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

• f 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, ω1] with properties Sz(X) < ω1 if and only if X ∗ separable, Sz(X) ≤ Sz(Y ) whenever X embeds into Y , sup{Sz(X) : X separable reflexive} = ω1.

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, ω1] with properties Sz(X) < ω1 if and only if X ∗ separable, Sz(X) ≤ Sz(Y ) whenever X embeds into Y , sup{Sz(X) : X separable reflexive} = ω1. Since Sz(X) ≤ Sz(E) for every separable reflexive X, we obtain Sz(E) = ω1, 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} < ω1 and sup{Sz(X ∗) : X ∈ C} < ω1.

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} < ω1 and sup{Sz(X ∗) : X ∈ C} < ω1. 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} < ω1.

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

• rdinal index o(X) ∈ [1, ω1] with properties
• (X) < ω1 if and only if X is not universal for all separable

Banach spaces,

• (X) ≤ o(Y ) whenever X embeds into Y ,

sup{o(X) : X separable reflexive} = ω1.

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

• rdinal index o(X) ∈ [1, ω1] with properties
• (X) < ω1 if and only if X is not universal for all separable

Banach spaces,

• (X) ≤ o(Y ) whenever X embeds into Y ,

sup{o(X) : X separable reflexive} = ω1. Since o(X) ≤ o(E) for every separable reflexive X, we obtain

• (E) = ω1, 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} < ω1.

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 SB =

• F ⊆ C(2N) : F is closed and linear
• 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(2N).

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 SB =

• F ⊆ C(2N) : F is closed and linear
• 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(2N). 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 x1, x2, . . . is called monotone if the associated partial sum projections Pk :

∞

• n=1

anxn →

k

• n=1

anxn satisfy Pk ≤ 1.

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 x1, x2, . . . is called monotone if the associated partial sum projections Pk :

∞

• n=1

anxn →

k

• n=1

anxn satisfy Pk ≤ 1. Definition A basis x1, x2, . . . of a Banach space X is said to be shrinking if X ∗ = span{x∗

1, x∗ 2, . . . }

where x∗

1, x∗ 2, . . . is the dual basic sequence x∗ k : ∞ n=1 anxn → ak.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Argyros, Dodos, 2007) Let P be one of the following classes of Banach spaces: the class of reflexive spaces with a basis, the class of spaces with a shrinking basis, the class of spaces with a basis which are not isomorphically universal for all separable Banach spaces. Let A be an analytic set of spaces from P. Then there exists a Banach space E which belongs to P and which contains a complemented isomorphic copy of every member of A.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Argyros, Dodos, 2007) Let P be one of the following classes of Banach spaces: the class of reflexive spaces with a basis, the class of spaces with a shrinking basis, the class of spaces with a basis which are not isomorphically universal for all separable Banach spaces. Let A be an analytic set of spaces from P. Then there exists a Banach space E which belongs to P and which contains a complemented isomorphic copy of every member of A. Question (Godefroy) Is there an isometric version of the Argyros-Dodos amalgamation theory?

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (K., 2015) Let P be one of the following classes of Banach spaces: the class of reflexive spaces with a monotone basis, the class of spaces with a monotone shrinking basis, the class of spaces with a monotone basis which are not isometrically universal for all separable Banach spaces, the class of strictly convex spaces with a monotone basis. Let A be an analytic set of spaces from P. Then there exists a Banach space E which belongs to P and which contains a 1-complemented isometric copy of every member of A.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Question Can the reliance on a basis be dropped?

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis. ... an alternative proof by Ghoussoub, Maurey and Schachermayer.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis. ... an alternative proof by Ghoussoub, Maurey and Schachermayer. ... its parameterized version by Bossard.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis. ... an alternative proof by Ghoussoub, Maurey and Schachermayer. ... its parameterized version by Bossard. Theorem (Bourgain, Pisier, 1983) Every separable Banach space X embeds isomorphically into a Banach space Y with a basis such that Y /X has the Schur property.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis. ... an alternative proof by Ghoussoub, Maurey and Schachermayer. ... its parameterized version by Bossard. Theorem (Bourgain, Pisier, 1983) Every separable Banach space X embeds isomorphically into a Banach space Y with a basis such that Y /X has the Schur property. ... if X is not isomorphically universal for separable Banach spaces, then Y is not universal as well.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Zippin, 1988) (1) Every separable reflexive Banach space X embeds isomorphically into a reflexive Banach space Y which has a basis. (2) Every Banach space X with a separable dual embeds isomorphically into a Banach space Y with a shrinking basis. ... an alternative proof by Ghoussoub, Maurey and Schachermayer. ... its parameterized version by Bossard. Theorem (Bourgain, Pisier, 1983) Every separable Banach space X embeds isomorphically into a Banach space Y with a basis such that Y /X has the Schur property. ... if X is not isomorphically universal for separable Banach spaces, then Y is not universal as well. ... a parameterized version by Dodos.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (Dodos, Ferenczi, 2007, Dodos, 2009) Let P be one of the following classes of Banach spaces: the class of reflexive spaces ✭✭✭✭✭

