Low distortion embeddings between C(K) spaces

Low distortion

embeddings between C (K ) spaces

joint work with Luis Sánchez González

Tony Procházka

Université de Franche-Comté

25-29 August 2014

First Brazilian Workshop in Geometry of Banach Spaces Maresias

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip <, thenKandLare homeomorphic ().

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

D Xif

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y). Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip <, thenKandLare homeomorphic ().

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

D Xif

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y). Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip <, thenKandLare homeomorphic ().

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

D Xif

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y). Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip <1 +ε, thenK andLare homeomorphic (Jarosz, 1989).

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

D Xif

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y). Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip <1 +ε, thenK andLare homeomorphic (Jarosz, 1989).

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

DX if

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y).

Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip < ¹⁷₁₆, thenKandLare homeomorphic (Dutrieux, Kalton, 2005).

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

DX if

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y).

Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

Introduction

• IfK andLare homeomorphic, thenC(K)andC(L)are isometrically isomorphic.

• IfC(K)andC(L)are isometrically isomorphic, thenKand Lare homeomorphic. (Banach-Stone, 1937).

• If there is a linear isomorphismf :C(K)→C(L)such that kfk

f⁻¹

<2, thenKandLare homeomorphic (Amir-Cambern, 1965).

• If there is a Lipschitz homeomorphismf :C(K)→C(L) such thatkfk_Lip

f⁻¹

Lip < ⁶₅, thenK andLare homeomorphic (Górak, 2011).

Definition (Lipschitz embedding/homeomorphism)

Let(M, d)be a metric space and(X,k·k)a Banach space. We denotef :M ,→

DX if

d(x, y)≤ kf(x)−f(y)k ≤Dd(x, y).

Equivalent tokfk_Lip

f⁻¹

_Lip≤Dup to a scalar multiple off.

• A countable compactK is always homeomorphic to [0, ω^α·n]whereK^(α+1)=∅and1≤n=

K^(α) (Mazurkiewicz-Sierpi ´nski).

• Górak for countable compacts: Ifα6=β orn6=m, then there is no Lipschitz homeomorphism

f :C([0, ω^α·n])→C([0, ω^β·m])such that kfk_Lip

f⁻¹

Lip < ⁶₅.

Theorem

Letα < ω₁ andβ < ω^α. Then there is no Lipschitz embedding f :C([0, ω^α])→C([0, β])such thatkfk_Lip

f⁻¹

Lip <2.

In particular ifα6=β < ω1 then, for anyn, m∈N, there is no Lipschitz homeomorphismf :C([0, ω^α·m])→C([0, ω^β·n]) such thatkfk_Lip

f⁻¹

Lip <2.

• A countable compactK is always homeomorphic to [0, ω^α·n]whereK^(α+1)=∅and1≤n=

K^(α) (Mazurkiewicz-Sierpi ´nski).

• Górak for countable compacts: Ifα6=β orn6=m, then there is no Lipschitz homeomorphism

f :C([0, ω^α·n])→C([0, ω^β·m])such that kfk_Lip

f⁻¹

Lip < ⁶₅.

Theorem

Letα < ω₁ andβ < ω^α. Then there is no Lipschitz embedding f :C([0, ω^α])→C([0, β])such thatkfk_Lip

f⁻¹

Lip <2.

In particular ifα6=β < ω1 then, for anyn, m∈N, there is no Lipschitz homeomorphismf :C([0, ω^α·m])→C([0, ω^β·n]) such thatkfk_Lip

f⁻¹

Lip <2.

• A countable compactK is always homeomorphic to [0, ω^α·n]whereK^(α+1)=∅and1≤n=

K^(α) (Mazurkiewicz-Sierpi ´nski).

• Górak for countable compacts: Ifα6=β orn6=m, then there is no Lipschitz homeomorphism

f :C([0, ω^α·n])→C([0, ω^β·m])such that kfk_Lip

f⁻¹

Lip < ⁶₅.

Theorem

Letα < ω₁ andβ < ω^α. Then there is no Lipschitz embedding f :C([0, ω^α])→C([0, β])such thatkfk_Lip

f⁻¹

Lip <2.

In particular ifα6=β < ω1 then, for anyn, m∈N, there is no Lipschitz homeomorphismf :C([0, ω^α·m])→C([0, ω^β ·n]) such thatkfk_Lip

f⁻¹

Lip <2.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t. M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω₁∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t. M ,→

D C(K),D <2⇒K^(α)6=∅. Corollary

IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t. M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω₁∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t. M ,→

D C(K),D <2⇒K^(α)6=∅. Corollary

IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t. M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω₁∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t. M ,→

D C(K),D <2⇒K^(α)6=∅. Corollary

IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t.

M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω₁∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t. M ,→

D C(K),D <2⇒K^(α)6=∅. Corollary

IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t.

M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω₁∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t. M ,→

D C(K),D <2⇒K^(α)6=∅. Corollary

IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t.

M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω1∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t.

M ,→

D C(K),D <2⇒K^(α)6=∅.

Corollary IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

Another angle

• Aharoni (1974),∃D≥2∀M separable metric sp.: M ,→

D c₀.

• Kalton-Lancien (2008)D= 2.

• Baudier (2013): what if we replacec₀byC(K)?

• A.P., L. Sánchez (2013): ∃M countable metric sp. s.t.

M ,→

D X,D <2⇒Xnot Asplund.

