Asymptotic properties of Banach spaces and coarse quotient maps
Author:
Sheng Zhang
Journal:
Proc. Amer. Math. Soc. 146 (2018), 4723-4734
MSC (2010):
Primary 46B80, 46B06
DOI:
https://doi.org/10.1090/proc/14097
Published electronically:
August 7, 2018
MathSciNet review:
3856140
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space 
is a coarse quotient of a subset of a Banach space 
, where the coarse quotient map is coarse Lipschitz, then the (
)-modulus of 
is bounded by the modulus of asymptotic uniform smoothness of 
up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of 
is bounded by the modulus of asymptotic uniform smoothness of 
up to some constants.
- [1] Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, 2nd ed., Graduate Texts in Mathematics, vol. 233, Springer, [Cham], 2016. With a foreword by Gilles Godefory. MR 3526021
- [2] 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, https://doi.org/10.1007/s000390050108
- [3] Florent P. Baudier and Sheng Zhang, (𝛽)-distortion of some infinite graphs, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 481–501. MR 3483124, https://doi.org/10.1112/jlms/jdv074
- [4] Yoav Benyamini, The uniform classification of Banach spaces, Texas functional analysis seminar 1984–1985 (Austin, Tex.), Longhorn Notes, Univ. Texas Press, Austin, TX, 1985, pp. 15–38. MR 832247
- [5] S. J. Dilworth, Denka Kutzarova, G. Lancien, and N. L. Randrianarivony, Asymptotic geometry of Banach spaces and uniform quotient maps, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2747–2762. MR 3209329, https://doi.org/10.1090/S0002-9939-2014-12001-6
- [6] S. J. Dilworth, D. Kutzarova, G. Lancien, and N. L. Randrianarivony, Equivalent norms with the property (𝛽) of Rolewicz, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111 (2017), no. 1, 101–113. MR 3596040, https://doi.org/10.1007/s13398-016-0278-2
- [7] S. J. Dilworth, Denka Kutzarova, and N. Lovasoa Randrianarivony, The transfer of property (𝛽) of Rolewicz by a uniform quotient map, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6253–6270. MR 3461033, https://doi.org/10.1090/S0002-9947-2015-06553-2
- [8] S. J. Dilworth, Denka Kutzarova, N. Lovasoa Randrianarivony, J. P. Revalski, and N. V. Zhivkov, Lenses and asymptotic midpoint uniform convexity, J. Math. Anal. Appl. 436 (2016), no. 2, 810–821. MR 3446981, https://doi.org/10.1016/j.jmaa.2015.11.061
- [9] 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, https://doi.org/10.1112/S0024611502013400
- [10] Nigel J. Kalton and N. Lovasoa Randrianarivony, The coarse Lipschitz geometry of 𝑙_{𝑝}⊕𝑙_{𝑞}, Math. Ann. 341 (2008), no. 1, 223–237. MR 2377476, https://doi.org/10.1007/s00208-007-0190-3
- [11] Denka Kutzarova, An isomorphic characterization of property (𝛽) of Rolewicz, Note Mat. 10 (1990), no. 2, 347–354. MR 1204212
- [12] Denka Kutzarova, 𝑘-𝛽 and 𝑘-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162 (1991), no. 2, 322–338. MR 1137623, https://doi.org/10.1016/0022-247X(91)90153-Q
- [13] Vegard Lima and N. Lovasoa Randrianarivony, Property (𝛽) and uniform quotient maps, Israel J. Math. 192 (2012), no. 1, 311–323. MR 3004085, https://doi.org/10.1007/s11856-012-0025-0
- [14] 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
- [15] Nirina Lovasoa Randrianarivony, Nonlinear classification of Banach spaces, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–Texas A&M University. MR 2707820
- [16] S. Rolewicz, On Δ-uniform convexity and drop property, Studia Math. 87 (1987), no. 2, 181–191. MR 928575, https://doi.org/10.4064/sm-87-2-181-191
- [17] Sheng Zhang, Coarse quotient mappings between metric spaces, Israel J. Math. 207 (2015), no. 2, 961–979. MR 3359724, https://doi.org/10.1007/s11856-015-1168-6
-distortion of some infinite graphs, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 481-501. 
of Rolewicz, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111 (2017), no. 1, 101-113. 
of Rolewicz by a uniform quotient map, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6253-6270. 
, Math. Ann. 341 (2008), no. 1, 223-237. 
of Rolewicz, Note Mat. 10 (1990), no. 2, 347-354. 
-
and 
-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162 (1991), no. 2, 322-338. 
and uniform quotient maps, Israel J. Math. 192 (2012), no. 1, 311-323. 
-uniform convexity and drop property, Studia Math. 87 (1987), no. 2, 181-191. Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B80, 46B06
Retrieve articles in all journals with MSC (2010): 46B80, 46B06
Additional Information
Sheng Zhang
Affiliation:
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, People’s Republic of China
Email:
sheng@swjtu.edu.cn
DOI:
https://doi.org/10.1090/proc/14097
Received by editor(s):
October 2, 2017
Received by editor(s) in revised form:
November 26, 2017, and January 11, 2018
Published electronically:
August 7, 2018
Additional Notes:
The author was supported by the Fundamental Research Funds for the Central Universities, Grant Number 2682017CX060
Communicated by:
Thomas Schlumprecht
Article copyright:
© Copyright 2018
American Mathematical Society
