The Ramsey property for Banach spaces and Choquet simplices

We show that the Gurarij space $\mathbb{G}$ 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 $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and T\"{o}rnquist. We 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 several Kechris-Pestov-Todorcevic correspondences, by establishing 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.


Introduction
Given a topological group G, a compact G-space or G-flow is a compact Hausdorff space X endowed with a continuous action of G.Such a G-flow X is called minimal when every orbit is dense.There is a natural notion of morphism between G-flows, given by a G-equivariant continuous map (factor).A minimal G-flow is universal if it factors onto any minimal G-flow.It is a classical fact that any topological group G admits a unique (up to isomorphism of G-flows) universal minimal flow, usually denoted by M (G) [13,26].For any locally compact non compact Polish group G, the universal minimal G-flow is nonmetrizable.At the opposite end, non locally compact topological groups often have metrizable universal minimal flows, or even reduced to a single point.A topological group for which M (G) is a singleton is called extremely amenable.(Amenability of G is equivalent to the assertion that every compact G-space has an invariant Borel measure.Thus any extremely amenable group is, in particular, amenable.) The universal minimal flow has been explicitly computed for a number of topological groups, typically given as automorphism groups of naturally arising mathematical structures.Examples of extremely amenable Polish groups include the group of order automorphisms of Q [46], the group of unitary operators on the separable infinite-dimensional Hilbert space [24], the automorphism group of the hyperfinite II 1 factor and of infinite type UHF C * -algebras [11,15], or the isometry group of the Urysohn space [47].Examples of nontrivial metrizable universal minimal flows include the universal minimal flow of the group of orientation preserving homeomorphisms of the circle, which is equivariantly homeomorphic to the circle itself [46], the universal minimal flow of the group S ∞ of permutations of N, which can be identified with the space of linear orders on N [16], and the universal minimal flow of the homeomorphism group Homeo(2 N ) of the Cantor set 2 N , which can be seen as the canonical action of Homeo(2 N ) on the space of maximal chains of closed subsets of 2 N [17, 28,51].
There are essentially two known ways to establish extreme amenability of a given topological group.The first method involves the phenomenon of concentration of measure, and can be applied to topological groups that admit an increasing sequence of compact subgroups with a dense union [24,48,Chapter 4].The second method applies to automorphism groups of discrete ultrahomogeneous structures or, more generally, approximately ultrahomogeneous metric structures [48,Chapter 6].A metric structure is approximately ultrahomogeneous if any partial isomorphism between finitely generated substructures is the pointwise limit of maps that are restrictions of automorphisms.It is worth noting that any Polish group can be realized as the automorphism group of an approximately ultrahomogeneous metric structure [40,Theorem 6].For the automorphism group Aut(M ) of an approximately ultrahomogeneous structure M , extreme amenability is equivalent to the approximate Ramsey property of the class of finitely generated substructures of M .This criterion is known as the Kechris-Pestov-Todorcevic (KPT) correspondence, first established in [28] for discrete structures, and recently generalized to the metric setting in [42].The discrete KPT correspondence has been used extensively in the last decade.In this paper the KPT correspondence is directly used for the first time to obtain new natural extreme amenability results.
In all the known examples of computations of metrizable universal minimal flows, the argument hinges on extreme amenability of a suitable subgroup and the following result due to Nguyen Van Thé [45] based on previous work of Pestov [46].Suppose that G is a topological group with an extremely amenable closed subgroup H.If the completion X of the homogeneous space G/H endowed with the quotient of the right uniformity on G is a minimal compact G-space, then X is the universal minimal flow of G.It was recently shown in [8,41] that, whenever the universal minimal flow of G is metrizable, it can be realized as the completion of G/H for a suitable closed subgroup H of G.
In this paper we compute the universal minimal flows of the automorphism groups of structures coming from functional analysis and Choquet theory: the Gurarij space G and the Poulsen simplex P. Recall that the Gurarij space is the unique separable approximately ultrahomogeneous Banach space that contains ℓ ∞ n for every n ∈ N [36], while P is the unique nontrivial metrizable Choquet simplex with dense extreme boundary [33].The group Aut(G) of surjective linear isometries of the Gurarij space is shown to be extremely amenable by establishing the approximate Ramsey property of the class of finite-dimensional Banach spaces.This answers a question of Melleray and Tsankov from [42] Similarly, the stabilizer Aut p (P) of an extreme point p of P is proven to be extremely amenable by establishing the approximate Ramsey property of the class of Choquet simplices with a distinguished point.It is then deduced from this that the universal minimal flow of Aut(P) is P itself, endowed with the canonical action of Aut(P).This answers Question 4.4 from [10].More generally, we prove that for any closed face F of P, the pointwise stabilizer Aut F (P) is extremely amenable.The analogous result holds in the Banach space setting as well.A Lazar simplex is a compact absolutely convex set that arises as the unit ball of the dual of a Lindenstrauss space.The Lusky simplex L is the Lazar simplex that arises in this fashion from the Gurarij space.The group Aut(G) can be identified with the group Aut(L) of symmetric affine homeomorphisms of L. It is proven in [35,Theorem 1.2] that L plays the same role among Lazar simplices as the Poulsen simplex plays in the class of Choquet simplices, where closed faces are replaced with closed bifaces.We prove that, for any closed proper biface H of L, the corresponding pointwise stabilizer Aut H (L) is extremely amenable.In the particular case when H is the trivial biface, this recovers the extreme amenability of Aut(G).
Recall that a function system is a closed subspace V of the space of continuous C-valued functions C(T ) of some compact Hausdorff space T containing the function constantly equal to 1 and such that if f ∈ V then the function f * defined by f * (t) = f (t) also belongs to V .In particular, when K is a compact convex set, the space A(K) of continuous complex-valued affine functions on K is a function system, and in fact any function system V ⊆ C(T ) arises in this way from a suitable compact convex set K. Precisely, K is the compact convex set of states of V , that is, the contractive functionals on V that are unital, i.e. map the unit of C(T ) to 1 ([1, Theorem II.1.8]).Furthermore, the map K → A(K) is a contravariant isomorphism of categories from the category of compact convex sets and continuous affine maps, to the category of function systems and unital linear contractions (Kadison correspondence).A metrizable compact convex set K is a simplex if and only if A(K) is a separable Lindenstrauss space, which means that the identity map of A(K) is the pointwise limit of a sequence of unital completely contractive maps that factor through finite-dimensional (abelian) C * -algebras.The function system A(P) corresponding to the Poulsen simplex is the unique separable approximately ultrahomogeneous function system that contains unital copies of ℓ ∞ n for n ∈ N [35, Theorem 1.1].The automorphism group Aut(A(P)) can be identified with the group of affine homeomorphisms of P. The Poulsen simplex P is then equivariantly homeomorphic to the state space of A(P).
The main tool to establish the results mentioned above will be the Dual Ramsey Theorem of Graham and Rothschild [23].This is a powerful pigeonhole principle known to imply many other results, such as the Hales-Jewett theorem, and the Ramsey theorem.It can be seen to be equivalent to a factorization result for colorings of Boolean matrices, which implies the celebrated Graham-Leeb-Rothschild theorem on Grassmannians over a finite field [22].In fact, it is shown in [4,5] that this is again a particular case of a factorization result for colorings of matrices over a finite field, stating that the coloring of matrices only depends on the invertible matrix needed to transform a given matrix into one in reduced column echelon form.In this paper, we provide factorization theorems for colorings of matrices and Grassmannians over the real or complex numbers.We prove in particular that colorings of matrices depend only on the canonical norm that a given matrix determines, while colorings of Grassmannians are determined by the Banach-Mazur type of the given subspace.
The paper is organized as follows.We start in Subsection 2.1 by recalling some basic concepts such as extreme amenability.In Subsection 2.2 we recall and introduce different versions of ultrahomogeneity and Ramsey properties for Banach spaces, and we prove a version of the KPT correspondence in this setting (Theorem 2.12).In Subsection 2.3 we prove the approximate Ramsey property (ARP) of the class {ℓ n ∞ } n .This has as a consequence the extreme amenability of the group of isometries of the Gurarij space.In Subsection 2.4 we prove the (ARP) of the class of polyhedral finite-dimensional spaces, and the class of all finite-dimensional Banach spaces.Using this, in Subsection 2.5 we give a direct proof of the (ARP) for the class of finite metric spaces.This provides a new proof of extreme amenability of the isometry group of the Urysohn space [47].Subsection 2.6 studies closed bifaces of Lusky simplices.We prove that group stabilizers of closed proper bifaces of the Lazar simplex are extremely amenable.This is done by establishing the corresponding (ARP) and a (KPT)correspondence, introduced in Subsubsection 2.6.1.In Section 3 we study Choquet simplices (with a distinguished face), and we prove that pointwise stabilizer of any closed proper face of the Poulsen simplex is extremely amenable.We conclude in Subsection 3.4 we prove that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex P is the canonical action on P.

