Ramsey properties and ultrahomogeneity for Banach and operator - - PowerPoint PPT Presentation

Ramsey properties and ultrahomogeneity for Banach and

• perator spaces.
• J. Lopez-Abad

Instituto de Ciencias Matem´ aticas,CSIC, Madrid joint work with D. Bartoˇ sov´ a, M. Lupini and B. Mbombo

Transfinite methods in Banach spaces and algebras of operators. Bedlewo, July 22, 2016

• J. Lopez-Abad (ICMAT)

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Outline

1 Ramsey properties of Grassmannians

Grassmannians over a finite field Fields F = Q, R, C

2 Approximate Ramsey, Ultrahomogeneity and Topological dynamics

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Main Results to discuss

(I) The group of isometries of the Gurarij space G with its SOT is extremely amenable (II) The universal minimal flow of the group of affine homeomorphism of the Poulsen simplex P is the natural action of it in P. (III) Analogues for the non-commutative case. (IV) A version of the Graham-Leeb-Rothschild Theorem for Grassmannians over Q, R, C. (V) The Approximate Ramsey Property of the finite dimensional normed spaces.

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Ramsey properties of Grassmannians Grassmannians over a finite field

Given a vector space V over F, and k ∈ N, let Gr(k, V ) be the collection

• f all subspaces of V of dimension exactly k.
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Ramsey properties of Grassmannians Grassmannians over a finite field

Given a vector space V over F, and k ∈ N, let Gr(k, V ) be the collection

• f all subspaces of V of dimension exactly k.

Theorem (Graham-Leeb-Rothschild (1972))

For every k, m ∈ N and r ∈ N there exists n such that for every coloring c : Gr(k, Fn) → {1, 2, . . . , r} there exists V ∈ Gr(m, Fn) such that c is constant on Gr(k, V ).

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

The pigeonhole principle is a consequence of a factorization result.

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

The pigeonhole principle is a consequence of a factorization result. (i) Let embn×k(F) (or emb(Fk, Fn)) be the collection of n × k-matrices

• f rank k.
• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

The pigeonhole principle is a consequence of a factorization result. (i) Let embn×k(F) (or emb(Fk, Fn)) be the collection of n × k-matrices

• f rank k.

(ii) Let GLk(F) be the group of invertible k × k-matrices.

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

The pigeonhole principle is a consequence of a factorization result. (i) Let embn×k(F) (or emb(Fk, Fn)) be the collection of n × k-matrices

• f rank k.

(ii) Let GLk(F) be the group of invertible k × k-matrices. (iii) Given A ∈ embn×k, let A = red(A) · τ(A) be the unique decomposition of A by the reduced column echelon form of A and a unique invertible matrix τ(A) ∈ GLk(F).

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

The pigeonhole principle is a consequence of a factorization result. (i) Let embn×k(F) (or emb(Fk, Fn)) be the collection of n × k-matrices

• f rank k.

(ii) Let GLk(F) be the group of invertible k × k-matrices. (iii) Given A ∈ embn×k, let A = red(A) · τ(A) be the unique decomposition of A by the reduced column echelon form of A and a unique invertible matrix τ(A) ∈ GLk(F). (iv) Let En×k := {A ∈ embn×k : red(A) = A}.

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

Theorem (Ramsey degree of full rank matrices)

For every k, m ∈ N and every r ∈ N there exists n such that for every coloring f : Embn×k(F) → {1, . . . , r} there exists R ∈ En×m and g : GLk(F) → {1, . . . , r} such that

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Grassmannians over a finite field

Theorem (Ramsey degree of full rank matrices)

For every k, m ∈ N and every r ∈ N there exists n such that for every coloring f : Embn×k(F) → {1, . . . , r} there exists R ∈ En×m and g : GLk(F) → {1, . . . , r} such that R · Embm×d(F) {1, . . . , r} GLk(F) f τ g

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Ramsey properties of Grassmannians Grassmannians over a finite field

Theorem (Ramsey degree of full rank matrices)

For every k, m ∈ N and every r ∈ N there exists n such that for every coloring f : Embn×k(F) → {1, . . . , r} there exists R ∈ En×m and g : GLk(F) → {1, . . . , r} such that R · Embm×d(F) {1, . . . , r} GLk(F) f τ g

• Since every subspace W ∈ Gr(k, Fn) is the image of a matrix A ∈ En×k,

this result gives the Graham-Leeb-Rothschild Theorem.

