The uniform structure of Banach spaces
type: Item
instance of: scholarly article
title: The uniform structure of Banach spaces
zbMATH Open document ID: 1268.46018
published in: Mathematische Annalen
publication date: 2012-12-20T00:00:00Z
review text: This paper explores the existence of uniformly continuous sections for quotient maps. The paper consists of eight parts, the first being the introduction and the next two covering preliminaries from linear and nonlinear Banach space theory. In Section 4, the author introduces the following nonlinear version of the approximation property: A complete metric space \((M,d)\) is approximable if there is a sub-additive map \(\omega: [0,\infty] \to [0,\infty]\) with \(\lim_{t \to 0} \omega(t) = 0\) such that, for every finite set \(E \subset M\) and every \(\varepsilon > 0\), we can find a uniformly continuous map \(\psi: M \to M\) such that \(d(x,\psi(x)) < \varepsilon\) for every \(x \in E\), \(\psi(M)\) is relatively compact, and \(\psi\)'s modulus of continuity satisfies \(\omega_\psi \leq \omega\). If \(X=M\) is a Banach space and the maps \(\psi\) can be chosen to be linear operators with finite rank and \(\omega(t) = \lambda t\), for some constant \(\lambda\), we recover the classical bounded approximation property (BAP). \textit{G. Godefroy} and \textit{N. J. Kalton} [Stud. Math. 159, No. 1, 121--141 (2003; Zbl 1059.46058)] showed that a Banach space \(X\) has the BAP if and only if \(X\) is Lipschitz approximable (i.e., \(\omega(t) = Lt\) for some constant \(L\)). Theorem~4.4 shows that \(X\) is approximable if and only if \(X\) is almost Lipschitz approximable. But Theorem~4.6 says that, if \(X\) has separable dual, then both \(X\) and \(X^*\) are approximable. These ideas are used to show that there are subspaces of \(c_0\) which are uniformly homeomorphic but not linearly isomorphic. (One has a finite dimensional decomposition (FDD), while the other fails the approximation property.) Furthermore, if \(X\) is separable, then \(X\) is approximable if and only if \(UB\) and \(UB \oplus X\) are uniformly homeomorphic (\(UB\) is Pełczyński's universal basis space.) In Section 5, the author considers the question asked by Benyamini and Lindenstrauss: Is every Banach …
reviewed by: Vegard Lima
Mathematics Subject Classification ID: 46B80
zbMATH DE Number: 6117678
zbMATH Keywords: Banach space
zbMATH Keywords: uniform homeomorphism
zbMATH Keywords: coarse homeomorphism
zbMATH Keywords: uniformly continuous section
zbMATH Keywords: nonlinear Banach space theory
zbMATH Keywords: approximation property
zbMATH Keywords: Lipschitz retract
author: N. J. Kalton
MaRDI profile type: Publication
full work available at URL: s00208-011-0743-3
OpenAlex ID: W2124057264
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DOI: 10.1007/S00208-011-0743-3
description: scientific article; zbMATH DE number 6117678