The Ramsey property of Banach spaces
The goal of this section is to introduce different notions of "Ramsey property" for several classes of structures.We show that, in the setting we are interested in, such notions are equivalent to each other.We furthermore establish an analogue of the Kechris-Pestov-Todorcevic correspondence in this section.We then establish the (stable) Ramsey property for the class of Banach spaces {ℓ n ∞ } n .From this, we infer that that the group of isometries of the Gurarij space is extremely amenable.

2.1.
Colorings and extreme amenability.We introduce some terminology to be used in the following.A metric coloring of a pseudo-metric space M is a 1-Lipschitz map from M to a metric space (K, d K ).A metric coloring with target space (K, d K ) will also be called a K-coloring.Following [42], a continuous coloring is a metric coloring whose target space is the closed unit interval [0, 1].A compact coloring is a metric coloring whose target space is a compact metric space.For a subset X of a compact metric space (K, d K ) and ε > 0, the ε-fattening K ε denotes the set of elements of K at distance at most ε from some element of X.
The oscillation osc F (c) of a compact coloring c : M → (K, d K ) on a subset F of M is the supremum of d K (c(y), c(y ′ )) where y, y ′ range within F .If osc F (c) ≤ ε, then we say that c ε-stabilizes on F , or that F is ε-monochromatic for c.A finite coloring of M is a function from M to a finite set X. When the target space is a natural number r (identified with the set {0, 1, . . ., r − 1} of its predecessors), we will say that c is an r-coloring.A subset F of M is monochromatic for c if c(p) = c(q) for every p, q ∈ F , and ε-monochromatic for c if there exists some x ∈ X such that for every p ∈ F there is q ∈ M such that c(q) = x and d M (p, q) ≤ ε.If F is ε-monochromatic, then we also say that c ε-stabilizes on F .
Given a Polish group G and a continuous action G M of G on a metric space (M, d M ), we write [p] G to denote the closure of the G-orbit of p ∈ M , and M//G to denote the space of closures of G-orbits of M .Since G acts by isometries the formula defines the quotient pseudometric induced by the quotient map π M,G : M → M//G, and since we consider closures of orbits, d G,M is a metric.It is easy to see that d G,M is complete when d M is complete.
When M is endowed with an action of a Polish group G we say that M is a metric G-space.A compact coloring c : (M, d M ) → (K, d K ) is finitely G-factorizable when there is a K-coloring c : M//G → K defined on the space M//G of closed G-orbits of M such that for every ε > 0 and every compact subset Similarly, c is finitely oscillation stable [48,Definition 1.1.8]if for every compact subset F of M and ε > 0 there exists g ∈ G such that c ε-stabilizes on g • F .We say that the action of G on M is finitely oscillation stable if every continuous coloring of M is finitely oscillation stable [48,Definition 1.1.11].
Given a compact metric space (K, d K ), we let Lip((M, d M ), (K, d K )) be the collection of all Kcolorings of M .With the topology of pointwise convergence Lip((M, Lemma 2.1.Suppose that G is a Polish group, and that M is a metric G-space.Let F be a ⊆-directed family of compact subsets of M whose union is M .The following assertions are equivalent: 1) Every compact coloring of M is finitely G-factorizable.
2) For every F ∈ F, every compact metric space K and every ε > 0 there is a H ∈ F such that for every coloring c : Proof.Suppose that 1) holds but not 2).Fix the counterexample K, F, M , ε > 0 and F ∈ F and for each H ∈ F containing F we fix a bad coloring c H : H → K.For each V ∈ F, let V be the collection of those W ∈ F containing V .Choose a non-principal ultrafilter U on F containing each V .This is possible since F is ⊆-directed.Define c U : M → K by declaring c U (p) := U − lim c H (p).This is well defined because there is some H ∈ F such that p ∈ H. Let c : M//G → K be the corresponding factorization, and let g be such that d K ( c([p] G ), c U (p)) ≤ ε/2 for every p ∈ g • F .Choose H ∈ F to be such that p • F ⊆ H and such that d K (c H (p), c U (p)) ≤ ε/2 for every p ∈ g • F .Then the restriction c : H//G → K disproves that c H is a bad color.Suppose now that 2) holds but not 1).This means that there is some c : M → K that cannot be finitely G-factorized, so we fix the corresponding ε > 0.
For every F ∈ F we use 2) for it, K, and for ε/2 to find the corresponding H F ∈ F, and then we apply the property of it to the restriction c : H → K to find e F : H//G → K. Now define c : M//G → K as the U -limit of (e F ) F .Since c does not finitely G-factorize c there must be a bad compact A witnessing this.Without loss of generality we may assume that A is a finite set.Let F ∈ F be such that A ⊆ F , and let contradicting the defining property of A.
Recall that a topological group G is called extremely amenable if every continuous action of G on a compact Hausdorff space has a fixed point.The following characterization of extreme amenability will be used extensively in this paper.Proposition 2.2.Suppose that G is a Polish group.The following assertions are equivalent.1) G is extremely amenable.2) For every left-invariant compatible metric d G on G, the left translation of G on (G, d G ) is finitely oscillation stable.
3) Every compact coloring of a metric G-space is finitely G-factorizable.4) Let M be a metric G-space, and let F be a ⊆-directed family of compact subsets of M whose union is M .For every F ∈ F, every compact metric space K and every ε > 0 there is an H ∈ F such that for every coloring c : H → K there is a coloring c : H//G → K and g ∈ G such that gF ⊆ H and such that d Proof.The equivalence of 1) and 2) can be found in [48,Theorem 2.1.11].The implication 3)⇒2) is immediate, since orbit space G//G is one point.We now establish the implication 1)⇒3): By the extreme amenability of G, there is some The equivalence of (3) and (4) follows from Lemma 2.1.

2.2.
The Ramsey property and the KPT correspondence for Banach spaces.In this section, we provide a characterization of extreme amenability of the isometry group of a Banach space (endowed with the topology of pointwise convergence).This can be seen as an analogue in this context of the Kechris-Pestov-Todorcevic from [28].A more general KPT correspondence for arbitrary metric structures is the topic of [42].
We introduce some basic terminology on Banach spaces.Let F be equal to R or to C. Given n ∈ N, and 1 ≤ p < ∞, let ℓ n p be the normed space (F n , • p ) where (a j ) j<n p = ( j<n |a j | p ) 1/p is the pnorm; similarly, let ℓ n ∞ = (F n , • ∞ ) where (a j ) j<n ∞ := max j<n |a j |.Given a Banach space (X, • ), let Ball(X) := {x ∈ X : x ≤ 1}, Sph(X) := {x ∈ X : x = 1} be the unit ball an the unit sphere of X, respectively.Recall that given two Banach spaces X, Y , a contraction T : X → Y is a bounded linear mapping T : X → Y such that T := max x ≤1 T (x) ≤ 1.Given δ ≥ 0, let Emb δ (X, Y ) be the space of contractions T : X → Y such that T x ≥ x /(1 + δ), endowed with its norm metric, d(T, U ) := T − U := max x ≤1 T (x) − U (x) ; when δ = 0, Emb(X, Y ) := Emb 0 (X, Y ) is the space of isometric embeddings from X into Y .Dually, when X and Y are finite-dimensional, a quotient map T : X → Y is a linear mapping such that T (Ball(X)) = Ball(Y ).The space of those quotient maps is denoted by Quo(X, Y ).It is well-known that T ∈ Emb(X, Y ) if and only if its dual operator T * : Y * → X * is a quotient map, and this assignment is an isometry.Finally, given a Banach space E, let Iso(E) be the group of surjective isometries of E, endowed with its strong operator topology (SOT), and observe that Iso(E) acts continuously on Emb δ (X, E) by left composition, g • T := g • T .
