The Ramsey property for Banach spaces and Choquet simplices

Author(s)

Bartosova, Dana ; Lopez-Abad, Jordi ; Lupini, Martino ; Mbombo, Brice

• Document type: Journal article
• Source: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY; v. 24, n. 4, p. 36-pg., 2022-01-01.

Abstract

We show that the Gurarij space G has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex P and we prove that it consists of the canonical action on P itself. This answers a question of Conley and Tornquist. We 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 several Kechris-Pestov-Todorcevic correspondences, by estab-lishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and their versions with distinguished contractions. 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. (AU)

• FAPESP's process: 12/20084-1 - Universal topological groups
• Grantee: Brice Rodrigue Mbombo Dempowo
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 13/24827-1 - Ramsey theory methods in Banach spaces
• Grantee: Valentin Raphael Henri Ferenczi
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• FAPESP's process: 13/14458-9 - Algebra in the Cech-Stone compactification and its applications to topological dynamics, ergodic theory and Ramsey theory.
• Grantee: Dana Bartosova
• Support Opportunities: Scholarships in Brazil - Post-Doctoral