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Motivation . . . . . . . . . . . . . . . . . . . - - PowerPoint PPT Presentation

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Our Muse: Automorphisms of ( N )/ Fin Automorphisms of / c 0 On automorphisms of the Banach space / c 0 Cristbal R P Universidade de So Paulo and Universidad de Los Andes Bdlewo,

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

On automorphisms of the Banach space ℓ∞/c0

Cristóbal R P

Universidade de São Paulo and Universidad de Los Andes

Będlewo, July 2016

Joint work with Piotr Koszmider

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is primary? In other words, is it consistent that and neither nor is isomorphic to ? Theme Is it possible to develop a theory of automorphisms on the Banach space corresponding to the theory of automorphism of the Boolean Algebra Fin? , Fin Fin

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is ℓ∞/c0 primary? In other words, is it consistent that and neither nor is isomorphic to ? Theme Is it possible to develop a theory of automorphisms on the Banach space corresponding to the theory of automorphism of the Boolean Algebra Fin? , Fin Fin

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is ℓ∞/c0 primary? In other words, is it consistent that ℓ∞/c0 = A ⊕ B and neither A nor B is isomorphic to ℓ∞/c0? Theme Is it possible to develop a theory of automorphisms on the Banach space corresponding to the theory of automorphism of the Boolean Algebra Fin? , Fin Fin

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is ℓ∞/c0 primary? In other words, is it consistent that ℓ∞/c0 = A ⊕ B and neither A nor B is isomorphic to ℓ∞/c0? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ∞/c0 corresponding to the theory of automorphism of the Boolean Algebra ℘(N)/Fin? , Fin Fin

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is ℓ∞/c0 primary? In other words, is it consistent that ℓ∞/c0 = A ⊕ B and neither A nor B is isomorphic to ℓ∞/c0? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ∞/c0 corresponding to the theory of automorphism of the Boolean Algebra ℘(N)/Fin? ℓ∞/c0 ≡ C(N∗), S(℘(N)/Fin) = N∗ = βN \ N Fin

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Motivation

estion Is ℓ∞/c0 primary? In other words, is it consistent that ℓ∞/c0 = A ⊕ B and neither A nor B is isomorphic to ℓ∞/c0? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ∞/c0 corresponding to the theory of automorphism of the Boolean Algebra ℘(N)/Fin? ℓ∞/c0 ≡ C(N∗), S(℘(N)/Fin) = N∗ = βN \ N ℘(N) ∼ {χA ∈ ℓ∞ : A ⊆ N} ⊆ ℓ∞ ℘(N)/Fin ∼ {[χA]c0 ∈ ℓ∞/c0 : A ⊆ N} ⊆ ℓ∞/c0

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Outline

1

Our Muse: Automorphisms of ℘(N)/Fin

2

Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If is a permutation of , then

Fin Fin

defines an automorphism of Fin. (CH) There are nontrivial automorphisms of Fin (Rudin, 1956) (PFA) All automorphisms of Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If σ : N → N is a permutation of N, then

Fin Fin

defines an automorphism of Fin. (CH) There are nontrivial automorphisms of Fin (Rudin, 1956) (PFA) All automorphisms of Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If σ : N → N is a permutation of N, then h([A]Fin) = [σ−1(A)]Fin, ∀A ⊆ N defines an automorphism of ℘(N)/Fin. (CH) There are nontrivial automorphisms of Fin (Rudin, 1956) (PFA) All automorphisms of Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If σ : N → N is a permutation of N, then h([A]Fin) = [σ−1(A)]Fin, ∀A ⊆ N defines an automorphism of ℘(N)/Fin. (CH) There are nontrivial automorphisms of ℘(N)/Fin (Rudin, 1956) (PFA) All automorphisms of Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If σ : N → N is a permutation of N, then h([A]Fin) = [σ−1(A)]Fin, ∀A ⊆ N defines an automorphism of ℘(N)/Fin. (CH) There are nontrivial automorphisms of ℘(N)/Fin (Rudin, 1956) (PFA) All automorphisms of ℘(N)/Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