In particular, suppose that X is a finite-dimensional subspace of E. Given a finite-dimensional subspace Y of E containing X we can canonically identify Emb(X, Y ) with the collection of those isometric embeddings T : X → E such that Im T ⊆ Y , so in this way Emb(X, E) = X⊆Y ⊆E Emb(X, Y ), where each Emb(X, Y ) is a compact subset of Emb(X, E).Suppose that Iso(E) is extremely amenable.By applying Proposition 2.2 we obtain that given such X ⊆ Y , compact metric (K, d K ) and ε > 0 we can find some finite-dimensional subspace Z 0 of E such that for every coloring c : Emb(X, Z 0 ) → K there is some g ∈ Iso(E) such that there is a coloring c : Emb(X, Z 0 )// Iso(E) → K and max We consider on Lip(Emb(X, Z 0 ), K) the compatible metric defined for K-colorings c 1 and c 2 by d(c 1 , c 2 ) := max γ∈Emb(X,Z 0 ) d K (c 1 (γ), c 2 (γ)).Using that Lip(Emb(X, Z 0 ), K) is compact, we can find a finite ε/2-dense subset D of it, and for each c ∈ D we choose some g c ∈ Iso(E) witnessing (1).Let Z be a finite-dimensional subspace of E containing Y and c∈D g c Y .Then Z has the property that for every coloring c : Emb(X, Z) → K there is g ∈ Iso(E) and c : Emb(X, Z)// Iso(E) → K with the property that gY ⊆ Z and d K (c(g • γ), c([g] Iso(E) )) ≤ ε for every γ ∈ Emb(X, Y ).This means in particular that the oscillation of c in g • Emb(X, Y ) is determined by the diameter of Emb(X, E)// Iso(E).
Recall that an action G M of a group G on a metric space (M, d) is ε-transitive, for some ε > 0, when the diameter of M//G is at most ε, that this, when for every x, y ∈ M there is some g ∈ G such that d(g • x, y) ≤ ε.G M is approximately transitive when it is ε-transitive for every ε > 0, or equivalently, when M//G consists of a point.Definition 2.3.A Banach space E is called approximately ultrahomogeneous when for every finitedimensional subspace X of E one has that Iso(E) Emb(X, E) is approximately transitive.
Hence, we obtain the following.
Corollary 2.4.Suppose that E is approximately ultrahomogeneous, and suppose that Iso(E) is extremely amenable.Then for every finite-dimensional subspaces X ⊆ Y of E and every compact metric space (K, d K ) there is some finite-dimensional subspace Z of E containing Y with the property that every coloring c : Emb(X, Z) → K ε-stabilizes in some set of the form γ • Emb(X, Y ).
Up to now the list of known approximately ultrahomogeneous (real or complex) Banach spaces is: • Hilbert spaces (indeed, they are ultrahomogeneous, i.e. the algebraic quotients Emb(X, E)/G consist always of a point); • The Lebesgue spaces L p [0, 1] when p / ∈ 2N, proved by W. Lusky in [38]; The original characterization of the Gurarij space considered by Gurarij [25] and Lusky [36,37,39] is as the unique separable Banach space satisfying the following extension property: for every finitedimensional Banach spaces E ⊆ F , linear contraction ϕ : E → G, and ε > 0, there exists an extension φ : F → G satisfying || φ|| < 1 + ε.The fact that such a space is indeed approximately ultrahomogeneous as in Definition 2.3 is proved by I. Ben Yaacov in [6].
The isometry groups (endowed with the strong operator topology) of the Banach spaces in the list above have very special topological dynamical properties.The groups Iso(L p (0, 1)) are extremely amenable for every 1 ≤ p < ∞, which was proved in the case of p = 2 by M. Gromov and V. D. Milman [24] and for p = 2 by T. Giordano and V. Pestov [15].Both of the cases use the method of concentration of measure.In this paper we prove the following.
Theorem 2.5.The group of isometries of the Gurarij space endowed with the strong operator topology is extremely amenable.
Our proof is not based on concentration of measure, but on a combinatorial property, the approximate Ramsey property, that characterizes the extreme amenability of certain isometry groups.With a similar approach this has been extended in [4] to the context of operator spaces.We now to introduce several variants of the Ramsey property for Banach spaces.Definition 2.6 (Approximate Ramsey Property).Let F be a family of finite-dimensional Banach spaces.a) F satisfies the approximate Ramsey property (ARP) if for any X, Y ∈ F and ε > 0 there exists Z ∈ F such that any continuous coloring of Emb(X, Z) ε-stabilizes on γ • Emb(X, Y ) for some γ ∈ Emb(Y, Z). b) F satisfies the compact approximate Ramsey property when for any X, Y ∈ F, ε > 0 and compact metric space (K, d K ) there exists Z ∈ F such that any K-coloring of Emb(X, Z) ε-stabilizes on γ • Emb(X, Y ) for some γ ∈ Emb(Y, Z). c) F satisfies the discrete approximate Ramsey property when for every X, Y ∈ F, r ∈ N and ε > 0 there is some Z ∈ F such that any r-coloring of Emb(X, Z) ε-stabilizes on γ • Emb(X, Y ) for some γ ∈ Emb(Y, Z).
So, rephrasing Corollary 2.4, if E is an approximately ultrahomogeneous Banach space whose isometry group is extremely amenable, then the class Age(E) of finite-dimensional subspaces of E has the approximate Ramsey property.Conversely, we will see in Theorem 2.12 that in fact the (ARP) of Age(E) characterizes the extreme amenability of Iso(E) for approximately ultrahomogeneous spaces E. Now we show that the different versions of the Ramsey property are in fact equivalent.
Proposition 2.7.The following are equivalent for a class F of finite-dimensional Banach spaces: 1) F satisfies the (ARP).
Proof.The compact (ARP) obviously implies the (ARP).Suppose that F satisfies the (ARP), and let us prove that F satisfies the discrete (ARP).This is done by induction on r ∈ N. The case r = 1 is trivial.Suppose that we have shown that F satisfies the discrete (ARP) for r-colorings.Consider X, Y ∈ F and ε > 0. Then by the inductive hypothesis, there is Z 0 ∈ F such that every r-coloring of Emb (X, Z 0 ) ε-stabilizes on γ • Emb (X, Y ) for some γ ∈ Emb (Y, Z 0 ).Since by the assumption F satisfies the continuous (ARP), there is Z ∈ F such that every continuous coloring of Emb (X, Z) ε/2-stabilizes on γ • Emb (X, Z 0 ) for some γ ∈ Emb (Z 0 , Z).We claim that Z witnesses that F satisfies the discrete (ARP) for (r + 1)-colorings.Indeed, suppose that c is an (r + 1)-coloring of Emb (X, Z).Define . This is a continuous coloring, so by the choice of Z there exists Y ).This concludes the proof that the continuous (ARP) implies the discrete (ARP).
Finally, the discrete (ARP) implies the compact (ARP).In fact, given ε > 0 and a compact metric space K, one can find a finite ε-dense subset D ⊆ K. Thus if Z ∈ F witnesses the discrete (ARP) for X, Y , ε and D, then given a 1-Lipschitz We are going to see that, when E is approximately ultrahomogeneous, the extreme amenability of Iso(E) is equivalent to the (ARP) of Age(E) and, in fact, also to a stronger version of the Ramsey property for a rich subfamily of Age(E).To state this property we recall that for two k-dimensional Banach spaces X, Y , the Banach-Mazur (pseudo)distance is defined by Definition 2.8.Given a family F of finite-dimensional Banach spaces, let [F] be the class of all separable Banach spaces E such that F ⊆ Age(E), and such that every finite-dimensional subspace of E is the d BM -limit of a sequence of subspaces of elements of F.
For example, the spaces c 0 , C[0, 1] or the Gurarij space are in the class [{ℓ n ∞ } n ], where each ℓ n ∞ is the (real or complex) vector space F n endowed with the sup norm, (a 1 , . . ., a n ) is the class of separable Lindenstrauss spaces.In the next, by a modulus of stability we mean a function ̟ : [0, ∞[→ [0, ∞[ that is increasing and continuous at zero with value zero.Definition 2.9 (Fraïssé properties).Let E be a separable Banach space, and let F be a family of finite-dimensional spaces.
a) E satisfies the stable homogeneity property with respect to F with modulus of stability ̟ if Emb(X, E) is nonempty for every X ∈ F and if for every space with modulus of stability ̟ when E satisfies the stable homogeneity property with respect to Age(E).c) F satisfies the stable amalgamation property (SAP) with modulus ̟ when for every F is a stable Fraïssé class when F satisfies the (SAP) and the joint embedding property (JEP), that is, for every X, Y ∈ F there is Z ∈ F such that Emb(X, Z), Emb(Y, Z) are nonempty.
