683-690. MR 1180730 (93h:46020), https://doi.org/10.1216/rmjm/1181072759
  • [22] Edward W. Odell and Thomas Schlumprecht, Embedding into Banach spaces with finite dimensional decompositions, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 295-323 (English, with Spanish summary). MR 2267413 (2008e:46013)
  • [23] E. Odell and Th. Schlumprecht, A universal reflexive space for the class of uniformly convex Banach spaces, Math. Ann. 335 (2006), no. 4, 901-916. MR 2232021 (2007f:46012), https://doi.org/10.1007/s00208-006-0771-6
  • [24] Stanisław Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 507-521 (English, with Italian summary). MR 1010520 (91a:46017)
  • [25] M. Raja, On $ {\rm weak}^\ast $ uniformly Kadec-Klee renormings, Bull. Lond. Math. Soc. 42 (2010), no. 2, 221-228. MR 2601548 (2011d:46020), https://doi.org/10.1112/blms/bdp108
  • [26] M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), no. 2-3, 139-147. MR 770696 (86e:46015), https://doi.org/10.1007/BF02761159
  • [27] S. Rolewicz, On drop property, Studia Math. 85 (1987), 27-35. MR 0879413
  • [28] S. Rolewicz, On $ \Delta $-uniform convexity and drop property, Studia Math. 87 (1987), no. 2, 181-191. MR 928575 (90j:46024)

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Additional Information

S. J. Dilworth
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: dilworth@math.sc.edu

Denka Kutzarova
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Address at time of publication: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: denka@math.uiuc.edu

G. Lancien
Affiliation: Université de Franche-Comté, Laboratoire de Mathématiques UMR 6623, 16 route de Gray, 25030 Besançon Cedex, France
Email: gilles.lancien@univ-fcomte.fr

N. L. Randrianarivony
Affiliation: Department of Mathematics and Computer Science, St. Louis University, St. Louis, Missouri 63103
Email: nrandria@slu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12001-6
Received by editor(s): March 27, 2012
Received by editor(s) in revised form: August 24, 2012, and September 3, 2012
Published electronically: April 25, 2014
Additional Notes: The first author was partially supported by NSF grant DMS1101490
All authors were supported by the Workshop in Analysis and Probability at Texas A&M University in summer 2011
The fourth author was supported in part by a Young Investigator award from this NSF funded Workshop.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 By the authors

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