The uniform structure of Banach spaces
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 space \(X\) a Lipschitz retract of its bidual \(X^{**}\)? A non-separable counterexample was given by the author [Fundam. Math. 212, No. 1, 53--69 (2011; Zbl 1220.46014)]. Here, some positive results for separable \(X\) are given. For example, if \(X\) has an unconditional FDD or \(X\) is a separable order-continuous Banach lattice, then there does exist a Lipschitz retraction of \(X^{**}\) onto \(X\). Section 6 is a rework of some results on so-called good partitions, introduced in [Collect. Math. 55, No. 2, 171--217 (2004; Zbl 1069.46004)]. Section 7 gives some criteria for the existence of uniformly continuous (and coarsely continuous) sections. These are then applied in Section~8 to produce many examples. Section~8 contains results such as: There is a closed subspace \(Z\) of \(L_1\) such that the unit ball \(B_Z\) is not an absolute uniform retract (Theorem~8.3). There exists two separable \(\mathcal{L}_1\) spaces which are uniformly homeomorphic but not linearly isomorphic (Theorem~8.5). In fact, we can find these as subspaces of \(\ell_1\) (Theorem~8.6). There exist separable Banach spaces which are coarsely homeomorphic but not uniformly homeomorphic (Theorem~8.9). Several open problems are stated throughout the paper.
scientific article; zbMATH DE number 6117678
scientific article