on the c0-extension property

On the c0-extension property

Claudia CorreaUniversidade Federal do ABC – Brazil

This research was partially supported by FAPESP grants2018/09797-2 and 2019/08515-6

48th Winter School in Abstract Analysis,Svratka, Czech Republic, 11-18 January 2020

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Sobczyk’s theorem

QuestionLet X and Z be Banach spaces, Y be a closed subspace of X andT : Y → Z be a bounded operator. Does T admit a bounded andlinear extension T̃ : X → Z?

X

∃T̃?

��Y?�

OO

T // Z

Answer: No.

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Sobczyk’s theorem

QuestionLet X and Z be Banach spaces, Y be a closed subspace of X andT : Y → Z be a bounded operator. Does T admit a bounded andlinear extension T̃ : X → Z?

X

∃T̃?

��Y?�

OO

T // Z

Answer: No.

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Sobczyk’s theorem

Theorem (Phillips–1940)

The identity operator of c0 does not admit a bounded and linearextension defined on `∞.

`∞

c0?�

OO

Id // c0

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Sobczyk’s theorem

Theorem (Sobczyk–1941)

If X is a separable Banach space, then every c0-valued boundedoperator defined on a closed subspace of X admits a bounded andlinear extension defined on X .

X

∃T̃

��Y?�

OO

∀T // c0

The heart of Sobczyk’s theorem is the weak-star metrizability ofthe closed dual unit ball of the separable Banach space X .

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Sobczyk’s theorem

Theorem (Sobczyk–1941)

If X is a separable Banach space, then every c0-valued boundedoperator defined on a closed subspace of X admits a bounded andlinear extension defined on X .

X

∃T̃

��Y?�

OO

∀T // c0

The heart of Sobczyk’s theorem is the weak-star metrizability ofthe closed dual unit ball of the separable Banach space X .

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Sobczyk’s theorem

NotationGiven a Banach space X , we write:

BX∗ = {α ∈ X ∗ : ‖α‖ ≤ 1}.

We denote by w∗ the weak-star topology on X ∗.

PropositionA Banach space X is separable if and only if BX∗ is w∗-metrizable.

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Sobczyk’s theorem

NotationGiven a Banach space X , we write:

BX∗ = {α ∈ X ∗ : ‖α‖ ≤ 1}.

We denote by w∗ the weak-star topology on X ∗.

PropositionA Banach space X is separable if and only if BX∗ is w∗-metrizable.

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The c0-extension property

Definition (Claudia and Daniel–2014)

We say that a Banach space X has the c0-extension property(c0-EP) if every c0-valued bounded operator defined on a closedsubspace of X admits a c0-valued bounded extension defined on X .

ExampleEvery separable Banach space has the c0-EP.The space `∞ does not have the c0-EP.

QuestionIs there a nonseparable Banach space with the c0-EP?

Answer: Yes. Every Hilbert space has the c0-EP.

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ExampleEvery separable Banach space has the c0-EP.

The space `∞ does not have the c0-EP.



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WCG Banach spaces

DefinitionWe say that a Banach space is weakly compactly generated(WCG) if it contains a weakly compact subset that is linearly dense.

PropositionEvery separable Banach space is WCG.Every reflexive Banach space is WCG. Thus every Hilbertspace is WCG.c0(I ) is WCG, for every set I .

TheoremEvery WCG Banach space has the c0-EP.

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WCG Banach spaces


PropositionEvery separable Banach space is WCG.

Every reflexive Banach space is WCG. Thus every Hilbertspace is WCG.c0(I ) is WCG, for every set I .


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WCG Banach spaces


PropositionEvery separable Banach space is WCG.Every reflexive Banach space is WCG. Thus every Hilbertspace is WCG.

c0(I ) is WCG, for every set I .


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WCG Banach spaces




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WCG Banach spaces




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Tommaso Russo at the 47th Winter School

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WLD Banach spaces

DefinitionWe say that a Banach space X is weakly Lindelöf determined(WLD) if (BX∗ ,w∗) is a Corson compactum.

RemarkRecall that a compact space is metrizable if and only if it embedshomeomorphically in Rω.

DefinitionWe say that a compact space K is a Corson compactum if thereexists a set I such that K embeds homeomorphically in

Σ(I ) = {f ∈ R I : supp(f ) is countable}.

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WLD Banach spaces





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WLD Banach spaces





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WLD Banach spaces

PropositionEvery WCG Banach space is WLD.

Theorem (Claudia–2019)

Every WLD Banach space has the c0-EP.

