Banach function lattices withthe Daugavet property

María D. Acosta(joint work with

A. Kaminska and M. Mastyło)

Departamento de Análisis MatemáticoUniversidad de Granada

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 1 / 11

Introduction

X Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

Examples

C(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

ExamplesC(K ), K perfect (Daugavet, 1963)

L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

ExamplesC(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)

L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

ExamplesC(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

ExamplesC(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

IntroductionX Banach space, X ∗ its topological dual,

L(X ) := T : X −→ X : T is bounded and linear ,

F (X ) := T ∈ L(X ) : T is a finite-rank operator

DefinitionX does have the Daugavet property (DP) if

‖I + T‖ = 1 + ‖T‖, ∀T ∈ F (X ) .

ExamplesC(K ), K perfect (Daugavet, 1963)L1(0,1) µ atomless and σ-finite (Lozanovskii, 1966)L∞(µ) µ atomless

References:D. Werner, Irish Math. Soc. Bull. (2001).Abramovich and Aliprantis, An invitation to operator theory, AMS, 2002.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 2 / 11

Definitions

Definition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

DefinitionsDefinition

(Ω,A, µ) complete and σ-finite measure space,L0(µ) is the set of all real valued µ-measurable functions.

X is a Banach function lattice on (Ω,A, µ) if it is subspace of L0(µ), endowedwith a complete norm ‖ · ‖X , such that there exists u ∈ X with u > 0 and thefollowing condition is satisfied:

y ∈ L0(µ), x ∈ X , |y | ≤ |x | a.e. ⇒ y ∈ X and ‖y‖X ≤ ‖x‖X .

X ′ is the Kothe dual of X , i.e., G ∈ X ′ iff there is g ∈ L0(µ) such that

G(f ) :=

∫Ω

gf dµ, ∀f ∈ X

An element x ∈ X is called order continuous if for every 0 ≤ xn ≤ |x | such that(xn)↓ 0 a.e. it holds

(‖xn‖

)→ 0.

Xa= subset of order continuous elements of X .

X is order continuous if Xa = X .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 3 / 11

Definitions

DefinitionX satisfies the weak Fatou property whenever for any xn, x ∈ X such that0 ≤ xn ≤ x ,

(xn)↑ x a.e. then

(‖xn‖

)X→ ‖x‖X .

X satisfies the Fatou property if for any x ∈ L0(µ), xn ∈ X such that 0 ≤ xn ≤ x ,(xn)↑ x a.e. and sup ‖xn‖X <∞ we have that x ∈ X and

(‖xn‖

)X→ ‖x‖X .

order continuous ⇒ Fatou property ⇒ weak Fatou property

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 4 / 11

Definitions

DefinitionX satisfies the weak Fatou property whenever for any xn, x ∈ X such that0 ≤ xn ≤ x ,

(xn)↑ x a.e. then

(‖xn‖

)X→ ‖x‖X .

X satisfies the Fatou property if for any x ∈ L0(µ), xn ∈ X such that 0 ≤ xn ≤ x ,(xn)↑ x a.e. and sup ‖xn‖X <∞ we have that x ∈ X and

(‖xn‖

)X→ ‖x‖X .

order continuous ⇒ Fatou property ⇒ weak Fatou property

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 4 / 11

Definitions

DefinitionX satisfies the weak Fatou property whenever for any xn, x ∈ X such that0 ≤ xn ≤ x ,

(xn)↑ x a.e. then

(‖xn‖

)X→ ‖x‖X .

X satisfies the Fatou property if for any x ∈ L0(µ), xn ∈ X such that 0 ≤ xn ≤ x ,(xn)↑ x a.e. and sup ‖xn‖X <∞ we have that x ∈ X and

(‖xn‖

)X→ ‖x‖X .

order continuous ⇒ Fatou property ⇒ weak Fatou property

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 4 / 11

Definitions

DefinitionX satisfies the weak Fatou property whenever for any xn, x ∈ X such that0 ≤ xn ≤ x ,

(xn)↑ x a.e. then

(‖xn‖

)X→ ‖x‖X .

X satisfies the Fatou property if for any x ∈ L0(µ), xn ∈ X such that 0 ≤ xn ≤ x ,(xn)↑ x a.e. and sup ‖xn‖X <∞ we have that x ∈ X and

(‖xn‖

)X→ ‖x‖X .

order continuous ⇒ Fatou property ⇒ weak Fatou property

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 4 / 11

Definitions

Let (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.

Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E .