It is easy to see that if F satisfies the (SAP) and it has a least element with respect to inclusion, then F has the (JEP).Using the fact that {ℓ n ∞ } n is a stable Fraïssé class with modulus ̟(δ) = δ (see Proposition 2.18), it is proved in [35, § §6.1] that the Gurarij space is a stable Fraïssé Banach space with modulus ̟(δ) = δ.In fact, this approximate ultrahomogeneity is a direct consequence of the fact that the Gurarij space is the "generic" direct limit of the class of all finite-dimensional Banach spaces, an instance of the following Fraïssé correspondence for Banach spaces (see for instance [35,Subsection 2.6]).
Proposition 2.10.Suppose that F is a class of finite-dimensional Banach spaces, and E is a separable Banach space.Then, a) If E is a Fraïssé space with modulus ̟, then Age(E) is a stable Fraïssé class with modulus ̟. b) If F is a stable Fraïssé class with modulus ̟, then there is a unique separable E ∈ [F] that satisfies the stable homogeneity property with respect to F with modulus ̟.This space is called the Fraïssé limit of F and denoted by FLim F.
Consequently, the class of all finite-dimensional Banach spaces is stable with modulus δ.The classes {ℓ n p } n for 1 ≤ p ≤ ∞ are also stable: The case p = ∞ is rather easy to prove (see Proposition 2.18), as well as the case p = 2, where one can use the polar decomposition; for 1 < p < ∞, p = 2, one can use a result of G. Schechtman in [50] of approximation of δ-embeddings by isometric embeddings.Also, it is proved in [14] that for p = 4, 6, 8, . . ., the class Age(L p (0, 1)) has a weaker form of stable approximate ultrahomogeneity, namely one that may depend on the dimension.Several other examples of Fraïssé classes of structures in functional analysis are studied in [35].
As we mentioned before, we will see that for an approximately ultrahomogeneous space E, the (ARP) of its age is equivalent to the extreme amenability of the isometry group of E. Furthermore, when E = [F] for some stable Fraïssé class F, a stronger form of the (ARP) of F is also equivalent to the extreme amenability of the isometry group of E. Definition 2.11.A class F of finite-dimensional Banach spaces satisfies the stable approximate Ramsey property (SRP) with stability modulus ̟ if for any X, Y ∈ F, ε > 0, δ ≥ 0 there exists Z ∈ F such that every 1-Lipschitz mapping c : The compact (SRP) and discrete (SRP) are defined as the (ARP), by replacing continuous colorings with compact and finite colorings, respectively.Theorem 2.12 (KPT correspondence for Banach spaces).Let E be an approximately ultrahomogeneous Banach space.Then the following are equivalent: 1) Iso(E) is extremely amenable.2) Age(E) satisfies the approximate Ramsey property.
3) For every X, Y ∈ Age(E), every ε > 0 and every continuous coloring c of Emb(X, E) there is some If in addition F is a family that satisfies the stable amalgamation property such that E ∈ [F] and F Age(E), that is, every space in F can be isometrically embedded into E, then 1), 2), 3) above are also equivalent to 4) F satisfies the (SRP).
The equivalence of 1) and 2) is a particular instance a more general characterization of extreme amenability in terms of an approximate Ramsey property when Banach spaces are regarded as metric structures [7] as in [20,Appendix B] or [35, §8.1].Before we give a proof of the correspondence, we compare these Ramsey properties.Proposition 2.13.Suppose that F is a class of finite-dimensional spaces with the joint embedding embedding property, that is, for every X, Y ∈ F there is Z ∈ F such that Emb(X, Z), Emb(Y, Z) = ∅.Then the following assertions are equivalent: 1) F satisfies the (ARP) and the (SAP) with modulus ̟.
Proof.Trivially, the compact (SRP) with modulus ̟ implies the discrete (SRP) with modulus ̟, and a simple modification of the proof of the Proposition 2.7 gives that the discrete (SRP) with modulus ̟ implies the (SRP) with modulus ̟.Trivially, the (SRP) with modulus ̟ implies the (ARP).In addition, we have the following Claim 2.13.1.If F has the (SRP) with modulus ̟ then F has the (SAP) with modulus ̟.
Proof of Claim: Fix X, Y, Z ∈ F, ε > 0, δ ≥ 0 and γ ∈ Emb δ (X, Y ) and η ∈ Emb δ (X, Z).Find V ∈ F such that Emb(X, V ), Emb(Y, V ) and Emb(Z, V ) are non empty.Find W ∈ F witnessing the (SRP) for initial parameters X, V ∈ F, ε, δ.We claim that W also witnesses the (SAP) for γ, η, ε and δ.Choose Suppose that F has the (ARP) and the (SAP) with modulus ̟, and we prove that F has the compact (SRP).The next claim is not difficult to prove.Claim 2.13.2.F has the (SAP) with modulus ̟ if and only if for every X, Y ∈ F, δ ≥ 0 and ε > 0 there exist Z ∈ F and Fix X, Y ∈ F, δ, ε > 0 and a compact metric space K.We use the previous claim to find By Proposition 2.7, F satisfies the compact (ARP).Thus there exists some Z ∈ F such that every Proof of Theorem 2.12.Corollary 2.4 gives that 1) implies 2).Let us prove the reverse.Suppose that Age(E) has the (ARP).Let (X n ) n be an increasing sequence of finite-dimensional subspaces of E whose union is dense in E, and let d be the metric on Iso(E) defined by d(g, h) Observe that d is a left-invariant compatible metric on Iso(E).In order to prove the extreme amenability of Iso(E) we prove 2) in Proposition 2.2 for the distance d, that is, that the left translation of Iso(E) on (Iso(E), d) is finitely oscillation stable.We fix a 1-Lipschitz mapping c : Iso(E) → [0, 1], a finite subset F ⊆ Iso(E) and ε > 0. Let n be such that 2 n−2 ε ≥ 1 and let Y ⊆ E be a finite-dimensional subspace of E such that X n ∪ g∈F g(X n ) ⊆ Y .Let Y ⊆ Z ⊆ E be a finite-dimensional space witnessing the (ARP) of Age(E) for the parameters X n , Y and ε/8.For each γ ∈ Emb(X n , Z) we choose g γ ∈ Iso(E) such that γ − g γ ↾ X n ≤ ε/8, and now we define the (discrete) coloring c : Emb(X n , Z) → {1, . . ., 2 n+1 }, by c(γ) = j when j is the first integer i such that c(g γ ) ∈ J i , where 2) and 3) are equivalent by Claim 2.13.2, under the hypothesis that E is approximately ultrahomogeneous.
Suppose that F is a family such that F Age(E), E ∈ [F] and suppose that it satisfies the stable amalgamation property.We suppose first that 2) holds, that is, Age(E) has the (ARP), and we prove 4): By Proposition 2.13 and Proposition 2.7, it suffices to show that F satisfies the discrete (ARP).Fix X, Y in F, r ∈ N, and ε > 0. We know by the hypothesis and Proposition 2.7 that Age(E) satisfies the discrete (ARP).Thus, we can find Z 0 ∈ Age(E) containing a copy of Y and such that every r-coloring of Emb(X, Z 0 ) has an ε-monochromatic subset of the form γ • Emb(X, Y ) for some γ ∈ Emb(Y, Z 0 ).Let δ ≤ ε be such that ̟(δ) < ε.Let Z 1 ∈ F for which there exists an δ-embedding θ : Z 0 → Z 1 .By the (SAP) of F we can find Z ∈ F and I ∈ Emb(Z 1 , Z) such that for every φ ∈ Emb δ (X, Z 1 ) there is φ ∈ Emb(X, Z) such that I • φ − φ ≤ ε, and similarly for the elements of Emb δ (Y, Z 1 ).We claim that Z witnesses the discrete (ARP) for the given X, Y, ε, r.Fix a coloring c : Emb(X, Z) → r.Define b : Emb(X, Z 0 ) → r, by choosing for each φ ∈ Emb(X, Z 0 ) an element φ ∈ Emb(X, Z) such that I • θ • φ − φ ≤ ε and declaring b(φ) := c( φ).By the choice of Z 0 from the discrete (ARP) of Age(E), there exist α ∈ Emb(Y, Z 0 ) and j < r such that α Finally, suppose that 4) holds, that is, F has the stable approximate Ramsey property with modulus ̟, and let us prove 3): Let F E be the collection of subspaces of E that are isometric to some element of F. Obviously, F E also has the (ARP).Fix X, Y ∈ Age(E) and ε > 0. We consider 0 < δ ≤ 1 such that ̟(δ) < ε and X 0 ∈ F E such that there is θ ∈ Emb δ (X, X 0 ).Choose also a finite ε-dense subset D of Emb(X, Y ), and for each γ ∈ D some g γ ∈ Iso(E) such that g γ ↾ X − γ ≤ ε.Let now X 1 ∈ F E be such that for every γ ∈ D there is η ∈ Emb δ (X 0 , X 1 ) such that g γ ↾ X 0 − η ≤ ε.Let Y 0 ∈ F E and ι ∈ Emb(X 1 , Y 0 ) be such that ι • Emb δ (X 0 , X 1 ) ⊆ (Emb(X 0 , Y 0 )) ε .We use now the (ARP) of F E when applied to X 0 , Y 0 and ε/2 to find the corresponding Z ∈ F E .Fix a continuous coloring c : Emb(X, E) → [0, 1], and we define a continuous coloring e : Emb(X 0 , Z) → [0, 1] as follows: Fix a non-principal ultrafilter U on N. Given γ ∈ Emb(X 0 , Z) we choose a sequence Since D is ε-dense, it follows from the previous inequality that Osc(h • Emb(X, Y )) ≤ 23ε.