Trivial automorphisms of ℘(N)/Fin

Remark If σ : N → N is a permutation of N, then h([A]Fin) = [σ−1(A)]Fin, ∀A ⊆ N defines an automorphism of ℘(N)/Fin. (CH) There are nontrivial automorphisms of ℘(N)/Fin (Rudin, 1956) (PFA) All automorphisms of ℘(N)/Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of ℘(N)/Fin are trivial (Veličković, 1993)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Trivial

h a Boolean automorphism of ℘(N)/Fin is trivial if there a permutation

  • f

such that for all we have

Fin Fin

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is trivial if there is a nonzero real and a permutation

  • f

such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Trivial

h a Boolean automorphism of ℘(N)/Fin h is trivial if there a permutation σ of N such that for all A ⊆ N we have h([A]Fin) = [σ−1(A)]Fin T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is trivial if there is a nonzero real and a permutation

  • f

such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Trivial

h a Boolean automorphism of ℘(N)/Fin h is trivial if there a permutation σ of N such that for all A ⊆ N we have h([A]Fin) = [σ−1(A)]Fin T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator T is trivial if there is a nonzero real r ∈ R and a permutation σ of N such that for all f ∈ ℓ∞ we have T([f]c0) = [rf ◦ σ]c0

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Liable

h a Boolean automorphism of ℘(N)/Fin is liable to if there exists a Boolean automorphism

  • f

such that for all we have

Fin Fin

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is liable to if there is a linear bounded such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Liable

h a Boolean automorphism of ℘(N)/Fin h is liable to ℘(N) if there exists a Boolean automorphism H of ℘(N) such that for all A ⊆ N we have h([A]Fin) = [H(A)]Fin

℘(N) ℘(N) ℘(N)/Fin ℘(N)/Fin H q q h

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is liable to if there is a linear bounded such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Liable

h a Boolean automorphism of ℘(N)/Fin h is liable to ℘(N) if there exists a Boolean automorphism H of ℘(N) such that for all A ⊆ N we have h([A]Fin) = [H(A)]Fin

℘(N) ℘(N) ℘(N)/Fin ℘(N)/Fin H q q h

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator T is liable to ℓ∞ if there is a linear bounded S : ℓ∞ → ℓ∞ such that for all f ∈ ℓ∞ we have T([f]c0) = [S(f)]c0

ℓ∞ ℓ∞ ℓ∞/c0 ℓ∞/c0 S Q Q T

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: determined by c0

h a Boolean automorphism of ℘(N)/Fin is induced by an automorphism of FinCofin if there is an automorphism FinCofin FinCofin, such that for all we have

Fin Fin

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is a matrix operator if there is an operator given by a real matrix such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: determined by c0

h a Boolean automorphism of ℘(N)/Fin h is induced by an automorphism of FinCofin(N) if there is an automorphism G : FinCofin → FinCofin, such that for all A ⊆ N we have h([A]Fin) = [∪{G(n) : n ∈ A}]Fin T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator is a matrix operator if there is an operator given by a real matrix such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: determined by c0

h a Boolean automorphism of ℘(N)/Fin h is induced by an automorphism of FinCofin(N) if there is an automorphism G : FinCofin → FinCofin, such that for all A ⊆ N we have h([A]Fin) = [∪{G(n) : n ∈ A}]Fin T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator T is a matrix operator if there is an operator S : c0 → c0 given by a real matrix (bij)(i,j)∈N2 such that for all f ∈ ℓ∞ we have T([f]c0) = [(∑

j∈N bijf(j))i∈N]c0

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Canonizable

h a Boolean automorphism of ℘(N)/Fin If we identify points of N∗ with ultrafilters in ℘(N)/Fin, the Stone duality gives that: there is a continuous map such that for every we have

