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

Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 On automorphisms of the Banach space ℓ ∞ / c 0 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 ∞ / c 0

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 Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

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 Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation estion Is ℓ ∞ / c 0 primary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

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 Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation estion Is ℓ ∞ / c 0 primary? In other words, is it consistent that ℓ ∞ / c 0 = A ⊕ B and neither A nor B is isomorphic to ℓ ∞ / c 0 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

, Fin Fin Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation estion Is ℓ ∞ / c 0 primary? In other words, is it consistent that ℓ ∞ / c 0 = A ⊕ B and neither A nor B is isomorphic to ℓ ∞ / c 0 ? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ ∞ / c 0 corresponding to the theory of automorphism of the Boolean Algebra ℘ ( N )/ Fin ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

Fin Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation estion Is ℓ ∞ / c 0 primary? In other words, is it consistent that ℓ ∞ / c 0 = A ⊕ B and neither A nor B is isomorphic to ℓ ∞ / c 0 ? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ ∞ / c 0 corresponding to the theory of automorphism of the Boolean Algebra ℘ ( N )/ Fin ? S ( ℘ ( N )/ Fin ) = N ∗ = β N \ N ℓ ∞ / c 0 ≡ C ( N ∗ ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Motivation estion Is ℓ ∞ / c 0 primary? In other words, is it consistent that ℓ ∞ / c 0 = A ⊕ B and neither A nor B is isomorphic to ℓ ∞ / c 0 ? Theme Is it possible to develop a theory of automorphisms on the Banach space ℓ ∞ / c 0 corresponding to the theory of automorphism of the Boolean Algebra ℘ ( N )/ Fin ? S ( ℘ ( N )/ Fin ) = N ∗ = β N \ N ℓ ∞ / c 0 ≡ C ( N ∗ ) , ℘ ( N ) ∼ { χ A ∈ ℓ ∞ : A ⊆ N } ⊆ ℓ ∞ ℘ ( N )/ Fin ∼ { [ χ A ] c 0 ∈ ℓ ∞ / c 0 : A ⊆ N } ⊆ ℓ ∞ / c 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Outline Our Muse: Automorphisms of ℘ ( N )/ Fin 1 Automorphisms of ℓ ∞ / c 0 2 Localizations Fountains and funnels OCA+MA vs. CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

(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) Remark If is a permutation of , then Fin Fin defines an automorphism of Fin . Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Trivial automorphisms of ℘ ( N )/ Fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

(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) Fin Fin defines an automorphism of Fin . Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 Trivial automorphisms of ℘ ( N )/ Fin Remark If σ : N → N is a permutation of N , then . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

(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) Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

(PFA) All automorphisms of Fin are trivial (Shelah-Steprāns, 1988) (OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993) Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 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) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

(OCA+MA) All automorphisms of Fin are trivial (Veličković, 1993) Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 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) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0

Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 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 ∞ / c 0

is trivial if there a is trivial if there is a permutation of such nonzero real and a that for all we have permutation of such that for all we have Fin Fin Our Muse: Automorphisms of ℘ ( N )/ Fin Automorphisms of ℓ ∞ / c 0 In search for the right notion: Trivial h a Boolean automorphism of T : ℓ ∞ / c 0 → ℓ ∞ / c 0 a linear ℘ ( N )/ Fin bounded operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Rodríguez Porras On automorphisms of the Banach space ℓ ∞ / c 0