The approximate Ramsey property of {ℓ n
∞ } n .The content of this part is the proof of the approximate Ramsey property of the family {ℓ n ∞ } n , and consequently of the class of all finitedimensional Banach spaces, over F = R, C. Our proof is based on the Dual Ramsey Theorem (DRT) of R. L. Graham and B. L. Rothschild [23].For convenience, we present its formulation in terms of rigid surjections between finite linear orderings.Given two linear orderings (R, < R ) and (S, < S ), a surjective map f : R → S is called a rigid surjection when min R f −1 (s 0 ) < min R f −1 (s 1 ) for every s 0 , s 1 ∈ S such that s 0 < S s 1 .Let Epi(R, S) be the collection of rigid surjections from R to S. Theorem 2.14 ((DRT) [23]).For every finite linear orderings R and S such that |R| < |S| and every r ∈ N there exists an integer n > |S| such that, considering n naturally ordered, every r-coloring of Epi(n, R) has a monochromatic set of the form Epi(S, R) We prove the following.It follows from the KPT correspondence in Theorem 2.12 and Proposition 2.13 the announced result and Corollary 2.16.
Theorem 2.5.The group of isometries of the Gurarij space endowed with the strong operator topology is extremely amenable.
We will give a direct proof of the (ARP) of the class of all finite-dimensional Banach spaces later.Coming back to Theorem 2.15, by means of Proposition 2.13 we need to prove that {ℓ n ∞ } satisfies the stable amalgamation property with modulus δ, and that it has the (ARP).Observe that a linear map γ : ℓ d ∞ → ℓ n ∞ is a δ-isometric embedding if and only if its dual operator γ * : ) is called a quotient map.A simple argument using extreme points shows that this is equivalent to saying that {u j } j<d ⊆ S 1 (F) • {σ(u j )} j<n , where S 1 (F) = {a ∈ F : |a| = 1}, and where u j is the j th unit vector whose only non-zero coordinate has value 1 and it is on the j th position.Let Quo(ℓ n 1 , ℓ d 1 ) be the metric space of quotients.Finally, observe that the dual functor is an isometric bijection.This means that the (ARP) of {ℓ n ∞ } n is equivalent to the assertion of the following lemma.
Lemma 2.17.For every d, m ∈ N and ε > 0 there is some n ∈ N such that every continuous coloring of Quo Lemma 2.17 will be proved later using the Dual Ramsey Theorem.
Proof.Suppose that γ : )), and η * (Ball(ℓ n 1 )) ⊆ Ball(ℓ d 1 ) ⊆ η * ((1 + δ) Ball(ℓ n 1 )).We define σ : ℓ m+n 1 → ℓ m 1 and τ : ℓ m+n 1 → ℓ n 1 as follows.For each j < m, choose y j ∈ ℓ n 1 with 1 ≤ y j ≤ 1 + δ such that η * (y j ) = γ * (u j ), and for Now for each j < m, let σ(u j ) := u j and τ (u j ) := y j / y j , and for k < n, let σ(m + k) = x k / x k and τ (u m+k ) = u k .Then clearly σ(Ball(ℓ m+n 1 )) = Ball(ℓ m 1 ) and τ (Ball(ℓ m+1 )) = Ball(ℓ n 1 ) and Our proof of the (ARP) of {ℓ n ∞ } n uses crucially the Dual Ramsey theorem.The case d = 1 was first proved by Gowers [21], indirectly, as it follows easily via a compactness argument from the oscillation stability of the space c 0 .We start by presenting a simple proof of this result for positive embeddings in the real case.Given integers k and n, let FIN k (n) be the collection of all mappings from n into k + 1 = {0, 1, . . ., k − 1, k} such that k is in its range.Let T : Proposition 2.19 (Gowers).For every k, m and every r there is some n such that every r-coloring of FIN k (n) has a monochromatic set of the form f i i<m for some disjointly supported sequence In the next, let GR(d, m, r) be the minimal n so that (DRT) holds for the parameters d, m and r.
Proof of Proposition 2.19.Fix k, m and r.We claim that n = GR(k + 1, km + 1, r) works.Fix an r-coloring c of FIN k (n).We consider k + 1, mk + 1, and n canonically ordered.For a subset A of n, we let ½ A be the indicator function of A. Let Φ : Epi(n, k + 1) → FIN k (n) be defined by Φ(σ) := i≤k i • ½ σ −1 (i) .By the Ramsey property of n there is some rigid surjection ̺ : . Then c is constant on f j j<m .To see this, given f = l<m T k−j l f l ∈ f j j<m we define σ : mk + 1 → k + 1 by σ(0) = 0 and σ(lk + i) := max{i − k + j l , 0} for l < m and 1 ≤ i ≤ k.Then for 0 < i 0 one has that min σ −1 (i 0 ) = kl 0 + (i 0 + k − j l 0 ) where l 0 = min{l < m : i 0 ≤ j l }, so σ is a rigid surjection.It is not difficult to see that Φ(σ • ̺) = f , so c(f ) = r.
Proof of Lemma 2.17.We start by the following simple fact.Claim 2.19.1.There is a finite ε-dense subset D of Ball(ℓ d 1 ) containing {u j } j<d such that for every non-zero x ∈ Ball(ℓ d 1 ) there is y ∈ D such that y − x 1 ≤ ε and y 1 < x 1 .
Proof of Claim: Let D be a finite ε/2-dense subset of the unit sphere of ℓ d 1 containing {u j } j<d , and Fix such a ε-dense set D, and let ≺ be any linear ordering of D such that if x 1 < y 1 then x ≺ y.Let emb(d, m) be the collection of all 1-1 mappings f : d → m, and let S be a finite ε-dense subset of S 1 (F).For each (f, θ) ∈ emb(d, m) × S d , let h f,θ : ℓ d 1 → ℓ m 1 be the linear map obtained by setting h f,θ (u j ) := θ j • u f (j) .Then clearly h f,θ is an isometric embedding from ℓ d 1 into ℓ m 1 .
We see that φ : ∆ → D is a rigid surjection.First, min φ −1 (0) = (0, f 0 , θ 0 ), where (f 0 , θ 0 ) is the minimum of emb(d, m) × S d .Now suppose that v ∈ D is a non zero vector.We prove that min φ −1 (v) = (v, f , θ): Suppose that φ(u, f, θ) = v, and (f, θ) = ( f , θ).By the definition of φ, because T is a contraction and h f,θ is an isometric embedding.Hence, v ≺ u, and since in ∆ we are considering the lexicographic ordering, Finally, we estimate Φ(φ . Fix ξ < n, and suppose that γ 0 (ξ) = (v, f, θ).Then by definition, (Φ(φ 2.4.(ARP) of Polyhedral spaces and finite-dimensional spaces.We give an explicit proof of the approximate Ramsey property of the class of finite-dimensional polyhedral spaces.This is done by using injective envelopes of polyhedral spaces, and then by reducing colorings of polyhedral spaces to colorings of ℓ n ∞ -spaces.We also use this to explicitly prove the (ARP) of the class of all finitedimensional Banach spaces.In this way, knowing the number of extreme points of the dual unit ball of given spaces, one can estimate upper bounds of the corresponding Ramsey numbers.For simplicity, we present the proof in the case of real Banach spaces.Thus, all the Banach spaces are assumed to be real in this section.Definition 2.20.A finite-dimensional space F is called polyhedral when its unit ball Ball(F ) is a polyhedron, i.e., when the set ∂ e (Ball(F )) of extreme points of Ball(F ) is finite.