It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

DefinitionsLet (Ω,A, µ) be a complete and σ-finite measure space.Definition

Given x ∈ L0(µ), its distribution function is defined by

µx (λ) = µt ∈ Ω : |x(t)| > λ (λ ∈ R+).

A Banach function lattice E on (Ω,A, µ) is a rearrangement invariant space(r.i.) if y ∈ E and ‖x‖E = ‖y‖E whenever µx = µy and x ∈ E . It is well knownthat for any r.i. space E we have

L1(µ) ∩ L∞(µ) ⊂ E ⊂ L1(µ) + L∞(µ)

If µ is atomless, the fundamental function of E (φE ) is given by

φE (t) = ‖χA‖E ,

where t ∈ R+ satisfies t ≤ µ(Ω) and A ∈ A satisfies µ(A) = t .María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 5 / 11

Results for rearrangement invariant spaces(Kaminska, Mastyło, A., J. Convex Anal., 2012)

Let (Ω,A, µ) be a complete, σ-finite and atomless measure space.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ).If E is order continuous and has the Daugavet property then L1(µ) → E .If µ(Ω) <∞ then E is L1(µ), endowed with an equivalent norm.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ) and assume that µ isfinite.If E has the Daugavet property and E ′ is order continuous, then E isisomorphic to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 6 / 11

Results for rearrangement invariant spaces(Kaminska, Mastyło, A., J. Convex Anal., 2012)

Let (Ω,A, µ) be a complete, σ-finite and atomless measure space.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ).If E is order continuous and has the Daugavet property then L1(µ) → E .If µ(Ω) <∞ then E is L1(µ), endowed with an equivalent norm.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ) and assume that µ isfinite.If E has the Daugavet property and E ′ is order continuous, then E isisomorphic to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 6 / 11

Results for rearrangement invariant spaces(Kaminska, Mastyło, A., J. Convex Anal., 2012)

Let (Ω,A, µ) be a complete, σ-finite and atomless measure space.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ).If E is order continuous and has the Daugavet property then L1(µ) → E .If µ(Ω) <∞ then E is L1(µ), endowed with an equivalent norm.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ) and assume that µ isfinite.If E has the Daugavet property and E ′ is order continuous, then E isisomorphic to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 6 / 11

Results for rearrangement invariant spaces(Kaminska, Mastyło, A., J. Convex Anal., 2012)

Let (Ω,A, µ) be a complete, σ-finite and atomless measure space.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ).If E is order continuous and has the Daugavet property then L1(µ) → E .If µ(Ω) <∞ then E is L1(µ), endowed with an equivalent norm.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ) and assume that µ isfinite.

If E has the Daugavet property and E ′ is order continuous, then E isisomorphic to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 6 / 11

Results for rearrangement invariant spaces(Kaminska, Mastyło, A., J. Convex Anal., 2012)

Let (Ω,A, µ) be a complete, σ-finite and atomless measure space.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ).If E is order continuous and has the Daugavet property then L1(µ) → E .If µ(Ω) <∞ then E is L1(µ), endowed with an equivalent norm.

PropositionLet E be a r.i. space with the Fatou property on (Ω,A, µ) and assume that µ isfinite.If E has the Daugavet property and E ′ is order continuous, then E isisomorphic to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 6 / 11

Isometric results (Kadets, Martín, Merí, D. Werner,Canadian J. Math., 2013)

TheoremThe only separable r.i. space on [0,1] with the (DP) is L1[0,1], endowed withits canonical norm.

Indeed, if (Ω,A, µ) is a finite measure space and E is a separable r.i. spaceon it with the (DP), then E is isometric to L1(µ).

Question: What about L∞(µ)?

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 7 / 11

Isometric results (Kadets, Martín, Merí, D. Werner,Canadian J. Math., 2013)

TheoremThe only separable r.i. space on [0,1] with the (DP) is L1[0,1], endowed withits canonical norm.

Indeed, if (Ω,A, µ) is a finite measure space and E is a separable r.i. spaceon it with the (DP), then E is isometric to L1(µ).

Question: What about L∞(µ)?

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 7 / 11

Isometric results (Kadets, Martín, Merí, D. Werner,Canadian J. Math., 2013)

TheoremThe only separable r.i. space on [0,1] with the (DP) is L1[0,1], endowed withits canonical norm.

Indeed, if (Ω,A, µ) is a finite measure space and E is a separable r.i. spaceon it with the (DP), then E is isometric to L1(µ).

Question: What about L∞(µ)?

