Title: The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs,
Abstract: 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
Törnquist. 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.