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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
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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 $ Y$ is a coarse quotient of a subset of a Banach space $ X$, where the coarse quotient map is coarse Lipschitz, then the ($ \beta $)-modulus of $ X$ is bounded by the modulus of asymptotic uniform smoothness of $ Y$ up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of $ X$ is bounded by the modulus of asymptotic uniform smoothness of $ Y$ up to some constants.


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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