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Ramsey properties of Grassmannians Grassmannians over a finite field

The factorization result is a consequence of the Dual Ramsey Theorem.

Definition

Let Pk(n) be the collection of all k-partitions of {1, . . . , n}; that is, partitions with exactly k-many pieces. Given a partition Π ∈ Pm(n), let Πk the collection of all k-partitions formed by joining the pieces of Π.

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Ramsey properties of Grassmannians Grassmannians over a finite field

The factorization result is a consequence of the Dual Ramsey Theorem.

Definition

Let Pk(n) be the collection of all k-partitions of {1, . . . , n}; that is, partitions with exactly k-many pieces. Given a partition Π ∈ Pm(n), let Πk the collection of all k-partitions formed by joining the pieces of Π.

Theorem (Dual Ramsey Theorem; Graham and Rothschild (1971))

For every k, m and every r there is n such that for every coloring c : Pk(n) → {1, . . . , r} there is Π ∈ Pm(n) such that c ↾ Πk is constant.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr(k, Fn) a metric space.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr(k, Fn) a metric space. (P2) Maybe there is no “approximate” GLR Theorem.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr(k, Fn) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric dn on Gr(k, Fn).

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Ramsey properties of Grassmannians Fields F = Q, R, C

(P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr(k, Fn) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric dn on Gr(k, Fn). (G2) Find a compact metric space (Kk, d) and a Lipschitz map ν : Gr(k, Fn) → K such that:

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Fields F = Q, R, C

(P1) There is NO GLR Theorem in this case. Some approximative result is needed, so we need to make Gr(k, Fn) a metric space. (P2) Maybe there is no “approximate” GLR Theorem. Goal: (G1) For every n find a good metric dn on Gr(k, Fn). (G2) Find a compact metric space (Kk, d) and a Lipschitz map ν : Gr(k, Fn) → K such that: For every k, m, every ε > 0, C > 0 and every compact metric (L, ̺) there exists n such that for every Lipschitz coloring f : (Gr(k, Fn), dn) → (L, ̺) with Lip(f ) ≤ C there exists V ∈ Gr(m, Fn) and a C-Lipschitz ¯ f : (K , d) → (L, ̺) such that Gr(k, V ) L Kk f ν ¯ f

• ε
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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a norm M on Fn, we define the gap (opening) distance Λ(Fn,M)(V , W ) between V , W ∈ Gr(k, Fn) as the Hausdorff distance (with respect to M) between the unit balls of (V , M) and (W , M).

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a norm M on Fn, we define the gap (opening) distance Λ(Fn,M)(V , W ) between V , W ∈ Gr(k, Fn) as the Hausdorff distance (with respect to M) between the unit balls of (V , M) and (W , M). Note that this is non-complete (Cauchy sequences of k-dimensional subspaces might converge to a < k-dimensional subspace). Its completion is Gr(≤ k, Fn) is now compact.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a norm M on Fn, we define the gap (opening) distance Λ(Fn,M)(V , W ) between V , W ∈ Gr(k, Fn) as the Hausdorff distance (with respect to M) between the unit balls of (V , M) and (W , M). Note that this is non-complete (Cauchy sequences of k-dimensional subspaces might converge to a < k-dimensional subspace). Its completion is Gr(≤ k, Fn) is now compact. Instead of discrete colorings, now we can talk about “metric colorings”, i.e. Lipschitz maps into a (compact) metric space.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Recall that given two normed spaces E, F of dimension k, the Banach-Mazur pseudo-distance between E and F is dBM(V , W ) := log inf

T TE,F · T −1E,F

where T runs over all isomorphisms T : E → F. The corresponding quotient Bk by dBM = 0 is the k-Banach-Mazur compactum

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Recall that given two normed spaces E, F of dimension k, the Banach-Mazur pseudo-distance between E and F is dBM(V , W ) := log inf

T TE,F · T −1E,F

where T runs over all isomorphisms T : E → F. The corresponding quotient Bk by dBM = 0 is the k-Banach-Mazur compactum (which is compact).