The spaces ℓ n ∞ and ℓ n 1 are polyhedral.In fact, a finite-dimensional space is polyhedral if and only if its dual ball is polyhedral.It follows from this, a separation argument, and the Milman theorem, that a finite-dimensional space F is polyhedral if and only if there is a finite set A ⊆ Sph(F * ) such that x = max f ∈A f (x) for every x ∈ F .Also, every subspace of a polyhedral space is polyhedral, and every finite-dimensional polyhedral space embeds into ℓ n ∞ for some n ∈ N. Definition 2.22 (Injective envelope of a polyhedral space).The injective envelope of a polyhedral space F is a pair (n F , Ψ F ), where n F is an integer and Ψ F ∈ Emb(F, ℓ n F ∞ ) such that for every isometric embedding T : F → ℓ n ∞ there is an isometric embedding U : and r < r be such that Let

2.4.1.
Approximate Ramsey property for finite-dimensional normed spaces.We give an explicit, constructive proof of approximate Ramsey property arbitrary finite-dimensional normed spaces.The proof is based on the approximate Ramsey property of polyhedral spaces and the well known fact that the finite-dimensional polyhedral spaces are dense in the class of finite-dimensional normed spaces with respect to the Banach-Mazur distance.In fact, we have the following.Proposition 2.24.Suppose that dim X = k.For every 0 < ε < 1 there is a polyhedral space [43,Lemma 2.6]).On X we define the polyhedral norm N (x) := max f ∈D |f (x)|.It follows that X 0 := (X, N ) ∈ Pol d with d ≤ #D, and d BM (X, X 0 ) ≤ ε.Definition 2.25.Given X of finite dimension, and θ ≥ 1, let Emb θ (X, Y ) be the collection of all 1-1 mappings T : X → Y such that 1 ≤ T , T −1 and T • T −1 ≤ θ.
Let (X i ) i≤n be a sequence of Banach spaces.We say that a pair (Y, J) of a Banach space Y and J ∈ Emb(X n , Y ) is (θ, τ )-correcting for X (1 < θ < τ ) when every X i isometrically embeds into Y , and for every j < n and every γ ∈ Emb θ (X j , X n ) there exists Proposition 2.26.Every finite sequence of finite-dimensional spaces (X i ) i≤n and every 1 < θ < τ has a (θ, τ )-correcting pair (Y, J).Moreover, when each X j is polyhedral, then Y can be taken polyhedral.
Proof.The proof is by induction on n ≥ 1. Suppose first that n = 1.A simple inductive argument, where the case #N = 1 is proved by Kubis and Solecki in [30, Lemma 2.1], gives the following.
Theorem 2.27.The class of all finite-dimensional Banach spaces FdBa has the (SRP).
Proof.We know that FdBa is a stable Fraïssé class, so we only have the proof that it satisfies the discrete (ARP).Fix finite-dimensional spaces F , G, r ∈ N, ε > 0, and set δ := ε/5.Let F 0 ∈ Pol d , G 0 be polyhedral, and surjective isomorphisms Φ F : Notice that d can be taken such that d ≤ ((10 + 3ε)/ε) dim F .Let (i) (H 0 , Θ 0 ) be a (1 + ε/5, 1 + ε/4)-correcting pair for (F 0 , G 0 ) with H 0 ∈ Pol m , and let (ii) (H, Θ 1 ) be a (1+ε/5, 1+ε/4)-correcting pair for the triple (F, G, ℓ n ∞ ) where n := n pol (d, m, r, ε/4).We claim that H works. Fix c : 3) and the fact that the operator T satisfies that T = 1, that This is the diagram: 2.5.Finite metric spaces.Recall that the Urysohn space U is the unique (up to isometry) ultrahomogeneous universal separable complete metric space.Pestov proved in [47] that the group Iso(U) of surjective isometries of U is extremely amenable, using the method of concentration of measure.It is also proved a version of the (KPT) correspondence for Iso(U), that gives as a consequence the following the (ARP) of finite metric spaces.
Theorem 2.28.For every finite metric spaces M and N , r ∈ N and ε > 0 there exists a finite metric space P such that every r-coloring emb(M, P ) has a ε-monochromatic set of the form σ • emb(M, N ) for some σ ∈ emb(N, P ).
In the previous statement emb(M, P ) is the collection of all isometric embeddings from (M, d M ) into (N, d N ), endowed with the uniform metric d(σ, τ ) := max x∈M d N (σ(x), τ (x)).Later, Nešetřil established the (exact) Ramsey property of finite ordered metric spaces [44], that is, for every finite ordered metric spaces X and Y and every r ∈ N there exists a finite ordered metric space Z such that for every r-coloring of the set Z X < of order isometric copies of X in Z there exists an order isometric copy Y 0 of Y in Z such that Y 0 X < is monochromatic.This gives another proof of the extreme amenability of Iso(U).We present here a third proof, which uses the approximate Ramsey property of the class of finite-dimensional polyhedral spaces.
Recall that a pointed metric space (X, d, p) is a metric space (X, d) with a distinguished point p ∈ X.Given two pointed metric spaces (M, p) and (N, q), let emb 0 (M, N ) be the set of pointed isometric embeddings, that is, all isometric embeddings from M into N sending p to q. Recall that when X and Y are normed spaces, we use Emb(X, Y ) to denote linear isometric embeddings.Definition 2.29.Given a pointed metric space (M, d, p), let Lip 0 (M, p) be the Banach space of all Lipschitz maps f : M → R such that f (p) = 0 endowed with the Lipschitz norm, Let F(M, p) be the (Lipschitz) free space over the pointed metric space (M, p) defined as the closed linear span of the molecules {δ x − δ p } x∈M in the dual space Lip 0 (M, p) * , where δ x for x ∈ X denotes the evaluation functional at x.It turns out that F(M, p) * is isometric to Lip 0 (M, p).
It is well-known that Lip 0 (M, p) does not depend, isometrically, on the choice of the point p, so the corresponding predual will be denoted by F(M ).The space F(M ) is also known as the Arens-Eells space.More information on Arens-Eells spaces can be found in [52,Section 2.2].It is easy to see that the mapping x ∈ M → δ x ∈ F(M ) is an isometric embedding.Given finite metric spaces M and N such that M isometrically embeds into N , let Proposition 2.30.Suppose that M and N are metric spaces.Then every isometric embedding σ : M → N extends to a unique linear isometric embedding The proof is a straightforward use of a standard duality argument, the McShane-Whitney extension Theorem for Lipschitz functions [52, Theorem 1.5.6], and the fact that Proposition 2.31.If M is a finite metric space, then F(M ) is a finite-dimensional polyhedral space.
Proof.Observe that for each x = y in M , µ x,y := (δ x − δ y )/d(x, y) has norm 1 in Lip 0 (M ) since clearly µ x,y ≤ 1, and the mapping d x (t) := d(x, t) for each t ∈ M is 1-Lipschitz and µ x,y (d x ) = 1.It follows from the definition of the Lipschitz norm that the convex hull of {µ x,y } x =y in M is equal to B F (M ) .
2.6.The closed bifaces of the Lusky simplex and R-Banach spaces.There is a natural correspondence between Banach spaces and those compact spaces which are absolutely convex.In the real case, by a compact absolutely convex set we mean a compact subset of a locally convex topological real vector space that is closed under absolutely convex combinations of the form µx + λy for λ, µ ∈ R such that |λ| + |µ| ≤ 1.Any compact absolutely convex set K has a canonical involution σ mapping x to −x.A real-valued continuous function f on K is symmetric if f • σ = −f .Similarly, a continuous affine function between compact absolutely convex sets is symmetric if it commutes with the given involutions.So, given a Banach space X, the unit ball Ball(X * ) of the dual space of X is a compact absolutely convex set when endowed with the w*-topology.Any compact absolutely convex set K is of this form, where X is the Banach space A σ (K) of real-valued symmetric affine continuous functions on K endowed with the supremum norm.Each contraction T : X → Y induces a symmetric affine continuous function T * : B Y * → B X * , and vice versa, a given symmetric affine continuous function ξ : K → L induces a contraction ξ : A σ (L) → A σ (K) by composition.Furthermore, such a correspondence is functorial, and induces an equivalence of categories.The following definition has been introduced in [35, Section 6.1].Definition 2.33.A Lazar simplex is any compact absolutely convex that is affinely homeomorphic to the unit ball of the dual of a Lindenstrauss space.