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 7 / 11

Isometric results (Kadets, Martín, Merí, D. Werner,Canadian J. Math., 2013)

TheoremThe only separable r.i. space on [0,1] with the (DP) is L1[0,1], endowed withits canonical norm.

Indeed, if (Ω,A, µ) is a finite measure space and E is a separable r.i. spaceon it with the (DP), then E is isometric to L1(µ).

Question: What about L∞(µ)?

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 7 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

TheoremLet E be a r.i. space on a finite measure space (Ω,A, µ) with the weak Fatouproperty.If E has the Daugavet property then either E = L1(µ) or E = L∞(µ)isometrically.

Some facts used in the proof:

If supp Ea = Ω, then Ea has (DP) (AKM, J. Convex Anal.).

Ea is order continuous and has (DP), so Ea coincides with L1(µ) as aBanach space if we assume that ‖χΩ‖E = µ(Ω) (KMMW, Canad. J.Math.).

Hence E is isometric to L1(µ).María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 8 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

If supp Ea = ∅ we have φE (0+) > 0.

SinceφE (t)φE′(t) = t , ∀0 ≤ t < µ(Ω)

Then φE ′(0+) = 0.

PropositionLet E be a r.i. space over (Ω,A, µ) with the weak Fatou property.If φE ′(0+) = 0 and E has (DP) then E is isometric to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 9 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

If supp Ea = ∅ we have φE (0+) > 0.

SinceφE (t)φE′(t) = t , ∀0 ≤ t < µ(Ω)

Then φE ′(0+) = 0.

PropositionLet E be a r.i. space over (Ω,A, µ) with the weak Fatou property.If φE ′(0+) = 0 and E has (DP) then E is isometric to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 9 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

If supp Ea = ∅ we have φE (0+) > 0.

SinceφE (t)φE′(t) = t , ∀0 ≤ t < µ(Ω)

Then φE ′(0+) = 0.

PropositionLet E be a r.i. space over (Ω,A, µ) with the weak Fatou property.If φE ′(0+) = 0 and E has (DP) then E is isometric to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 9 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

If supp Ea = ∅ we have φE (0+) > 0.

SinceφE (t)φE′(t) = t , ∀0 ≤ t < µ(Ω)

Then φE ′(0+) = 0.

PropositionLet E be a r.i. space over (Ω,A, µ) with the weak Fatou property.If φE ′(0+) = 0 and E has (DP) then E is isometric to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 9 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

If supp Ea = ∅ we have φE (0+) > 0.

SinceφE (t)φE′(t) = t , ∀0 ≤ t < µ(Ω)

Then φE ′(0+) = 0.

PropositionLet E be a r.i. space over (Ω,A, µ) with the weak Fatou property.If φE ′(0+) = 0 and E has (DP) then E is isometric to L∞(µ).

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 9 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?

E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?

E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP)

(Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP)

(Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)

LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)

(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such thatx∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.

(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.

(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Isometric results (Kaminska, Mastyło, A.Trans. Amer. Math. Soc.)

A few words about the proof of the Proposition:

1 Where to use the assumption of weak Fatou property?E has the weak Fatou property ⇒ E ′ is norming.

2 Characterization of (DP) (Kadets, Shvidkoy, Sirotkin, Werner, TAMS,2000)LemmaTFAE:(i) X has (DP)(ii) For every x0 ∈ SX , y∗0 ∈ SX∗ and ε > 0 there is x∗ ∈ SX∗ such that

x∗(x0) > 1− ε and ‖x∗ + y∗0 ‖ > 2− ε.(ii)′ Given a norming set B ⊂ SX∗ with −B ⊂ B,

for every x0 ∈ SX , x∗0 ∈ SX∗ and ε > 0 there is b∗ ∈ B such thatb∗(x0) > 1− ε and ‖b∗ + x∗0 ‖ > 2− ε.

We use condition (ii)′ for B = SE ′ and the argument in the proof byKadets, Martín, Merí and Werner.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 10 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11

Some open problems

What happens in case of infinite measure?

Assume that (Ω,A, µ) is an atomless σ-finite measure space andE is a r.i. space on (Ω,A, µ) with (DP).Is E isometric to L1(µ) or L∞(µ)?

Assume that E is a r.i. space on (Ω,A, µ) and A ∈ A.If E has (DP), does E(A) also has the Daugavet property?E(A) = xχA : x ∈ E.

María D. Acosta (Universidad de Granada) The Daugavet property Valencia, 05/06/2013 11 / 11