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Recall that given two normed spaces E, F of dimension k, the Banach-Mazur pseudo-distance between E and F is dBM(V , W ) := log inf

T TE,F · T −1E,F

where T runs over all isomorphisms T : E → F. The corresponding quotient Bk by dBM = 0 is the k-Banach-Mazur compactum (which is compact). Given a Banach space X, let Bk(X) be the set of all classes E of subspaces E of X. This is in general not closed, but in certain cases is, for example when X = Lp, 1 ≤ p < ∞, p = 4, 6, 8, .. or X = G is the Gurarij space.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a Banach space X, we define the following quantity on Bk(X): γX(E, F) := inf

T,U ΛX(TE, UF)

where the infimum runs over all isometric embeddings T : E → X, U : F → X. When X is universal for f.d. spaces, we will write γ.

• J. Lopez-Abad (ICMAT)

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a Banach space X, we define the following quantity on Bk(X): γX(E, F) := inf

T,U ΛX(TE, UF)

where the infimum runs over all isometric embeddings T : E → X, U : F → X. When X is universal for f.d. spaces, we will write γ.

Proposition

(i) k−1γ(E, F) ≤ dBM(E, F) ≤ 3k log(k)γX(E, F)

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Given a Banach space X, we define the following quantity on Bk(X): γX(E, F) := inf

T,U ΛX(TE, UF)

where the infimum runs over all isometric embeddings T : E → X, U : F → X. When X is universal for f.d. spaces, we will write γ.

Proposition

(i) k−1γ(E, F) ≤ dBM(E, F) ≤ 3k log(k)γX(E, F) (ii) When X = Lp, 1 ≤ p < ∞, p = 4, 6, 8, .. or X = G is the Gurarij space, then γX is a metric and we call it X-Kadets metric.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Fix a Banach space X and a norm M such that (Fn, M) isometrically embeds into X. Let τX : (Gr(k, Fn), Λ(Fn,M)) → (Bk(X), γ) be the mapping that assigns to V ∈ Gr(k, Fn) the isometric type (V, M). This is a 1-Lipschitz map.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Definition

Fix a Banach space X and a norm M such that (Fn, M) isometrically embeds into X. Let τX : (Gr(k, Fn), Λ(Fn,M)) → (Bk(X), γ) be the mapping that assigns to V ∈ Gr(k, Fn) the isometric type (V, M). This is a 1-Lipschitz map. Let Xp = Lp if 1 ≤ p < ∞, G the Gurarij space if p = ∞.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Theorem (p-factorization of Grassmannians)

Let p = 4, 6, 8, . . . . For every k, m C > 0, ε > 0 and every (K, d) compact metric there is n such that for every C-Lipschitz map f : (Gr(k, Fn), Λ·p) → (K, d) there exists R ∈ Gr(m, Fn) such that (R, · p) is isometric to (Fm, · p) and there exists ¯ f : Bk(Xp) → (K, d) such that Gr(k, R) K Bk(Xp) f τXp ¯ f

• ε
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Ramsey properties of Grassmannians Fields F = Q, R, C

Corollary (p-GLR for F = Q, R, C)

Let p = 4, 6, 8, . . . . For every k, m C > 0, ε > 0 and every (K, d) compact metric there is n such that for every C-Lipschitz map f : (Gr(k, Fn), Λ·p) → (K, d) there exists H ∈ Gr(m, Fn) such that Osc(f ↾ Gr(k, H)) < ε.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Corollary (p-GLR for F = Q, R, C)

