Amalgamations of classes of Banach spaces with or without a monotone - PowerPoint PPT Presentation

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Amalgamations of classes of Banach spaces with or without a monotone basis Ondˇ rej Kurka Charles University in Prague Transfinite methods in Banach spaces and algebras of operators B¸ edlewo, July 18th, 2016

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces ABSTRACT: In the talk, we will introduce isometric counterparts of results concerning universality questions in separable Banach space theory and their natural connection with descriptive set theory. Our purpose is to construct small isometrically universal spaces for small families of Banach spaces. A particular result is for example the existence of a separable reflexive space which contains an isometric copy of every separable super-reflexive space. Our technique is a revision of a technique of S. A. Argyros and P. Dodos and uses also a parameterized version of Zippin’s embedding theorem due to B. Bossard.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Szlenk, 1968) If a Banach space E is isomorphically universal for all separable reflexive Banach space, then its dual E ∗ is not separable.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Szlenk, 1968) If a Banach space E is isomorphically universal for all separable reflexive Banach space, then its dual E ∗ is not separable. Sketch of the proof. For every separable Banach space X , Szlenk defined an ordinal index Sz( X ) ∈ [1 , ω 1 ] with properties Sz( X ) < ω 1 if and only if X ∗ separable, Sz( X ) ≤ Sz( Y ) whenever X embeds into Y , sup { Sz( X ) : X separable reflexive } = ω 1 .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Szlenk, 1968) If a Banach space E is isomorphically universal for all separable reflexive Banach space, then its dual E ∗ is not separable. Sketch of the proof. For every separable Banach space X , Szlenk defined an ordinal index Sz( X ) ∈ [1 , ω 1 ] with properties Sz( X ) < ω 1 if and only if X ∗ separable, Sz( X ) ≤ Sz( Y ) whenever X embeds into Y , sup { Sz( X ) : X separable reflexive } = ω 1 . Since Sz( X ) ≤ Sz( E ) for every separable reflexive X , we obtain Sz( E ) = ω 1 , and so E ∗ is not separable.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Odell, Schlumprecht, Zs´ ak, 2007) For a class C of separable reflexive Banach spaces, the following assertions are equivalent: There exists a separable reflexive Banach space, isomorphically universal for C . sup { Sz( X ) : X ∈ C } < ω 1 and sup { Sz( X ∗ ) : X ∈ C } < ω 1 .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Odell, Schlumprecht, Zs´ ak, 2007) For a class C of separable reflexive Banach spaces, the following assertions are equivalent: There exists a separable reflexive Banach space, isomorphically universal for C . sup { Sz( X ) : X ∈ C } < ω 1 and sup { Sz( X ∗ ) : X ∈ C } < ω 1 . Theorem (Dodos, Ferenczi, 2007) For a class C of separable Banach spaces, the following assertions are equivalent: There exists a Banach space with a separable dual, isomorphically universal for C . sup { Sz( X ) : X ∈ C } < ω 1 .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Bourgain, 1980) If a separable Banach space E is isomorphically universal for all separable reflexive Banach spaces, then E is actually isomorphically universal for all separable Banach spaces.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Bourgain, 1980) If a separable Banach space E is isomorphically universal for all separable reflexive Banach spaces, then E is actually isomorphically universal for all separable Banach spaces. Sketch of the proof. For every separable Banach space X , it is possible to define an ordinal index o ( X ) ∈ [1 , ω 1 ] with properties o ( X ) < ω 1 if and only if X is not universal for all separable Banach spaces, o ( X ) ≤ o ( Y ) whenever X embeds into Y , sup { o ( X ) : X separable reflexive } = ω 1 .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Bourgain, 1980) If a separable Banach space E is isomorphically universal for all separable reflexive Banach spaces, then E is actually isomorphically universal for all separable Banach spaces. Sketch of the proof. For every separable Banach space X , it is possible to define an ordinal index o ( X ) ∈ [1 , ω 1 ] with properties o ( X ) < ω 1 if and only if X is not universal for all separable Banach spaces, o ( X ) ≤ o ( Y ) whenever X embeds into Y , sup { o ( X ) : X separable reflexive } = ω 1 . Since o ( X ) ≤ o ( E ) for every separable reflexive X , we obtain o ( E ) = ω 1 , and so E is universal.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Dodos, 2009) For a class C of separable Banach spaces, the following assertions are equivalent: There exists a separable Banach space which is universal for C but still not for all separable Banach spaces.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Theorem (Dodos, 2009) For a class C of separable Banach spaces, the following assertions are equivalent: There exists a separable Banach space which is universal for C but still not for all separable Banach spaces. sup { o ( X ) : X ∈ C } < ω 1 .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Definition A subset A of a Polish space X is called analytic if A = f ( Y ) for a Polish space Y and a Borel mapping f : Y → X .

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Definition A subset A of a Polish space X is called analytic if A = f ( Y ) for a Polish space Y and a Borel mapping f : Y → X . Definition The standard Borel space of separable Banach spaces is defined as the set F ⊆ C (2 N ) : F is closed and linear � � SB = equipped with the Effros Borel structure, i.e., the σ -algebra generated by the sets { F ∈ SB : F ∩ U � = ∅} where U varies over open subsets of C (2 N ).

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Definition A subset A of a Polish space X is called analytic if A = f ( Y ) for a Polish space Y and a Borel mapping f : Y → X . Definition The standard Borel space of separable Banach spaces is defined as the set F ⊆ C (2 N ) : F is closed and linear � � SB = equipped with the Effros Borel structure, i.e., the σ -algebra generated by the sets { F ∈ SB : F ∩ U � = ∅} where U varies over open subsets of C (2 N ). This σ -algebra is the Borel σ -algebra of a Polish topology on SB. It is therefore possible to talk about analytic subsets of SB.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Fact Let E be a separable Banach space. Then the set of all X ∈ SB which are isomorphic to a subspace of E is analytic.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Fact Let E be a separable Banach space. Then the set of all X ∈ SB which are isomorphic to a subspace of E is analytic. Theorem (Bossard, 2002) Every analytic subset A of SB containing all separable reflexive spaces up to isomorphism must also contain an element which is isomorphically universal for all separable Banach spaces.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces Fact Let E be a separable Banach space. Then the set of all X ∈ SB which are isomorphic to a subspace of E is analytic. Theorem (Bossard, 2002) Every analytic subset A of SB containing all separable reflexive spaces up to isomorphism must also contain an element which is isomorphically universal for all separable Banach spaces. Remark Bossard ⇒ Bourgain.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces In what follows, a basis means a Schauder basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces In what follows, a basis means a Schauder basis. Definition A basis x 1 , x 2 , . . . is called monotone if the associated partial sum projections k ∞ � � P k : a n x n �→ a n x n n =1 n =1 satisfy � P k � ≤ 1.