Fin

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Since , we obtain is canonizable along if is a surjective continuous mapping and there is a nonzero real such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Canonizable

h a Boolean automorphism of ℘(N)/Fin If we identify points of N∗ with ultrafilters in ℘(N)/Fin, the Stone duality gives that: there is a continuous map ψ : N∗ → N∗ such that for every A ⊆ N we have χh([A]Fin)∗ = χA∗ ◦ ψ T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Since , we obtain is canonizable along if is a surjective continuous mapping and there is a nonzero real such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Canonizable

h a Boolean automorphism of ℘(N)/Fin If we identify points of N∗ with ultrafilters in ℘(N)/Fin, the Stone duality gives that: there is a continuous map ψ : N∗ → N∗ such that for every A ⊆ N we have χh([A]Fin)∗ = χA∗ ◦ ψ T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Since ℓ∞/c0 ≡ C(N∗), we obtain ˆ T : C(N∗) → C(N∗) is canonizable along if is a surjective continuous mapping and there is a nonzero real such that for all we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Canonizable

h a Boolean automorphism of ℘(N)/Fin If we identify points of N∗ with ultrafilters in ℘(N)/Fin, the Stone duality gives that: there is a continuous map ψ : N∗ → N∗ such that for every A ⊆ N we have χh([A]Fin)∗ = χA∗ ◦ ψ T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Since ℓ∞/c0 ≡ C(N∗), we obtain ˆ T : C(N∗) → C(N∗) T is canonizable along ψ : N∗ → N∗ if ψ is a surjective continuous mapping and there is a nonzero real r such that for all g ∈ C(N∗) we have ˆ T(g) = rg ◦ ψ

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Continuous

h a Boolean automorphism of ℘(N)/Fin A special feature of liings: Every automorphism of is continuous with respect to the product topology on T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Is every liing continuous in the product topology of ?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Continuous

h a Boolean automorphism of ℘(N)/Fin A special feature of liings: Every automorphism of ℘(N) is continuous with respect to the product topology on ℘(N) ∼ {0, 1}N T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator Is every liing continuous in the product topology of ?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Continuous

h a Boolean automorphism of ℘(N)/Fin A special feature of liings: Every automorphism of ℘(N) is continuous with respect to the product topology on ℘(N) ∼ {0, 1}N T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator This sugests the question: Is every liing continuous in the product topology of ?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0

In search for the right notion: Continuous

h a Boolean automorphism of ℘(N)/Fin A special feature of liings: Every automorphism of ℘(N) is continuous with respect to the product topology on ℘(N) ∼ {0, 1}N T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator This sugests the question: Is every liing S : ℓ∞ → ℓ∞ continuous in the product topology of ℓ∞ ⊆ RN?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

h a Boolean automorphism of ℘(N)/Fin

Trivial automorphism Automorphism induced by an automorphism

  • f FinCofin

Automorphism with a continuous lifting Liftable automorphism Automorphism ⇔ ⇔ ⇔

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator

Trivial automorphism Automorphic matrix operator Automorphic liftable operator with a lifting continuous

  • n Bℓ∞

Automorphic liftable operator Automorphism ⇒

  • ⇔

⇒

  • ⇒
  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

h a Boolean automorphism of ℘(N)/Fin

Trivial automorphism Automorphism induced by an automorphism

  • f FinCofin

Automorphism with a continuous lifting Liftable automorphism Automorphism ⇔ ⇔ ⇔

T : ℓ∞/c0 → ℓ∞/c0 a linear bounded operator

Trivial automorphism Automorphic matrix operator Automorphic liftable operator with a lifting continuous

  • n Bℓ∞

Automorphic liftable operator Automorphism ⇒

  • ⇔

⇒

  • ⇒
  • The nontrivial parts of the second chart are noted in red.
  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that is a linear bounded operator and two infinite sets. The localization of to is the

  • perator

given by

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator and A, B ⊆ N two infinite sets. The localization of T to (A, B) is the

  • perator TB,A : ℓ∞(A)/c0(A) → ℓ∞(B)/c0(B) given by
  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator and A, B ⊆ N two infinite sets. The localization of T to (A, B) is the

  • perator TB,A : ℓ∞(A)/c0(A) → ℓ∞(B)/c0(B) given by

TB,A = PB ◦ T ◦ IA

ℓ∞/c0 ℓ∞/c0 ℓ∞(A)/c0(A) ℓ∞(B)/c0(B) T I P

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that is a linear bounded operator. Let be one of the properties “liable”, “matrix operator”, “trivial”, “canonizable”.