Let p = 4, 6, 8, . . . . For every k, m C > 0, ε > 0 and every (K, d) compact metric there is n such that for every C-Lipschitz map f : (Gr(k, Fn), Λ·p) → (K, d) there exists H ∈ Gr(m, Fn) such that Osc(f ↾ Gr(k, H)) < ε. Dvoretzky’s Theorem and the fact that the isometry type of a subspace of a Hilbert space is hilbertian.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Similarly one defines (i) a metric on emb(Fk, Fn) by fixing two norms M and N on Fk and Fn, and defining dM,N(A, B) := A − BM,N.

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Ramsey properties of Grassmannians Fields F = Q, R, C

Similarly one defines (i) a metric on emb(Fk, Fn) by fixing two norms M and N on Fk and Fn, and defining dM,N(A, B) := A − BM,N. (ii) Given a Banach space X, let Nk(X) be the collection of all norms M

• n Fk such that (Fk, M) can be isometrically embedded into X. For

a fixed norm M ∈ Nk(X) we consider the following ∂M,X(P, Q) := inf

T,U T − U(Fk,M),X

where the infimum is running on all isometric embeddings T : (Fk, P) → X and U : (Fk, Q) → X.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(iii) Given a Banach space X and N ∈ Nn(X), let νN : emb(Fk, Fn) → Nk(X) be defined as (νN(A))(x) := N(A(x)). (iv) For 1 ≤ λ, and two f.d. normed spaces E and F, let embλ(E, F) be the collection of all operators T : E → F such that 1 λxE ≤ TxF ≤ λxE.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(iii) Given a Banach space X and N ∈ Nn(X), let νN : emb(Fk, Fn) → Nk(X) be defined as (νN(A))(x) := N(A(x)). (iv) For 1 ≤ λ, and two f.d. normed spaces E and F, let embλ(E, F) be the collection of all operators T : E → F such that 1 λxE ≤ TxF ≤ λxE. (v) Similarly, let N λ

k (X) be the collection of all norms on Fk such that

λ−1Mk(x) ≤ M(x) ≤ λMk(x).

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Ramsey properties of Grassmannians Fields F = Q, R, C

Theorem (p-factorization of full rank matrices)

Let p = 4, 6, 8, . . . . Then for every k, m C > 0, ε > 0, λ ≥ 1 and every (K, d) compact metric there is n such that for every C-Lipschitz map f : (embλ(ℓk

p, ℓn p), Λ·p) → (K, d)

there exists an isometric embedding σ : (Fm, · p) → (Fn, · p) and there exists a C-Lipschitz map ¯ f : N λ

k (Xp) → (K, d) such that

σ ◦ embλ(ℓk

p, ℓm p )

K N λ

k (Xp)

f ν·p ¯ f

• ε
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Ramsey properties of Grassmannians Fields F = Q, R, C

(i) The p-factorization is false for p = 4, 6, . . . : There are f.d subspaces X of Lp such that X is well-complemented in Lp, and there is an isometric copy of X badly complemented. (B. Randrianantoanina proved that the uncomplemented subspace Yp of Lp of H. P. Rosenthal, which has a basis, is isometric to a certain complemented subspace Zp of Lp.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(i) The p-factorization is false for p = 4, 6, . . . : There are f.d subspaces X of Lp such that X is well-complemented in Lp, and there is an isometric copy of X badly complemented. (B. Randrianantoanina proved that the uncomplemented subspace Yp of Lp of H. P. Rosenthal, which has a basis, is isometric to a certain complemented subspace Zp of Lp. (ii) There is a factorization result for arbitrary target metric spaces, but the factor ¯ f is approximate Lipschitz.

