Amalgamation and Ramsey properties of Lp spaces(Article)
- aDepartamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão, 1010, São Paulo, SP 05508-090, Brazil
- bDepartamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, Madrid, 28040, Spain
- cInstitut de Mathématiques de Jussieu, UMR 7586, 2 place Jussieu, Case 247, Paris Cedex 05, 75222, France
- dDepartment of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada
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. © 2020
Funding details
| Funding sponsor | Funding number | Acronym |
|---|---|---|
| Universidade de São Paulo | USP | |
| University of Toronto | U of T | |
| Centre National de la Recherche Scientifique | UMR7586 | CNRS |
| Fundação de Amparo à Pesquisa do Estado de São Paulo See opportunities by FAPESP | 2016/25574-8,2013/24827-1,2013/11390-4,2012/20084-1 | FAPESP |
| Conselho Nacional de Desenvolvimento Científico e Tecnológico | 303034/2015-7,2013-7/31466UC,303721/2019-2 | CNPq |
| Ministerio de Economía y Competitividad | MTM2016-76808-P | MINECO |
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1
V. Ferenczi, J. Lopez-Abad and B. Mbombo were supported by FAPESP , projects 2012/20084-1 , 2013/11390-4 , 2013/24827-1 and 2016/25574-8 . V. Ferenczi was supported by CNPq , projects 303034/2015-7 and 303721/2019-2 . V. Ferenczi and S. Todorcevic were supported by USP Cofecub project number 2013-7/31466UC . J. Lopez-Abad was partially supported by the Ministerio de Econom\u00EDa y Competitividad under grant MTM2016-76808-P . S. Todorcevic was also supported by grants from NSERC ( 455916 ) and CNRS ( UMR7586 ). This work was initiated during a visit of J.L.-A. to the Universidade de Sao P\u00E3ulo in 2014, and continued during visits of J.L.-A. to the Fields Institute in the Fall 2014, a visit of all the authors at the Banff International Research Station in occasion of the Workshop on Homogeneous Structures in the Fall 2015, a visit of J.L.-A. to the University of Toronto, and a visit of V.F. to UNED in the winter of 2018. The hospitality of all these institutions is gratefully acknowledged.
- ISSN: 00018708
- Source Type: Journal
- Original language: English
- DOI: 10.1016/j.aim.2020.107190
- Document Type: Article
- Publisher: Academic Press Inc.
Lopez-Abad, J.; Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, Madrid, Spain;
© Copyright 2020 Elsevier B.V., All rights reserved.
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