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Amalgamation and Ramsey properties of \(L_p\) spaces. (English) Zbl 1465.46013

This remarkable article provides original and deep links between different fields. Model theory plays an important role through Fraïssé theory, which allows the construction of ultrahomogeneous structures \(\mathcal{M}\) in terms of properties of the set \(\operatorname{Age}(\mathcal{M})\) of finitely generated substructures. Combinatorics is present through the Kechris-Pestov-Todorcevic correspondence which relates the extreme amenability of \(\operatorname{Aut}(\mathcal{M})\) with the Ramsey property of \(\operatorname{Age}(\mathcal{M})\). Geometry of Banach spaces gets the leading part since the article focuses on the dynamics of the group of isometries of \(L_p\)-spaces. It turns out that the Banach spaces \(L_p\) have strong homogeneity properties: they are in particular almost transitive, and more is shown in the present work. However, they are not transitive if \(p\not=2\), and actually a famous open problem which goes back to Mazur asks if a separable transitive Banach space is isometric to \(L_2\). What follows now is merely a sample of the results shown in this paper.
The first section of the article is an introduction, where the main concepts are presented and the whole work is put in its proper frame. Section 2 is devoted to Fraïssé Banach spaces. This notion and related ones are defined in the crucial Definition 2.2. A Banach space \(E\) is Fraïssé if for every integer \(k\) and every \(\epsilon>0\), there exists \(\delta>0\) such that for any \(k\)-dimensional subspace \(X\) of \(E\), the canonical action of the group \(\operatorname{Iso}(E)\) of isometries on the set of \(\delta\)-embeddings of \(X\) into \(E\) is \(\epsilon\)-transitive on this set. This property is a strengthening of almost-transitivity which corresponds to the case \(k=1\). It will be shown later on (Theorem 4.1) that if \(1\leq p <\infty\) and \(p\not=2l\) with \(l\geq 2\), then the space \(L_p\) equipped with its natural norm is Fraïssé. Such spaces have remarkable isometric properties: for instance, a separable Banach space \(X\) is finitely representable in a Fraïssé space \(E\) if and only if it is isometric to a subspace of \(E\) (Proposition 2.13). The Fraïssé property can be relative to a family \(\mathcal{G}\) of finite-dimensional normed spaces. The Fraïssé limit \(\operatorname{Flim} \mathcal{G}\) of an appropriate class \(\mathcal{G}\) is defined in Definition 2.20. Fraïssé classes \(\mathcal{G}\) are defined in Definition 2.23, and such classes are exactly the collections of finite-dimensional subspaces of a “limit” Fraïssé space \(\operatorname{Flim} \mathcal{G}\) by the Fraïssé correspondence for normed spaces (Corollary 2.27). The Fraïssé limit of the sequence \((l_p^n)_n\) is \(L_p\) for all \(1\leq p<\infty\), while the limit of the sequence \((l_\infty^n)_n\) is the Gurarij space \(\mathbb{G}\), which is the unique separable Banach space with trivial cotype which is Fraïssé, and also the maximal separable Fraïssé space.
Section 3 provides a lattice version of the previous results. A motivation for this development is that when \(p\not=2\) is an even integer, the space \(L_p\) is not Fraïssé any more, but although its subspaces fail the relevant homogeneity property, its sublattices behave properly. The critical result (Theorem 3.8) of this section is a theorem due to G. Schechtman on perturbations of subspaces of \(L_p\) which are close to \(l_p^n\). The main result of Section 4 is the already quoted Theorem 4.1: the space \(L_p\) equipped with its natural norm is Fraïssé provided that \(1\leq p <\infty\) and \(p\not= 4, 6, 8,\dots\). The proof requests elaborate techniques, such as approximate equimeasurability principles of Hardin-Plotkin-Rudin type (see Theorem 4.4) and inversion formulas for computing a measure \(\mu\) when its \(p\)-characteristics are given (see Definition 4.12). To my taste, the inversion formula given by Lemma 4.22 is particularly neat.
Some ideas which go back to a seminal paper published by M. Gromov and V. D. Milman [Am. J. Math. 105, 843–854 (1983; Zbl 0522.53039)] are investigated in Section 5, devoted to the Approximate Ramsey Property (ARP) and related concepts. It may actually be so that all Fraïssé classes of finite dimensional spaces have the ARP (see Problem 5.8). Theorem 5.10 is a Kechris-Pestov-Todorcevic correspondence for normed spaces. The ARP of the sequence \((l_p^n)_n\) is translated into a multidimensional Borsuk-Ulam theorem (see Definition 5.14). The Mazur nonlinear mapping, Lévy families and concentration functions are also present in this Section and one feels that distortion and the Odell-Schlumprecht theorem might be somewhat related with this part of the work. Finally, Section 6 returns to the lattice considerations of Section 3. Fraïssé Banach lattices are defined in Definition 6.1: this definition is quite delicate since a Fraïssé Banach lattice is in particular a Banach lattice but not necessarily a Fraïssé Banach space as defined above. Indeed, all \(L_p\)-spaces (\(1\leq p <\infty)\) are Fraïssé Banach lattices. The main result of this section is the existence of a separable \(M\)-space which is a Fraïssé Banach lattice. This space is in fact a sublattice of a transitive (non separable!) \(M\)-space constructed by F. Cabello-Sánchez (Proposition 6.6), and it is actually \(\mathcal{C}([0, 1])\) equipped with some equivalent norm (which is not made explicit). Theorem 6.12 is a Kechris-Pestov-Todorcevic correspondence for Banach lattices, and thus also a device to construct new extremely amenable groups. An extensive bibliography concludes this important work.

MSC:

46B04 Isometric theory of Banach spaces
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
05C55 Generalized Ramsey theory
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
46B42 Banach lattices

Citations:

Zbl 0522.53039
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References:

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