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Ramsey properties of Grassmannians Fields F = Q, R, C

(i) The p-factorization is false for p = 4, 6, . . . : There are f.d subspaces X of Lp such that X is well-complemented in Lp, and there is an isometric copy of X badly complemented. (B. Randrianantoanina proved that the uncomplemented subspace Yp of Lp of H. P. Rosenthal, which has a basis, is isometric to a certain complemented subspace Zp of Lp. (ii) There is a factorization result for arbitrary target metric spaces, but the factor ¯ f is approximate Lipschitz. (iii) There is a factorization result for square matrices.

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Ramsey properties of Grassmannians Fields F = Q, R, C

The result follows from: (1) The Extreme Amenability of the corresponding group of isometries Iso(Xp):

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Ramsey properties of Grassmannians Fields F = Q, R, C

The result follows from: (1) The Extreme Amenability of the corresponding group of isometries Iso(Xp): Gromov-Milman for p = 2, Giordano-Pestov for 1 ≤ p < ∞, p = 2, and Bartosova-LA-Lupini-Mbombo for p = ∞.

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Ramsey properties of Grassmannians Fields F = Q, R, C

The result follows from: (1) The Extreme Amenability of the corresponding group of isometries Iso(Xp): Gromov-Milman for p = 2, Giordano-Pestov for 1 ≤ p < ∞, p = 2, and Bartosova-LA-Lupini-Mbombo for p = ∞. (2) The approximate Ultrahomogeneity of Xp:

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Ramsey properties of Grassmannians Fields F = Q, R, C

The result follows from: (1) The Extreme Amenability of the corresponding group of isometries Iso(Xp): Gromov-Milman for p = 2, Giordano-Pestov for 1 ≤ p < ∞, p = 2, and Bartosova-LA-Lupini-Mbombo for p = ∞. (2) The approximate Ultrahomogeneity of Xp: for p = 2 obvious, p = ∞ by definition and for p = 2, 4, 6, . . . , ∞ proved by W. Lusky, (see also Ferenczi-Mbombo-LA-Todorcevic)

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Definition

A Banach space X is called approximately ultrahomogeneous (AU) when for every finite dimensional subspace F of X, every isometric embedding σ : F → X and every ε > 0 there is g ∈ Iso(X) such that σ − g ↾ F < ε.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

By a continuous coloring of a metric space (X, d) we mean a 1-Lipschitz map c : X → [0, 1].

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

By a continuous coloring of a metric space (X, d) we mean a 1-Lipschitz map c : X → [0, 1].

Definition

We say that a collection of Banach spaces F has the Approximate Ramsey Property (ARP) when for every F, G ∈ F and ε > 0 there exists H ∈ F such that Emb(G, H) = ∅ and such that for every continuous coloring c of Emb(F, H) there exists ̺ ∈ Emb(G, H). with

• sc(c ↾ ̺ ◦ Emb(F, G)) < ε.
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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Definition

A topological group G is called extremely amenable when for every continuous action (flow) G K on a compact K has a fixed point; that is, there is p ∈ K such that g · p = p for all g ∈ G.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Definition

A topological group G is called extremely amenable when for every continuous action (flow) G K on a compact K has a fixed point; that is, there is p ∈ K such that g · p = p for all g ∈ G.

Definition

A flow G K is called minimal when every G-orbit is dense. It is called universal when every other flow G-factors through it. Universal Minimal flows M(G) exists and are unique.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

ℓ2 is ultrahomogeneous; the Gurarij space is (AU).

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

ℓ2 is ultrahomogeneous; the Gurarij space is (AU). Lp for p finite and not 4, 6, 8, . . . is (AU).

Problem

Find other (AU) Banach spaces.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

ℓ2 is ultrahomogeneous; the Gurarij space is (AU). Lp for p finite and not 4, 6, 8, . . . is (AU).

Problem

Find other (AU) Banach spaces.

Problem

Does every (AU) Banach space contain an isomorphic copy of ℓ2?

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Proposition

For (AU) spaces X the (ARP) is equivalent to the extreme amenability of Iso(X).

