A new Proof of Zippin’s Embedding Theorem and Applications

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A new Proof of Zippin’s Embedding Theorem and Applications

Th. Schlumprecht

First Brazilian Workshop in Geometry of Banach Spaces August 25-29, 2014

Maresias, Sao Paulo State, Brazil

Th. Schlumprecht A new Proof of Zippin’s Embedding Theorem and Applications

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In 1964 Aleksander Pe lczy´nski asked the following question:

Does a separable reflexive Banach space embed into a reflexive Banach space with a basis ?

Zippin, solved that problem positively in 1988 by proving: Theorem (Zippin, 1988)

Every Banach space with separable dual, embeds into a space Z with shrinking basis(e_j), i.e. the biorthogonal sequence(e_j^∗)is a basis for Z^∗.

and then applying

Theorem (Davis, Figiel, Johnson, and Pe lczy´nski, 1974)

A weakly compact operator from a Banach space X into a Banach space Z , which has a shrinking basis, factors through a reflexive space with a basis.

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Zippin, solved that problem positively in 1988 by proving: Theorem (Zippin, 1988)

and then applying

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Zippin, solved that problem positively in 1988 by proving:

Theorem (Zippin, 1988)

and then applying

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and then applying

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and then applying

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Universality and Embedding Problems

Universality Problems

Assume (P) is a property of (separable) Banach spaces.

Is there a separable Banach space X_u having property (P), (or some slightly weaker property (P’)) which is universal for all Banach spaces with property (P), i.e. every separable Banach space X with property (P) embeds (isomorphically) into X_u?

Often it is easier to solve a universality problem within the class of Banach spaces with (Schauder) bases, or with FDDs.

In that case a Universality Problem becomes anEmbedding Problem.

Embedding Problems

Does every Banach space X with property (P) embed into a Banach space Z with property (P) having a (certain) basis/FDD ?

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Universality and Embedding Problems

Is there a separable Banach space X_u having property (P), (or some slightly weaker property (P’)) which is universal for all Banach spaces with property (P), i.e. every separable Banach space X with property (P) embeds (isomorphically) into X_u? Often it is easier to solve a universality problem within the class of Banach spaces with (Schauder) bases, or with FDDs.

Embedding Problems

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Universality and Embedding Problems

Is there a separable Banach space X_u having property (P), (or some slightly weaker property (P’)) which is universal for all Banach spaces with property (P), i.e. every separable Banach space X with property (P) embeds (isomorphically) into X_u?

Often it is easier to solve a universality problem within the class of Banach spaces with (Schauder) bases, or with FDDs.

Embedding Problems

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Universality and Embedding Problems

Embedding Problems

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Universality and Embedding Problems

Embedding Problems

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Universality and Embedding Problems

Embedding Problems

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Examples Universality and Embedding Results

(Banach)C[0,1] is universal for all separable Banach spaces. (Szlenk, 1968) There is no separable reflexive space universal for all separable reflexive spaces.

Idea: define Sz(X)∈(0, ω₁] for separableX, so that: Sz(X)< ω1 ⇐⇒ X^∗ separable,

Sz(x) monotone with respect to isomorphic embeddings,

∀α < ω₁ ∃X separable & reflexive with Sz(X)> α.

(Odell & S, 2002) Characterization of subspaces of (⊕^∞_n=1Fn)`p, (Odell & S, 2006) Existence of a separable reflexive space universal for all separable super reflexive (uniform convex) spaces,

(Dodos & Ferenczi, 2007) Existence of spaces with separable dual, universal for spaces spaces with Szlenk index below a given countable ordinal α,

(Johnson & Zheng 2008, 2011) Characterization of subspaces of spaces having an unconditional basis.

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Examples Universality and Embedding Results

(Banach)C[0,1] is universal for all separable Banach spaces.

(Szlenk, 1968) There is no separable reflexive space universal for all separable reflexive spaces.