• Where doesM live?

Proposition

Forα < ω1∃M_α ⊂C([0, ω^α])countable uniformly discrete s.t.

M ,→

D C(K),D <2⇒K^(α)6=∅.

Corollary IfM_ω^α,→

D X,D <2, thenSz(X)≥ω^α+1.

More about M

Theorem

There exists a countable metric spaceM such that ifM ,→

D X, D <2, then`₁ ⊂X.

Also, ifM ,→

D `1, thenD≥2.

But...

there is an equivalent norm|·|on`1 such thatM ,→

1 (`1,|·|).

More about M

Theorem

There exists a countable metric spaceM such that ifM ,→

D X, D <2, then`₁⊂X.

Also, ifM ,→

D `1, thenD≥2.

But...

there is an equivalent norm|·|on`1 such thatM ,→

1 (`1,|·|).

More about M

Theorem

There exists a countable metric spaceM such that ifM ,→

D X, D <2, then`₁⊂X.

Also, ifM ,→

D `1, thenD≥2.

But...

there is an equivalent norm|·|on`1 such thatM ,→

1 (`1,|·|).

More about M

Theorem

There exists a countable metric spaceM such that ifM ,→

D X, D <2, then`₁⊂X.

Also, ifM ,→

D `1, thenD≥2.

But...

there is an equivalent norm|·|on`1 such thatM ,→

1 (`1,|·|).

More about M

Theorem

There exists a countable metric spaceM such that ifM ,→

D X, D <2, then`₁⊂X.

Also, ifM ,→

D `1, thenD≥2.

But...

there is an equivalent norm|·|on`1 such thatM ,→

1 (`1,|·|).

The unwieldy metric space M

∅

1 2 3 4

{3,4} {2,4}

{3,2}

{3,2,4} {1,4}

{1,3}

{1,3,4}

{1,2}

{1,2,4} {1,3,2}

{1,3,2,4}

M ={∅} ∪N∪F whereF ={A⊂N: 2≤ |A|<∞}

(a, b)is an edge iffa=∅andb∈N

ora∈N,b∈F anda∈b

The unwieldy metric graph M

∅

1 2 3 4

{3,4} {2,4}

{3,2}

{3,2,4} {1,4}

{1,3}

{1,3,4}

{1,2}

{1,2,4} {1,3,2}

{1,3,2,4}

M ={∅} ∪N∪F whereF ={A⊂N: 2≤ |A|<∞}

(a, b)is an edge iffa=∅andb∈N

ora∈N,b∈F anda∈b

The unwieldy metric graph M

∅

1 2 3 4

{3,4}

{2,4}

{3,2}

{3,2,4}

{1,4}

{1,3}

{1,3,4}

{1,2}

{1,2,4}

{1,3,2}

{1,3,2,4}

M ={∅} ∪N∪F whereF ={A⊂N: 2≤ |A|<∞}

(a, b)is an edge iffa=∅andb∈N

ora∈N,b∈F anda∈b

The unwieldy metric graph M

∅

1 2 3 4

{3,4}

{2,4}

{3,2}

{3,2,4}

{1,4}

{1,3}

{1,3,4}

{1,2}

{1,2,4}

{1,3,2}

{1,3,2,4}

M ={∅} ∪N∪F whereF ={A⊂N: 2≤ |A|<∞}

(a, b)is an edge iffa=∅andb∈N

ora∈N,b∈F anda∈b

The unwieldy metric graph M

∅

1 2 3 4

{3,4}

{2,4}

{3,2}

{3,2,4}

{1,4}

{1,3}

{1,3,4}

{1,2}

{1,2,4}

{1,3,2}

{1,3,2,4}

M ={∅} ∪N∪F whereF ={A⊂N: 2≤ |A|<∞}

(a, b)is an edge iffa=∅andb∈Nora∈N,b∈F anda∈b

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2. Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy. Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy. Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy. Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N.

Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}.

Then∀n∈N

Kn:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

K_n:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

K_n:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

K_n:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

K_n:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Proof of M , →

D X, D < 2 ⇒ ` ₁ ⊂ X

Assumef :M ,→

D X,D <2.

Putx_k:=f(k)∀k∈N

Claim: No subsequence of(x_k)is weakly Cauchy.

Indeed, let(kn)⊂Nbe given.

PutA_N ={k_2n:n≤N}andB_N ={k_2n−1 :n≤N} ∀N ∈N. Put

Xa,b={x^∗∈BX^∗ :hx^∗, f(a)−f(b)i ≥4−2D}. Then∀n∈N

K_n:= \

a∈A_n,b∈B_n

X_a,b6=∅.

(K_n)is a decreasing sequence of non-emptyw^∗-compacts

⇒ ∃x^∗ ∈T∞ n=1Kn

⇒

x^∗, x_k2n−x_k2n+1

≥4−2D

(x_kn)_nis not weakly Cauchy+Rosenthal’s theorem⇒`₁⊂X

Left open

• Fully non-linear Amir-Cambern? At least for scattered compacta?

• Is it true that`₁,→

D X,D <2, implies that`₁⊂X?

Left open

• Fully non-linear Amir-Cambern? At least for scattered compacta?

• Is it true that`₁,→

D X,D <2, implies that`₁⊂X?

Autumn 2014: Thematic trimester at the Université de Franche-Comté

“Geometric and noncommutative methods in

functional analysis”