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

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

Published 2017-08-03Version 1

We show that the Gurarij space $\mathbb{G}$ and its noncommutative analog $\mathbb{NG}$ both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex $\mathbb{P}$ and its noncommutative analogue $\mathbb{NP}$. The former is $\mathbb{P}$ itself, and the latter is the state space of the operator system associated with $\mathbb{NP}$. This answers a question of Conley and T\"{o}rnquist. We also show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ 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 $\mathbb{R}$ and ${\mathbb{C}}$, 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.