The Ramsey property for Operator spaces and noncommutative Choquet simplices

Abstract

The noncommutative Gurarij space $\mathbb{\mathbb{\mathbb{NG}}}$, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fra\"{\i}ss\'{e} 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 $\mathbb{\mathbb{NG}}$ 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 $\mathbb{NP}$ of the Fra\"{\i}ss\'{e} limit $A(\mathbb{NP})$ of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of $\mathbb{NP}$ on the compact set $\mathbb{NP}_1$ of unital linear functionals on $A(\mathbb{NP})$ is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of \textrm{Aut}$\left( \mathbb{NP}% \right) $.