✭ ❤❤❤❤❤ ❤

with a basis, the class of spaces with a ✭✭✭✭✭✭✭

❤❤❤❤❤❤❤

shrinking basis separable dual, the class of spaces ✭✭✭✭✭

✭ ❤❤❤❤❤ ❤

with a basis which are not isomorphically universal for all separable Banach spaces. Let A be an analytic set of spaces from P. Then there exists a separable Banach space E which belongs to P and which contains an ✭✭✭✭✭✭

✭ ❤❤❤❤❤❤ ❤

complemented isomorphic copy of every member of A.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Corollary (Odell, Schlumprecht, 2006) There exists a separable reflexive Banach space, isomorphically universal for all separable super-reflexive Banach spaces.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis. (2) Every Banach space X with a separable dual embeds isometrically into a Banach space Y with a monotone shrinking basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis. (2) Every Banach space X with a separable dual embeds isometrically into a Banach space Y with a monotone shrinking basis. ... “observation” because an already developed method works.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis. (2) Every Banach space X with a separable dual embeds isometrically into a Banach space Y with a monotone shrinking basis. ... “observation” because an already developed method works. Theorem (K.) (3) Every separable Banach space X which is not isometrically universal for all separable Banach spaces embeds isometrically into a Banach space Y which is also non-universal and has a monotone basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis. (2) Every Banach space X with a separable dual embeds isometrically into a Banach space Y with a monotone shrinking basis. ... “observation” because an already developed method works. Theorem (K.) (3) Every separable Banach space X which is not isometrically universal for all separable Banach spaces embeds isometrically into a Banach space Y which is also non-universal and has a monotone basis. (4) Every separable strictly convex Banach space X embeds isometrically into a strictly convex Banach space Y which has a monotone basis.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem/Observation (K.) (1) Every separable reflexive Banach space X embeds isometrically into a reflexive Banach space Y which has a monotone basis. (2) Every Banach space X with a separable dual embeds isometrically into a Banach space Y with a monotone shrinking basis. ... “observation” because an already developed method works. Theorem (K.) (3) Every separable Banach space X which is not isometrically universal for all separable Banach spaces embeds isometrically into a Banach space Y which is also non-universal and has a monotone basis. (4) Every separable strictly convex Banach space X embeds isometrically into a strictly convex Banach space Y which has a monotone basis. ... both results hold in a parameterized way.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Theorem (K., 2015) Let P be one of the following classes of Banach spaces: the class of reflexive spaces ✭✭✭✭✭✭✭✭✭✭

✭ ❤❤❤❤❤❤❤❤❤❤ ❤

with a monotone basis, the class of spaces with a ✭✭✭✭✭✭✭✭✭✭✭

✭ ❤❤❤❤❤❤❤❤❤❤❤ ❤

monotone shrinking basis separable dual, the class of spaces ✭✭✭✭✭✭✭✭✭✭

✭ ❤❤❤❤❤❤❤❤❤❤ ❤

with a monotone basis which are not isometrically universal for all separable Banach spaces, the class of strictly convex spaces ✭✭✭✭✭✭✭✭✭✭

✭ ❤❤❤❤❤❤❤❤❤❤ ❤

with a monotone basis. Let A be an analytic set of spaces from P. Then there exists a separable Banach space E which belongs to P and which contains an ✭✭✭✭✭✭✭

✭ ❤❤❤❤❤❤❤ ❤

1-complemented isometric copy of every member of A.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Corollary 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. There exists a separable reflexive Banach space, isometrically universal for C. sup{Sz(X) : X ∈ C} < ω1 and sup{Sz(X ∗) : X ∈ C} < ω1.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Corollary 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. There exists a separable reflexive Banach space, isometrically universal for C. sup{Sz(X) : X ∈ C} < ω1 and sup{Sz(X ∗) : X ∈ C} < ω1. Corollary 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. There exists a Banach space with a separable dual, isometrically universal for C. sup{Sz(X) : X ∈ C} < ω1.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

Corollary There exists a separable reflexive Banach space, isometrically universal for all separable super-reflexive Banach spaces.

Introduction Bossard’s approach Amalgamations of spaces with a basis Amalgamations of general separable spaces

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