1

We say that is somewhere if, and only if, there are infinite and such that has .

2

We say that is right-locally if, and only if, for every infinite there are infinite and such that has .

3

We say that is le-locally if, and only if, for every infinite there are infinite and such that has .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator. Let P be one of the properties “liable”, “matrix operator”, “trivial”, “canonizable”.

1

We say that is somewhere if, and only if, there are infinite and such that has .

2

We say that is right-locally if, and only if, for every infinite there are infinite and such that has .

3

We say that is le-locally if, and only if, for every infinite there are infinite and such that has .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator. Let P be one of the properties “liable”, “matrix operator”, “trivial”, “canonizable”.

1

We say that T is somewhere P if, and only if, there are infinite A ⊆ N and B ⊆ N such that TB,A has P.

2

We say that is right-locally if, and only if, for every infinite there are infinite and such that has .

3

We say that is le-locally if, and only if, for every infinite there are infinite and such that has .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator. Let P be one of the properties “liable”, “matrix operator”, “trivial”, “canonizable”.

1

We say that T is somewhere P if, and only if, there are infinite A ⊆ N and B ⊆ N such that TB,A has P.

2

We say that T is right-locally P if, and only if, for every infinite A ⊆ N there are infinite A1 ⊆ A and B ⊆ N such that TB,A1 has P.

3

We say that is le-locally if, and only if, for every infinite there are infinite and such that has .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

Definition Suppose that T : ℓ∞/c0 → ℓ∞/c0 is a linear bounded operator. Let P be one of the properties “liable”, “matrix operator”, “trivial”, “canonizable”.

1

We say that T is somewhere P if, and only if, there are infinite A ⊆ N and B ⊆ N such that TB,A has P.

2

We say that T is right-locally P if, and only if, for every infinite A ⊆ N there are infinite A1 ⊆ A and B ⊆ N such that TB,A1 has P.

3

We say that T is le-locally P if, and only if, for every infinite B ⊆ N there are infinite B1 ⊆ B and A ⊆ N such that TB1,A has P.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

In the class of automorphisms of ℓ∞/c0 we have: estion Are automorphisms on somewhere canonizable along a homeomorphism?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

In the class of automorphisms of ℓ∞/c0 we have:

Somewhere trivial Somewhere isomorphic matrix operator Somewhere isomorphic liftable operator with a lifting continuous

  • n Bℓ∞

Somewhere liftable isomorphism Automorphism ⇔ ⇔ ⇔ ?

estion Are automorphisms on somewhere canonizable along a homeomorphism?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

slide-44
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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

In the class of automorphisms of ℓ∞/c0 we have:

Somewhere trivial Somewhere isomorphic matrix operator Somewhere isomorphic liftable operator with a lifting continuous

  • n Bℓ∞

Somewhere liftable isomorphism Automorphism ⇔ ⇔ ⇔ ?

estion Are automorphisms on somewhere canonizable along a homeomorphism?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Localizations

In the class of automorphisms of ℓ∞/c0 we have:

Somewhere trivial Somewhere isomorphic matrix operator Somewhere isomorphic liftable operator with a lifting continuous

  • n Bℓ∞

Somewhere liftable isomorphism Automorphism ⇔ ⇔ ⇔ ?

estion Are automorphisms on ℓ∞/c0 somewhere canonizable along a homeomorphism?

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

We represent a bounded linear

  • perator

as a weakly continuous mapping Remark An operator is somewhere canonizable along a homeomorphism if, and only if, there exist infinite , a homeomorphism and some nonzero , such that for every we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

We represent a bounded linear

  • perator T : C(N∗) → C(N∗) as

a weakly∗ continuous mapping τ : N∗ → M(N∗) Remark An operator is somewhere canonizable along a homeomorphism if, and only if, there exist infinite , a homeomorphism and some nonzero , such that for every we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

We represent a bounded linear

  • perator T : C(N∗) → C(N∗) as

a weakly∗ continuous mapping τ : N∗ → M(N∗) y → τ(y) = T ⋆(δy) Remark An operator is somewhere canonizable along a homeomorphism if, and only if, there exist infinite , a homeomorphism and some nonzero , such that for every we have