Lazar simplices have been internally characterized by A. J. Lazar in [31] in terms of a uniqueness assertion for boundary representing measures, reminiscent of the analogous characterization of Choquet simplices due to Choquet [1, Section II.3]; see also Subsection 3.1 below.The Lazar simplex corresponding to the Gurarij space is denoted by L and called the Lusky simplex in [35,Section 6.1].It is proved in [35,36,39] that L plays the same role in the category of metrizable Lazar simplices as the Poulsen simplex P plays in the category of metrizable Choquet simplices (see next section 3).Recall that a closed subset H of a Lazar simplex is a biface or essential face if it is the absolutely convex hull of a (not necessarily closed) face [32].This is equivalent to the assertion that the linear span of H inside A σ (K) * is a w*-closed L-ideal [2,3].Relevant properties of L: • The Lusky simplex is the unique nontrivial metrizable Lazar simplex with dense extreme boundary (Lusky [36]); • the Lusky simplex is universal among metrizable Lazar simplices, in the sense that any metrizable Lazar simplex is symmetrically affinely homeomorphic to a closed biface of L (Lusky [39]); • the Lusky simplex is homogeneous: any symmetric affine homeomorphism between proper closed bifaces of L extends to a symmetric affine homeomorphisms of L (Lupini [35, Subsection 6.1]).
Our intention is to prove the following: Theorem 2.34.Suppose that H is a closed biface of the Lusky simplex L. Then the group Aut H (L) of symmetric affine homeomorphisms α of L such that α(p) = p for every p ∈ H is extremely amenable.
Remark 2.35.A similar result holds for complex Banach spaces.In this setting, one considers compact convex sets endowed with a continuous action of the circle group T (compact convex circled sets).The compact convex circled sets corresponding to complex Lindenstrauss spaces (Effros simplices) have been characterized by Effros in [12].Again, the unit ball of the dual space of the complex Gurarij space has canonical uniqueness, universality, and homogeneity properties within the class of Effros simplices; see [35,Subsection 6.2].Here one considers the natural complex analog of the notion of a closed biface (circled face).The same argument as above shows that, in the complex case, the pointwise stabilizer of any closed circled face of Ball(G * ) is extremely amenable.
Observe that in the particular case when H is the trivial biface {0}, such a statement recovers extreme amenability of the group of surjective linear isometries of G. Observe also that given a closed biface H of a Lazar simplex L, we have that g ∈ Aut H (L) if and only if g ∈ Iso i (A σ (L)), where i : H → L is the inclusion map and where, in general, given Banach spaces X and Y and an operator σ : X → Y by Iso σ (X) we mean the subgroup of isometries g of X so that σ • g = σ.This motivates our study of such pairs (X, σ).Definition 2.36 (R-Banach space).Given a Lindenstrauss space R, an R-Banach space is a couple X := (X, σ) when σ : X → R is a linear contraction, called R-functional.
Note that a classical result of Wojtaszczyk [53] asserts that the separable Lindenstrauss spaces are precisely the separable Banach spaces that are isometric to the range of a contractive projection on the Gurarij space G.The R-functional Ω R is called the generic contractive R-functional on G.The name is justified by the fact that the Iso(G)-orbit of Ω R is a dense G δ subset of the space of contractive R-functionals on G.The universality and homogeneity properties of L can be seen as consequences of the following result, established in [35,Subsection 6.1] using the theory of M -ideals in Banach spaces developed by Alfsen and Effros [2,3], and the Choi-Effros lifting theorem from [9].Proposition 2.38.Suppose that R is a separable Lindenstrauss space.A contraction s : G → R belongs to the Iso(G)-orbit of Ω R if and only if s is a non-trivial facial quotient, that is, if ker s = 0, and s * is an isometric embedding such that s * (Ball(R * )) is a closed biface of Ball(G * ).
In particular, suppose that H is a proper closed biface of L, i : H → L is the canonical inclusion and we identify canonically G and A σ (L).Then i : A σ (L) → A σ (H) is a non-trivial facial quotient, hence i ∈ Iso(G) • Ω Aσ(H) .This implies that Iso i (G) = Iso Ω Aσ (H) (G), and Theorem 2.34 can be rephrased as follows.
Theorem 2.39.The stabilizer of the generic contractive R-functional on the Gurarij space is extremely amenable for any separable Lindenstrauss Banach space R.
When R = {0}, we recover the extreme amenability of Iso(G).In fact, the proof of this extension is based on the approximate Ramsey property of finite-dimensional R-Banach spaces, by means of the KPT correspondence.The corresponding non-commutative version of the previous theorem is established in [4].
2.6.1.KPT correspondence and (ARP) of R-Banach spaces.We give a proof of Theorem 2.34.By the correspondence between the categories of Lazar simplices and that of R-Banach spaces, Theorem 2.34 is equivalent to the fact that Aut(G G G R ) is extremely amenable, which will be proved by means of a KPT correspondence and an appropriate approximate Ramsey property.Given an R-space X = (X, s), let Age(X) be the collection of pairs (F, s ↾ F ), where F ∈ Age(X).Given a family F of finite-dimensional R-Banach spaces, let [F] be the collection of all separable R-Banach spaces X such that for every F ∈ Age(X) and every δ > 0 there is some G ∈ F such that Emb δ (F, G) = ∅.
Theorem 2.40 (KPT correspondence for stable Fraïssé R-Banach spaces).Suppose that E = (E, Ω) is an approximately ultrahomogeneous R-Banach space.Then the following are equivalent 1) Aut(E) is extremely amenable.2) Age(E) satisfies the (ARP), that is for every X, Y ∈ Age(E) and ε > 0 there is Z ∈ Age(E) such that every continuous coloring of Emb(X, Z) ε-stabilizes on γ • Emb(X, Y) for some γ ∈ Emb(Y, Z).
Suppose that F is a family such that F Age(E), E ∈ [F].Then (1), (2), and (3) are equivalent to 3) F satisfies the stably approximate Ramsey property (SRP) with modulus ̟(δ), that is for every X, Y ∈ F, ε > 0 and δ ≥ 0 there is Z ∈ F such that every continuous coloring of Emb δ (X, Z) The proof of Theorem 2.40 is a straightforward modification of that of Theorem 2.12; we leave its details to the reader.

Proof of Claim
We prove now the (ARP) of F. Fix ℓ k ∞ -spaces X := (ℓ d ∞ , s) and Y := (ℓ m ∞ , u), and ε > 0. Let n ∈ N be witnessing the (ARP) of {ℓ r ∞ } r for the initial parameters d, m, and ε.Let π : ℓ n+k ∞ → ℓ k ∞ be the canonical second projection π((a j ) j<n+k ) := (a j ) n+k−1 j=n .We claim that Z := (ℓ n+k ∞ , π) works: For suppose that c : Emb(X, Z) Fix a Lindenstrauss space R, and choose an increasing sequence of subspaces (R n ) n whose union is dense in R and such that each R n is isometric to ℓ n ∞ .Let F be the class of R-Banach spaces (X, s) where X is isometric to some ℓ d ∞ and such that Im s ⊆ n R n .It follows easily from a) that F has the (SRP) with modulus 2δ.We know from Theorem 2.37 that G G G R = (G, Ω R ) is a stable Fraïssé R-Banach space such that age(G G G R ) consists of all finite-dimensional R-Banach spaces.On the other hand, G G G R ∈ [F], so it follows from a) and the KPT correspondence in Theorem 2.40 that age(G G G R ) satisfies the (SRP) with modulus 2δ.Theorem 2.41 and the characterization of extreme amenability in Theorem 2.40 give the previously announced result.
Theorem 2.39.The stabilizer of the generic contractive R-functional on the Gurarij space is extremely amenable for any separable Lindenstrauss Banach space R.

The Ramsey property of Choquet simplices and function systems
The main goal of this section is to establish the approximate (dual) Ramsey property for Choquet simplices with a distinguished point.We will then apply this to compute the universal minimal flow of the automorphisms group of the Poulsen simplex P. We will prove that the minimal compact Aut(P)-space is the Poulsen simplex P itself endowed with the canonical action of Aut(P), answering [10, Question 4.4] (the fact that such an action is minimal is a result of Glasner from [18]).This will be done by studying function systems with a distinguished unital positive map to a fixed separable Lindenstrauss function system R. Similarly as in the case of Banach spaces ( § §2.6), we will also consider function systems X with a distinguished state, a unital linear contraction s : X → R where R is a fixed separable Lindenstrauss function system.
3.1.Choquet simplices and function systems.Recall that a compact convex set K is a compact convex subset of some locally convex topological vector space.In a compact convex set one can define in the usual way the notion of convex combination.The extreme boundary ∂ e K of K is the set of extreme points of K, that is, points that cannot be written in a nontrivial way as a convex combination of points of K.When K is metrizable the boundary ∂ e K is a G δ subset.In this case, a boundary measure on K is a Borel probability measure on K that vanishes off the boundary of K. Choquet's representation theorem asserts that any point in a compact convex set can be realized as the barycenter of a boundary measure on K (representing measure).A compact convex set K where every point has a unique representing measure is called a Choquet simplex.In particular, any standard finite-dimensional simplex ∆ n for n ∈ N is a Choquet simplex.