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Proposition

For (AU) spaces X the (ARP) is equivalent to the extreme amenability of Iso(X). This characterization holds for many other metric structures (as operator spaces, systems) which are approximately ultrahomogeneous, and it has been introduced and studied by I. Ben Yaacov.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Proposition

For (AU) spaces X the (ARP) is equivalent to the extreme amenability of Iso(X). This characterization holds for many other metric structures (as operator spaces, systems) which are approximately ultrahomogeneous, and it has been introduced and studied by I. Ben Yaacov.

Proposition

The factorization result of full Rank matrices for 1 ≤ p ≤ ∞, p = 4, 6, . . . is equivalent to the (ARP) of Xp.

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• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

* Gromov-Milman using concentration of measure proved that the unitary group is a Levy group, which implies extreme amenability.

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Ramsey properties Bedlewo 25 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

* Gromov-Milman using concentration of measure proved that the unitary group is a Levy group, which implies extreme amenability. * The Banach-Lamperti representation of isometries on Lp-spaces, gives that for Lp[0, 1] its group of isometries is a semidirect product of two Levy groups, so extremely amenable (Giordano-Pestov).

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 25 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Theorem (Bartosova-LA-Lupini-Mbombo)

The class of all finite dimensional normed spaces has the (ARP).

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 26 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Theorem (Bartosova-LA-Lupini-Mbombo)

The class of all finite dimensional normed spaces has the (ARP). The case of coloring points was already proved by E. Odell, H. P. Rosenthal and Th. Schlumprecht.

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 26 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Theorem (Bartosova-LA-Lupini-Mbombo)

The class of all finite dimensional normed spaces has the (ARP). The case of coloring points was already proved by E. Odell, H. P. Rosenthal and Th. Schlumprecht. The key step in our proof is the following.

Theorem

For every k, m, and r, and every ε > 0 there is n such that whenever we color c : Emb(ℓk

∞, ℓn ∞) → {1, . . . , r} there exists σ ∈ Emb(ℓm ∞, ℓn ∞) and

i ∈ {1, . . . , r} such that σ ◦ Emb(ℓk

∞, ℓm ∞) ⊆ c−1(i).

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 26 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Theorem (Bartosova-LA-Lupini-Mbombo)

The class of all finite dimensional normed spaces has the (ARP). The case of coloring points was already proved by E. Odell, H. P. Rosenthal and Th. Schlumprecht. The key step in our proof is the following.

Theorem

For every k, m, and r, and every ε > 0 there is n such that whenever we color c : Emb(ℓk

∞, ℓn ∞) → {1, . . . , r} there exists σ ∈ Emb(ℓm ∞, ℓn ∞) and

i ∈ {1, . . . , r} such that σ ◦ Emb(ℓk

∞, ℓm ∞) ⊆ c−1(i).

This is is proved from the dual Ramsey theorem and analyzing how those isometric embeddings are.

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 26 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Definition

A compact and convex set K is called a (Choquet) simplex if for every x ∈ K there is a unique probability measure supported on Ext(K) such that the barycenter of µ is x, that is x =

• pdµ(p)

The Poulsen simplex P is the unique (up to affine homeomorphism) metrizable simplex such that Ext(P) is dense in P.

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 27 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

Definition

A compact and convex set K is called a (Choquet) simplex if for every x ∈ K there is a unique probability measure supported on Ext(K) such that the barycenter of µ is x, that is x =

• pdµ(p)

The Poulsen simplex P is the unique (up to affine homeomorphism) metrizable simplex such that Ext(P) is dense in P.

Theorem (Bartosova-LA-Lupini-Mbombo)

The universal minimal flow of the group G of affine homeomorphisms of P is the natural action of G P.

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 27 / 28

• Approx. Ramsey, Ultrahomogeneity, Topological dynamics

This can be done in a quite unified (functorial) way, by identifying the function space A(P) with P. Similarly one does with the Non-commutative Gurarij space NG introduced by T. Oikhberg, and more.

• J. Lopez-Abad (ICMAT)

Ramsey properties Bedlewo 28 / 28