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The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs

Bartošová, Dana and López-Abad, Jordi and Lupini, Martino and Mbombo, Brice (2017) The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs. . (Submitted)

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We show that the Gurarij space G and its noncommutative analog NG both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex P and its noncommutative analogue NP. The former is P itself, and the latter is the state space of the operator system associated with NP. This answers a question of Conley and Törnquist. We also show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable. These results are obtained via the Kechris--Pestov--Todorcevic correspondence, by establishing the approximate Ramsey property for several classes of finite-dimensional operator spaces and operator systems (with distinguished linear functionals), including: Banach spaces, exact operator spaces, function systems with a distinguished state, and exact operator systems with a distinguished state. This is the first direct application of the Kechris--Pestov--Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild. In the second part of the paper, we obtain factorization theorems for colorings of matrices and Grassmannians over R and C, which can be considered as continuous versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Additional Information:D.B. was supported by the grant FAPESP 2013/14458-9. J.L.-A. was partially supported by the grant MTM2012-31286 (Spain) and the Fapesp Grant 2013/24827-1 (Brazil). M.L. was partially supported by the NSF Grant DMS-1600186. R. Mbombo was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) postdoctoral grant, processo 12/20084-1. This work was initiated during a visit of J.L.-A. to the Universidade de Sao Pãulo in 2014, and continued during visits of D.B. and J.L.-A. to the Fields Institute in the Fall 2014, a visit of M.L. to the Instituto de Ciencias Matemáticas in the Spring 2015, and a visit of all the authors at the Banff International Research Station in occasion of the Workshop on Homogeneous Structures in the Fall 2015. The hospitality of all these institutions is gratefully acknowledged.
Funding AgencyGrant Number
Fundação de Amparo à Pesquisa do Estado de Sao Paulo (FAPESP)2013/14458-9
Ministerio de Ciencia e Innovación (MCINN)MTM2012-31286
Fundação de Amparo à Pesquisa do Estado de Sao Paulo (FAPESP)2013/24827-1
Fundação de Amparo à Pesquisa do Estado de Sao Paulo (FAPESP)12/20084-1
Subject Keywords:Gurarij space, Poulsen simplex, extreme amenability, Ramsey property, Banach space, Choquet simplex, operator space, operator system, oscillation stability, Dual Ramsey Theorem
Classification Code:2000 Mathematics Subject Classification. Primary 05D10, 46B04; Secondary 47L25, 46A55
Record Number:CaltechAUTHORS:20180410-131618091
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85727
Deposited By: Tony Diaz
Deposited On:11 Apr 2018 18:40
Last Modified:03 Oct 2019 19:34

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