Authors: Ferenczi, Valentin
Lopez-Abad, Jorge
Mbombo, Brice
Todorčević, Stevo 
Title: Amalgamation and Ramsey properties of Lp spaces
Journal: Advances in Mathematics
Volume: 369
First page: 107190
Issue Date: 5-Aug-2020
Rank: M21
ISSN: 0001-8708
DOI: 10.1016/j.aim.2020.107190
Abstract: 
We study the dynamics of the group of isometries of Lp-spaces. In particular, we study the canonical actions of these groups on the space of δ-isometric embeddings of finite dimensional subspaces of Lp(0,1) into itself, and we show that for every real number 1≤p<∞ with p≠4,6,8,… they are ε-transitive provided that δ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces Lp(0,1), p≠4,6,8,… and the Gurarij space. We also give a proof of the Ramsey property of the classes {ℓpn}n, p≠2,∞, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X.
Keywords: Amalgamation | Extreme amenability | Fraïssé theory | Isometries on L spaces p | Ramsey property | Ultrahomogeneity
Publisher: Elsevier
Project: FAPESP, projects 2012/20084-1, 2013/11390-4, 2013/24827-1 and 2016/25574-8
CNPq, projects 303034/2015-7 and 303721/2019-2
USP Cofecub project number 2013-7/31466UC
NSERC (455916)
CNRS (UMR7586)

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