The Ramsey property for Banach spaces and Choquet simplices, and applications

Dana Bartošová; Jordi Lopez-Abad; Martino Lupini; Brice Mbombo

doi:10.1016/j.crma.2017.11.001

Functional analysis/Dynamical systems

The Ramsey property for Banach spaces and Choquet simplices, and applications
[La propriété de Ramsey des espaces de Banach et des simplexes de Choquet, et applications]

Présenté par : Gilles Pisier

Dana Bartošová ¹ ; Jordi Lopez-Abad ² ; Martino Lupini ³ ; Brice Mbombo ⁴

¹ Department of Mathematical Sciences, Carnegie Mellon University, PA, USA
² Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, 28040 Madrid, Spain
³ Mathematics Department, California Institute of Technology, 1200 E. California Blvd, MC 253-37, Pasadena, CA 91125, USA
⁴ Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1242-1246.

Résumé
Abstract

Nous montrons que la classe des espaces de Banach de dimension finie et la classe des simplexes de Choquet de dimension finie ont la propriété de Ramsey. En guise d'application, nous montrons que le groupe $Aut (G)$ des isométries linéaires surjectives de l'espace de Gurarij $G$ est extrêmement moyennable, et que l'action canonique $Aut (P) ↷ P$ est le flot minimal universel du groupe $Aut (P)$ des homéomorphismes affines du simplexe de Poulsen $P$ . Ceci répond aux questions de Melleray–Tsankov et Conley–Törnquist.

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group $Aut (G)$ of surjective linear isometries of the Gurarij space $G$ is extremely amenable, and that the canonical action $Aut (P) ↷ P$ is the universal minimal flow of the group $Aut (P)$ of affine homeomorphisms of the Poulsen simplex $P$ . This answers questions of Melleray–Tsankov and Conley–Törnquist.

Reçu le : 2017-08-11
Accepté le : 2017-11-06
Publié le : 2017-11-13

DOI : 10.1016/j.crma.2017.11.001

Affiliations des auteurs :

Dana Bartošová ¹ ; Jordi Lopez-Abad ² ; Martino Lupini ³ ; Brice Mbombo ⁴

¹ Department of Mathematical Sciences, Carnegie Mellon University, PA, USA
² Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, 28040 Madrid, Spain
³ Mathematics Department, California Institute of Technology, 1200 E. California Blvd, MC 253-37, Pasadena, CA 91125, USA
⁴ Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

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     author = {Dana Barto\v{s}ov\'a and Jordi Lopez-Abad and Martino Lupini and Brice Mbombo},
     title = {The {Ramsey} property for {Banach} spaces and {Choquet} simplices, and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1242--1246},
     publisher = {Elsevier},
     volume = {355},
     number = {12},
     year = {2017},
     doi = {10.1016/j.crma.2017.11.001},
     language = {en},
}

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Dana Bartošová; Jordi Lopez-Abad; Martino Lupini; Brice Mbombo. The Ramsey property for Banach spaces and Choquet simplices, and applications. Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1242-1246. doi : 10.1016/j.crma.2017.11.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.001/

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