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

We represent a bounded linear

  • perator T : C(N∗) → C(N∗) as

a weakly∗ continuous mapping τ : N∗ → M(N∗) y → τ(y) = T ⋆(δy) Remark An operator T is somewhere canonizable along a homeomorphism if, and only if, there exist infinite A, B ⊆ N, a homeomorphism ψ : B∗ → A∗ and some nonzero r ∈ R, such that for every y ∈ B∗ we have T ⋆(δy)|A∗ = rδψ(y)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

Consider the partial multifunction given by All atoms of the measure We are looking for a mapping which is a homeomorphic selection from , defined between clopen sets. Remark An operator T is somewhere canonizable along a homeomorphism if, and only if, there exist infinite A, B ⊆ N, a homeomorphism ψ : B∗ → A∗ and some nonzero r ∈ R, such that for every y ∈ B∗ we have T ⋆(δy)|A∗ = rδψ(y)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

Consider the partial multifunction φT : N∗ → ℘(N∗) given by φT (y) = {All atoms of the measure T ⋆(δy)} We are looking for a mapping which is a homeomorphic selection from , defined between clopen sets. Remark An operator T is somewhere canonizable along a homeomorphism if, and only if, there exist infinite A, B ⊆ N, a homeomorphism ψ : B∗ → A∗ and some nonzero r ∈ R, such that for every y ∈ B∗ we have T ⋆(δy)|A∗ = rδψ(y)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

The method

Consider the partial multifunction φT : N∗ → ℘(N∗) given by φT (y) = {All atoms of the measure T ⋆(δy)} We are looking for a mapping ψ which is a homeomorphic selection from φT , defined between clopen sets. Remark An operator T is somewhere canonizable along a homeomorphism if, and only if, there exist infinite A, B ⊆ N, a homeomorphism ψ : B∗ → A∗ and some nonzero r ∈ R, such that for every y ∈ B∗ we have T ⋆(δy)|A∗ = rδψ(y)

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Fountains and funnels

Definition Let T : C(N∗) → C(N∗) be an operator, A, B ⊆ N infinite and F ⊆ N∗ be nowhere dense. The pair is called a fountain for if is concentrated on for every . The pair is called a funnel for if there is no proper closed subset of where all the measures for are concentrated.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Fountains and funnels

Definition Let T : C(N∗) → C(N∗) be an operator, A, B ⊆ N infinite and F ⊆ N∗ be nowhere dense. The pair (F, B∗) is called a fountain for T if T ⋆(δy) is concentrated on F for every y ∈ B∗ . The pair is called a funnel for if there is no proper closed subset of where all the measures for are concentrated.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Fountains and funnels

Definition Let T : C(N∗) → C(N∗) be an operator, A, B ⊆ N infinite and F ⊆ N∗ be nowhere dense. The pair (F, B∗) is called a fountain for T if T ⋆(δy) is concentrated on F for every y ∈ B∗ . The pair (A∗, F) is called a funnel for T if there is no proper closed subset of A∗ where all the measures T ⋆(δy)|A∗ for y ∈ F are concentrated.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Sufficient conditions for canonization

Theorem Every automorphism on which is fountainless is le-locally canonizable along a quasi-open mapping. Every automorphism on which is funnelless is right-locally canonizable along a quasi-open mapping.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Sufficient conditions for canonization

Theorem Every automorphism on ℓ∞/c0 which is fountainless is le-locally canonizable along a quasi-open mapping. Every automorphism on which is funnelless is right-locally canonizable along a quasi-open mapping.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Sufficient conditions for canonization

Theorem Every automorphism on ℓ∞/c0 which is fountainless is le-locally canonizable along a quasi-open mapping. Every automorphism on ℓ∞/c0 which is funnelless is right-locally canonizable along a quasi-open mapping.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

OCA+MA

Theorem (OCA+MA) Every fountainless automorphism of is le-locally trivial. (OCA+MA) Every funnelless automorphism of is right-locally trivial.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