The class of standard finite-dimensional simplices ∆ n for n ∈ N naturally form a projective Fraïssé class in the sense of [27]; see [29].The corresponding Fraïssé limit is the Poulsen simplex P. Initially constructed by Poulsen in [49], P is a nontrivial metrizable Choquet simplex with dense extreme boundary.It was later shown in [33] that there exists a unique nontrivial metrizable Choquet simplex with this property up to affine homeomorphism.Furthermore P is universal among metrizable Choquet simplices, in the sense that any metrizable Choquet simplex is affinely homeomorphic to a closed proper face of P. Also, the Poulsen simplex is ultrahomogeneous: any affine homeomorphism between closed proper faces of P extends to an affine homeomorphism of P.
The Poulsen simplex P can also be studied from the perspective of direct Fraïssé theory by considering the natural dual category to compact convex sets.For a compact convex set K, let A(K) be the space of complex-valued continuous affine functions on K.This is a closed subspace of the space C(K) of complex-valued continuous functions on K, endowed with the supremum norm.Furthermore, A(K) contains a distinguished element, its unit, that corresponds to the constant function equal to 1.In general, recall that a function system is a closed subspace V of C(T ) for some compact Hausdorff space T containing the function constantly equal to 1 and such that if f ∈ V then the function f * defined by f * (t) = f (t) also belongs to V .So, A(K) is a function system, and in fact any function system V ⊆ C(T ) arises in this way from a suitable compact convex set K. Precisely, K is the compact convex set of states of V , that is, the contractive functionals on V that are unital, i.e., that map the unit of C(T ) to 1.
As mentioned in the introduction, the assignment K → A(K) establishes a contravariant equivalence of categories from the category of compact convex sets and continuous affine maps to the category of function systems and unital contractive linear maps.The finite-dimensional function systems that are injective in such category are precisely the function systems A(∆ n ) = ℓ n ∞ corresponding to the standard finite-dimensional simplices ∆ n .The function systems that correspond to Choquet simplices are precisely those that are Lindenstrauss as Banach spaces, or equivalently, the function systems whose identity map is the pointwise limit of unital contractive linear maps that factor through finitedimensional injective function systems.
The function systems approach has been adopted in the work of Conley and Törnquist [10] and, independently, in [34,35], where it is shown that the class of finite-dimensional function systems is a Fraïssé class.Its limit can be identified with the function system A(P) corresponding to the Poulsen simplex, which we will call the Poulsen system.The model-theoretic properties of A(P) and their non-commutative analogues have been studied in [19].
Suppose that X is an function system.Recall that a state on X is a unital contractive linear map from X to C.More generally, if R is any separable Lindenstrauss function system, we call a unital contractive linear map from X to R an R-state on X.Let UC(X, R) be the space of R-states on X.We have that UC(X, R) is a Polish space endowed with a canonical continuous action of Aut(X).An R-function system is a pair X = (X, s X ) of a function system X and an R-state s X on X.In the following, we regard UC (X, R) as an Aut (X)-space with respect to the canonical action Aut (X) UC (X, R) given by (α, s) → s • α −1 .Given R-function systems X = (X, s X ) and Y = (Y, s Y ) and given δ ≥ 0, let Emb δ (X, Y) be the collection of unital δ-isometric embeddings γ : X → Y such that s Y • γ − s X X,R ≤ δ.Given an R-function system X = (X, s X ), let Age(X) be the collection of all finite-dimensional R-function subsystems Y = (Y, s Y ) of X, that is, Y ⊆ X and s Y = s X ↾ Y .Given a class F of R-function systems, let [F] be the class of all separable R-function systems X = (X, s X ) such that for every Y and every δ > 0 there is Z ∈ F such that Emb δ (Y, Z) = ∅.Let Aut(X, s X ) be the stabilizer of s X ∈ UC (X, R) in Aut(X).Given a family A of function systems, let A R be the collection of R-function systems (X, s X ) where X ∈ A.
Proposition 3.1.Let R be a separable Lindenstrauss function system.Then the class FdBa R of finite-dimensional R-function systems is a stable Fraïssé class with stability modulus ̟(δ) = 2δ and A(P) A(P) A(P) R := (A(P), Ω R ) is its Fraïssé limit, that is, A(P) A(P) A(P) R is a stable Fraïssé R-function system such that Age(A(P) A(P) A(P) R ) = FdBa R .
As in the case of operator spaces, the R-state Ω R as in Proposition 3.1 is called the generic R-state on A(P).This is the unique R-state on A(P) whose Aut(A(P))-orbit is a dense G δ subset of the space UC(A(P), R).The elements of the Aut(A(P))-orbit of Ω R can be characterized as follows (see [35, § §6.3].Proposition 3.2.Suppose that R is a separable Lindenstrauss function system.A unital quotient map s : A(P) → R belongs to the Aut(A(P))-orbit of Ω R if and only if s is a unital facial quotient, i.e., s is unital and s * is an isometric embedding such that s * (Ball(R * )) is a closed proper face of P.
The intention is to prove the following Theorem 3.3.For every metrizable Choquet simplex F the stabilizer Aut(A(P) A(P) A(P) A(F ) ) of the generic A(F )-state Ω A(F ) on the Poulsen system A(P) is extremely amenable.

Approximate Ramsey property and extreme amenability.
The following result provides a correspondence between extreme amenability and Ramsey properties in the context of R-function systems.The proof is analogous to the one for Banach spaces, and is left to the reader.Theorem 3.4 (KPT correspondence for (aUH) and stable Fraïssé R-function systems).Suppose that X = (X, Ω) is an approximately ultrahomogeneous R-function system.Then the following are equivalent: 1) Aut(X) is extremely amenable.2) Age(X) satisfies the approximate Ramsey property.
If in addition F is a family that satisfies the stable amalgamation property such that E ∈ [F] and F Age(X), that is, every R-function system in F can be isometrically R-embedded into E, then the previous are equivalent to 3) F satisfies the (SRP).Theorem 3.5.Suppose that (R k ) k is a sequence of function subsystems of R, each R k isometric to ℓ k ∞ , and with a dense union.The following classes of R-function systems have the (SRP) with modulus 2δ: 2) For every k ∈ N the class of R k -function systems (X, s) where X is isometric to some ℓ n ∞ .
3) The class B R of R-function systems (X, s) where X is isometric to some ℓ n ∞ and s : X → k R k .4) The class of all R-function systems.
To prove this Theorem we will use (and prove) the (ARP) of the class {ℓ n ∞ } n with respect to positive embeddings.Its proof is similar to that of Lemma 2.17.We present the details for the reader's convenience.Let Emb + (ℓ d ∞ , ℓ n ∞ ) be the space of positive isometric embeddings from ℓ d ∞ into ℓ n ∞ .Dually, let Quo + (ℓ n 1 , ℓ d 1 ) be the space of corresponding positive quotient mappings.
Lemma 3.6.For every d, m, r ∈ N, and ε > 0 there is some n such that every r-coloring of Emb + (ℓ d ∞ , ℓ n ∞ ) has an ε-monochromatic set of the form γ

Definition 2 . 21 (
Polyhedral spaces).Given an integer d, let Pol d be the class of all polyhedral spaces F such that #∂ e (B F * ) = 2d.Given d, m ∈ N, r ∈ N and ε > 0, let n pol (d, m, r, ε) be the minimal integer n ≥ m such that for every F ∈ Pol d and G ∈ Pol m , every r-coloring of Emb(F, ℓ n ∞ ) has an ε-monochromatic set of the form T • Emb(F, G) for some T ∈ Emb(G, ℓ n ∞ ).

Theorem 2 . 41 .
The following classes have the (SRP) with modulus of stability 2δ: a) For every k ∈ N, the class of ℓ k ∞ -Banach spaces (X, s) where X = ℓ n ∞ for some n ∈ N. b) For every separable Lindenstrauss space R the class of all finite-dimensional R-Banach spaces.Proof.As for the case of Banach spaces in Proposition 2.13, a class of R-finite dimensional spaces has the (SRP) with modulus ̟ if and only if it satisfies the (ARP) and it has the corresponding stable amalgamation property with modulus ̟. a): Claim 2.41.1.The family F of ℓ k ∞ -spaces of the form (ℓ n ∞ , s) for some n has the stable amalgamation property with modulus 2δ.