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A metric interpretation of reflexivity for Banach spaces. (English) Zbl 1386.46024

J. Bourgain [Isr. J. Math. 56, 222–230 (1986; Zbl 0643.46013)] initiated the search for purely metric characterizations of local isomorphic properties of Banach spaces and found such a characterization for superreflexivity. Bourgain called this study the Ribe program, because it was motivated by the following result of M. Ribe [Ark. Mat. 14, 237–244 (1976; Zbl 0336.46018)]: If Banach spaces \(X\) and \(Y\) are uniformly homeomorphic, then there exists a constant \(1\leq C<\infty\), depending only on \(X\) and \(Y\), such that, for each finite-dimensional subspace \(E_n\) in one of the spaces, there is a finite-dimensional subspace \(F_n\) in the other space such that the Banach-Mazur distance between \(E_n\) and \(F_n\) does not exceed \(C\). See [A. Naor, Jpn. J. Math. (3) 7, No. 2, 167–233 (2012; Zbl 1261.46013)] for a survey of the Ribe program.
In [“Texas 2009 nonlinear problems”, Report from the Workshop in Analysis and Probability, August 2009, Texas A & M University (2009), http://facpub.stjohns.edu/ostrovsm/Texas2009nonlinearproblems.pdf], William B. Johnson suggested to consider similar problems for non-local properties, first of all, for reflexivity and the Radon-Nikodým property. This direction of research is not considered as a part of the Ribe program because these properties are not preserved under uniform homeomorphisms.
Metric characterizations of non-local properties were obtained (1) for the class of reflexive Banach spaces which do not admit any equivalent asymptotically uniformly smooth norm or any equivalent asymptotically uniformly convex norm [F. Baudier et al., Stud. Math. 199, No. 1, 73–94 (2010; Zbl 1210.46017)]; (2) for reflexivity and the Radon-Nikodým property [the reviewer, J. Topol. Anal. 6, No. 3, 441–464 (2014; Zbl 1307.46007)]. The characterization of reflexivity obtained by the reviewer is of the following type: there is a set of pairs \(S\) in \(\ell_1\times\ell_1\) such that a Banach space \(X\) is non-reflexive if and only if \(\ell_1\) admits an embedding into \(X\) which satisfies the bilipschitz condition on pairs of \(S\) (but possibly not on other pairs).
The main goal of the present paper is to get another metric characterization of reflexivity. The main result of the paper is based on the following definitions.
The Schreier families \(\mathcal{S}_\alpha\) are families of finite subsets of \(\mathbb{N}\), defined using transfinite induction for all \(\alpha<\omega_1\) as follows: \(\mathcal{S}_0=\big\{ \{n\}: n\in\mathbb{N}\big\}\cup\{\emptyset\}\). If \(\alpha=\gamma+1\), that is, \(\alpha\) is not a limit ordinal, then \(\mathcal{S}_{\alpha} =\{ \bigcup_{j=1}^n E_j: n\leq\min(E_1),\, E_1< E_2<\dots< E_n,\, E_j\in\mathcal{S}_{\gamma},\, j=1,2,\dots,n\}\). Finally, if \(\alpha\) is a limit ordinal, we choose a fixed sequence \(\big(\lambda(\alpha,n):n\in\mathbb{N}\big)\subset[1,\alpha)\) which increases to \(\alpha\) and set \(\mathcal{S}_\alpha=\{ E: \exists \, k\leq \min(E), \text{ with } E\in\mathcal{S}_{\lambda(\alpha,k)} \}\). After that, the authors define repeated averages on Schreier sets, which are positive elements of the unit sphere of \(\ell_1\), indexed by pairs \((\alpha, A)\), where \(\alpha\) is a countable ordinal and \(A\) is a maximal element of \(\mathcal{S}_\alpha\) (the definition is too technical to be reproduced here).
These averages are used to introduce two distances on \(\mathcal{S}_\alpha\). (1) The weighted tree distance on \(\mathcal{S}_\alpha\). For \(A,B\) in \(\mathcal{S}_\alpha\), let \(C\) be the largest common initial segment of \(A\) and \(B\), and then let \(d_{1,\alpha}(A,B)= \sum_{a\in A\setminus C} z_{(\alpha,A)}(a)+\sum_{b\in B\setminus C} z_{(\alpha,B)}(b) .\) (2) The weighted interlacing distance on \(\mathcal{S}_\alpha\). For \(A,B\in \mathcal{S}_\alpha\), say \(A=\{a_1,a_2,\dots,a_l\}\) and \(B=\{b_1,b_2,\dots, b_m\}\), such that \(a_1<a_2<\dots <a_l\) and \(b_1<b_2<\dots<b_m\), we put \(a_0=b_0=0\) and \(a_{l+1}=b_{m+1}=\infty\), and define \(d_{\infty,\alpha}(A,B)= \max_{i=1,\dots, m+1} \sum_{a\in A, b_{i-1}<a<b_i }z_{(\alpha,A)}(a)+ \max_{i=1,\dots, l+1} \sum_{b\in B, a_{i-1}<b<a_i} z_{(\alpha,B)}(b)\).
The main result of the paper reads: A Banach space \(X\) is reflexive if and only if there is \(\alpha<\omega_1\) so that there is no mapping \(\Phi:\mathcal{S}_\alpha\to X\) for which, for some \(C\geq c>0\), \( cd_{\infty,\alpha}(A,B)\leq \|\Phi(A)-\Phi(B)\|\leq C d_{1,\alpha}(A,B)\) for all \(A,B\in\mathcal{S}_\alpha\).

MSC:

46B80 Nonlinear classification of Banach spaces; nonlinear quotients
46B03 Isomorphic theory (including renorming) of Banach spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
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