OCA+MA

Theorem (OCA+MA) Every fountainless automorphism of ℓ∞/c0 is le-locally trivial. (OCA+MA) Every funnelless automorphism of is right-locally trivial.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

OCA+MA

Theorem (OCA+MA) Every fountainless automorphism of ℓ∞/c0 is le-locally trivial. (OCA+MA) Every funnelless automorphism of ℓ∞/c0 is right-locally trivial.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Continuum Hypothesis

Theorem (CH) There is an automorphism (with fountains) such that it is not le-locally canonizable along any continuous map. (CH) There is an automorphism (with funnels) such that it is not right-locally canonizable along any continuous map. (CH) There is an automorphism which is nowhere canonizable along a quasi-open map, in particular along a homeomorphism. (CH) There is a fountainless and funnelless everywhere present isomorphic embedding globally canonizable along a quasi-open map which is nowhere canonizable along a homeomorphism.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

slide-63
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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Continuum Hypothesis

Theorem (CH) There is an automorphism (with fountains) such that it is not le-locally canonizable along any continuous map. (CH) There is an automorphism (with funnels) such that it is not right-locally canonizable along any continuous map. (CH) There is an automorphism which is nowhere canonizable along a quasi-open map, in particular along a homeomorphism. (CH) There is a fountainless and funnelless everywhere present isomorphic embedding globally canonizable along a quasi-open map which is nowhere canonizable along a homeomorphism.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

slide-64
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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Continuum Hypothesis

Theorem (CH) There is an automorphism (with fountains) such that it is not le-locally canonizable along any continuous map. (CH) There is an automorphism (with funnels) such that it is not right-locally canonizable along any continuous map. (CH) There is an automorphism which is nowhere canonizable along a quasi-open map, in particular along a homeomorphism. (CH) There is a fountainless and funnelless everywhere present isomorphic embedding globally canonizable along a quasi-open map which is nowhere canonizable along a homeomorphism.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

slide-65
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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Continuum Hypothesis

Theorem (CH) There is an automorphism (with fountains) such that it is not le-locally canonizable along any continuous map. (CH) There is an automorphism (with funnels) such that it is not right-locally canonizable along any continuous map. (CH) There is an automorphism which is nowhere canonizable along a quasi-open map, in particular along a homeomorphism. (CH) There is a fountainless and funnelless everywhere present isomorphic embedding globally canonizable along a quasi-open map which is nowhere canonizable along a homeomorphism.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Continuum Hypothesis

Theorem (CH) There is an automorphism (with fountains) such that it is not le-locally canonizable along any continuous map. (CH) There is an automorphism (with funnels) such that it is not right-locally canonizable along any continuous map. (CH) There is an automorphism which is nowhere canonizable along a quasi-open map, in particular along a homeomorphism. (CH) There is a fountainless and funnelless everywhere present isomorphic embedding globally canonizable along a quasi-open map which is nowhere canonizable along a homeomorphism.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Open problems

Is it consistent (does it follow from PFA or OCA+MA) that every automorphism is somewhere trivial? If yes, then is not embeddable into .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Open problems

Is it consistent (does it follow from PFA or OCA+MA) that every automorphism is somewhere trivial? If yes, then is not embeddable into .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Open problems

Is it consistent (does it follow from PFA or OCA+MA) that every automorphism is somewhere trivial? If yes, then is not embeddable into .

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

Open problems

Is it consistent (does it follow from PFA or OCA+MA) that every automorphism is somewhere trivial? If yes, then ℓ∞(ℓ∞/c0) is not embeddable into ℓ∞/c0.

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0

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Our Muse: Automorphisms of ℘(N)/Fin Automorphisms of ℓ∞/c0 Localizations Fountains and funnels OCA+MA vs. CH

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  • P. Koszmider and C. Rodríguez-Porras, On automorphisms of the Banach space

ℓ∞/c0. Fund. Math. 235 (2016), 49-99. Available at the journal’s website and at: http://arxiv.org/abs/1501.03466?context=math

  • C. Rodríguez Porras

On automorphisms of the Banach space ℓ∞/c0