The Ramsey property for operator spaces and noncommutative Choquet simplices

Author(s)

Bartosova, Dana ^[1] ; Lopez-Abad, Jordi ^[2] ; Lupini, Martino ^[3] ; Mbombo, Brice ^{[4, 5]}

Affiliation

^[1] Univ Florida, Dept Math, POB 118105, Gainesville, FL 32611 - USA

^[2] UNED, Fac Ciencias, Dept Matemat Fundamentales, Madrid 28040 - Spain

^[3] Victoria Univ Wellington, Sch Math & Stat, POB 600, Wellington 6140 - New Zealand

^[4] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[5] Coll Lionel Groulx, 100 Rue Duquet, Sainte Therese, PQ J7E 3G6 - Canada

• Document type: Journal article
• Source: JOURNAL OF FUNCTIONAL ANALYSIS; v. 281, n. 9 NOV 1 2021.
• Web of Science Citations: 0

Abstract

The noncommutative Gurarij space NG, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fraisse limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of NG is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris-Pestov-Todorcevic correspondence in the setting of operator spaces. Recent work of Davidson and Kennedy, building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space NP of the Fraisse limit A(NP) of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of NP on the compact set NP1 of unital linear functionals on A(NP) is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow ofAut(NP). (C) 2021 Elsevier Inc. All rights reserved. (AU)

• 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: 12/20084-1 - Universal topological groups
• Grantee: Brice Rodrigue Mbombo Dempowo
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• 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
• FAPESP's process: 16/25574-8 - Geometry of Banach Spaces
• Grantee: Valentin Raphael Henri Ferenczi
• Support Opportunities: Research Projects - Thematic Grants