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

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

@article{CRMATH_2017__355_12_1242_0,
     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},
}

TY  - JOUR
AU  - Dana Bartošová
AU  - Jordi Lopez-Abad
AU  - Martino Lupini
AU  - Brice Mbombo
TI  - The Ramsey property for Banach spaces and Choquet simplices, and applications
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 1242
EP  - 1246
VL  - 355
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2017.11.001
LA  - en
ID  - CRMATH_2017__355_12_1242_0
ER  -

%0 Journal Article
%A Dana Bartošová
%A Jordi Lopez-Abad
%A Martino Lupini
%A Brice Mbombo
%T The Ramsey property for Banach spaces and Choquet simplices, and applications
%J Comptes Rendus. Mathématique
%D 2017
%P 1242-1246
%V 355
%N 12
%I Elsevier
%R 10.1016/j.crma.2017.11.001
%G en
%F CRMATH_2017__355_12_1242_0

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/

Bibliographie
Cité par

[1] D. Bartosova; J. Lopez-Abad; M. Lupini; B. Mbombo The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs | arXiv

[2] I. Ben Yaacov Fraïssé limits of metric structures, J. Symb. Log., Volume 80 (2015) no. 1, pp. 100-115

[3] C. Conley; A. Törnquist A Fraïssé approach to the Poulsen simplex, Sets and Computations, World Scientific, 2017, pp. 11-24

[4] S. Glasner Distal and semisimple affine flows, Amer. J. Math., Volume 109 (1987) no. 1, pp. 115-131

[5] R.L. Graham; B.L. Rothschild Ramsey's theorem for n-parameter sets, Trans. Amer. Math. Soc., Volume 159 (1971), pp. 257-291

[6] M. Gromov; V.D. Milman A topological application of the isoperimetric inequality, Amer. J. Math., Volume 105 (1983) no. 4, pp. 843-854

[7] V.I. Gurariĭ Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sib. Math. J., Volume 7 (1966), pp. 1002-1013

[8] W. Herer; J.P.R. Christensen On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann., Volume 213 (1975), pp. 203-210

[9] A.S. Kechris; V. Pestov; S. Todorcevic Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 106-189

[10] W. Kubi; S. Solecki A proof of uniqueness of the Gurariĭ space, Isr. J. Math., Volume 195 (2013) no. 1, pp. 449-456

[11] J. Lindenstrauss; G. Olsen; Y. Sternfeld The Poulsen simplex, Ann. Inst. Fourier, Volume 28 (1978) no. 1, pp. 91-114

[12] M. Lupini Fraïssé limits in functional analysis, 2015 | arXiv

[13] M. Lupini Uniqueness, universality, and homogeneity of the noncommutative Gurarij space, Adv. Math., Volume 298 (2016), pp. 286-324

[14] J. Melleray Some problems related to the themes of the HIM trimester “Homogeneity and Universality”, Selected Problems Universality Trimester, Hausdorff Institute of Mathematics, Bonn, Germany, 2013

[15] J. Melleray; T. Tsankov Extremely amenable groups via continuous logic, 2014 | arXiv

[16] V. Pestov On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc., Volume 350 (1998) no. 10, pp. 4149-4165

[17] E.T. Poulsen A simplex with dense extreme points, Ann. Inst. Fourier, Volume 11 (1961), pp. 83-87

Cité par Sources :

Commentaires - Politique