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WLD Banach spaces




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WLD Banach spaces




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Going beyond WLD spaces

Proposition (Claudia and Daniel–2014)

The space C [0, ω1] has the c0-EP.

QuestionIs the space C [0, ω1] WLD?

Answer: No.

QuestionWhat does the space C [0, ω1] have in common with the WLDspaces?

Answer: If X = C [0, ω1] or X is WLD, then (BX∗ ,w∗) ismonolithic.

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Answer: No.



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Answer: No.



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Answer: No.



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Answer: No.



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DefinitionWe say that a compact and Hausdorff space K is monolithic ifevery closed and separable subspace of K is metrizable.

PropositionEvery Corson compactum is monolithic. Therefore, if X isWLD, then (BX∗ ,w∗) is monolithic.

(Claudia–2019) If X = C [0, ω1], then (BX∗ ,w∗) is monolithic.

QuestionLet X be a Banach space. If (BX∗ ,w∗) is monolithic, then X hasthe c0-EP?

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Let X be a Banach space. If (BX∗ ,w∗) is monolithic, then X hasthe c0-EP.

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The amazing C (K ) world

Open Problem

Let K be a compact and Hausdorff space. When(BC(K)∗ ,w

∗) ismonolithic?

Remark

If(BC(K)∗ ,w

∗) is monolithic, then K is monolithic.

DefinitionWe say that a compact and Hausdorff space K has Property (M)if every measure in M(K ) has separable support.

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Open Problem


∗) ismonolithic?

Remark

If(BC(K)∗ ,w



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Open Problem


∗) ismonolithic?

Remark

If(BC(K)∗ ,w



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Proposition (Claudia–2019)

If K is a monolithic compact space with Property (M), then(BC(K)∗ ,w

∗) is monolithic.

Corollary

If K is a monolithic compact space with Property (M), then C (K )has the c0-EP.

Example

Every metrizable compact space has Property (M).Every Eberlein compactum has Property (M).

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∗) is monolithic.

Corollary


Example


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∗) is monolithic.

Corollary


Example

Every metrizable compact space has Property (M).

Every Eberlein compactum has Property (M).

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∗) is monolithic.

Corollary


Example


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QuestionIf K is a Corson compactum, then C (K ) has the c0-EP?

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QuestionIf K is a Corson compactum, then C (K ) has the c0-EP?

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Theorem (Argyros, Mercourakis and Negrepontis–1988)

Assume MA + ¬CH. Every Corson compactum has Property (M).

Corollary

Assume MA + ¬CH. If K is a Corson compactum, then C (K ) hasthe c0-EP.


Assume CH. There exists a Corson compactum without Property(M).

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Corollary




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Corollary




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If K is the Corson compact space built by Argyros, Mercourakisand Negrepontis under CH, then C (K ) does not have the c0-EP.

Sketch of the proof:C (K ) contains an isomorphic copy of `1(ω1).(Claudia–2019) `1(ω1) does not have the c0-EP.The c0-EP is hereditary for closed subspaces.

Corollary

The existence of a Corson compactum K such that C (K ) does nothave the c0-EP is independent from the axioms of ZFC .

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Sketch of the proof:C (K ) contains an isomorphic copy of `1(ω1).

(Claudia–2019) `1(ω1) does not have the c0-EP.The c0-EP is hereditary for closed subspaces.

Corollary


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Sketch of the proof:C (K ) contains an isomorphic copy of `1(ω1).(Claudia–2019) `1(ω1) does not have the c0-EP.

The c0-EP is hereditary for closed subspaces.

Corollary


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Corollary


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Corollary


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Scattered spaces

Proposition

Every scattered compact space has Property (M).

Corollary

If K is a monolithic scattered compact space, then C (K ) has thec0-EP.

Open Problem

If K is a scattered compact space such that C (K ) has the c0-EP,then K is monolithic?

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Scattered spaces

Proposition


Corollary


Open Problem


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Scattered spaces

Proposition


Corollary


Open Problem


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Scattered spaces


Let K be a scattered compact space with height at most ω + 1. IfC (K ) has the c0-EP, then K is monolithic.

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Bibliography I

A. Argyros, S. Mercourakis, and S. Negrepontis.Functional-analytic properties of corson-compact spaces.Studia Math., 89:197–229, 1988.

C. Correa.On the c0-extension property.https://arxiv.org/pdf/1912.08564.pdf.

C. Correa and D. V. Tausk.Compact lines and the sobczyk property.J. Func. Anal., 266 (9):5765–5778, 2014.

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on the c0-extension property

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