Idea: define Sz(X)∈(0, ω₁] for separableX, so that: Sz(X)< ω1 ⇐⇒ X^∗ separable,

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Examples Universality and Embedding Results

Idea: define Sz(X)∈(0, ω₁] for separable X, so that: Sz(X)< ω1 ⇐⇒ X^∗ separable,

∀α < ω₁ ∃X separable & reflexive with Sz(X)> α. (Odell & S, 2002) Characterization of subspaces of (⊕^∞_n=1Fn)`p, (Odell & S, 2006) Existence of a separable reflexive space universal for all separable super reflexive (uniform convex) spaces,

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Examples Universality and Embedding Results

Idea: define Sz(X)∈(0, ω₁] for separable X, so that:

Sz(X)< ω1 ⇐⇒ X^∗ separable,

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Examples Universality and Embedding Results

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Examples Universality and Embedding Results

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Examples Universality and Embedding Results

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Examples Universality and Embedding Results

(Odell & S, 2002) Characterization of subspaces of (⊕^∞_n=1Fn)`p,

(Odell & S, 2006) Existence of a separable reflexive space universal for all separable super reflexive (uniform convex) spaces,

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Examples Universality and Embedding Results

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Examples Universality and Embedding Results

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Examples Universality and Embedding Results

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Quantified Versions of Zippin’s Theorem

Problem (Pe lczy´nski, 2005)

Assume X has Szlenk index Sz(X) =ω^α,α < ω1.

•Does X embed into a space Z with a basis having the same Szlenk index?

•If X is reflexive, can Z be chosen to be reflexive, with Sz(Z^∗) =Sz(X^∗)?

•Does the class of spaces X , for which Sz(X)≤ω^α, admit a universal space Xα, for which Sz(Xα) =ω^α+1?

•Does the class of reflexive separable spaces X , for which Sz(X),Sz(X^∗)≤ω^α , admit a universal space X_α, which is reflexive and for which Sz(Xα),Sz(X_α^∗) =ω^α+1?

Answers:

Odell, Zsak & S (2007) and Freeman, Odell, Zsak & S (2009), yes to all questions ifα=βω,

Causey (2013 and 2014), yes (for allα < ω1) but for Szlenk index ofZ in embedding problem we have: Sz(Z) =ω^α+1, resp.

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1.

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Quantified Versions of Zippin’s Theorem

Answers:

Sz(Z) =ω^α+1. Th. Schlumprecht A new Proof of Zippin’s Embedding Theorem and Applications

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Remark

The proofs of these embedding results start by using Zippin’s Theorem and embed our given space X into a space Y with shrinking basis, respectively, in the reflexive case, with shrinking and boundedly complete basis.

Then, using the special assumptions on X , renorm the space Y , not necessarily in an equivalent way, but so that on the subspace of Y which is isomorphic to X , the norm stays equivalent to the original one.

Our goal:

A new proof of Zippin’s Embedding Theorem, in which for a given spaceX, with X^∗ separable, orX separable and reflexive, the spaceY, in whichX embeds, inherits as many properties from X as possible.

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Remark

Our goal:

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Remark

Our goal:

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Remark

All known proofs of Zippin’s Theorem (Zippin’s original proof and a proof by Ghoussoub, Maurey, and Schachermayer, as well as proof by Bossard) start by embedding X into Z =C(∆),

∆ =Cantor set (which has a basis), and then modifying Z until the modification has a shrinking basis but still contains X .

Disadvantage of that approach:

Not much else is really known about the space Z . Our approach will be different:

We start with aMarkushevich basis (e_i) of X (every separable space has such a basis) or more generally, aFinite Dimensional Markushevich Decomposition(FMD), andaugment it just enough to produce a space Z with a shrinking Finite Dimensional

Decomposition (FDD), which contains X .

Then we use a construction of Lindenstrauss and Tzafriri to embed Z in a space W with a shrinking basis.

As we will see, several properties of X will be automatically inherited by Z and W .

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Remark

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Remark

Not much else is really known about the space Z .

Our approach will be different:

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Remark

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Remark

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Remark

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Remark

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Main Result

Theorem

Assume that X is a Banach space with separable dual. Then X embeds into a space W with a shrinking basis(wi), so that

a) Sz(W) =Sz(X), where Sz(X) is the Szlenk index ofX , b) if X is reflexive then W is reflexive and Sz(X^∗) =Sz(W^∗),

c) if X^∗ has thew^∗-Unconditional Tree Propertythen (w_i)is unconditional, and

d) if X is reflexive and has the w -Unconditional Tree Property then (wi) is unconditional.

w^∗-Unconditional Tree Property(Johnson-Zheng): Everyw^∗-null tree inSX^∗, (inf. countably branching, inf. countable height) has a branch which is unconditional.

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Main Result

Theorem

Assume that X is a Banach space with separable dual. Then X embeds into a space W with a shrinking basis(w_i), so that

a) Sz(W) =Sz(X), where Sz(X) is the Szlenk index ofX , b) if X is reflexive then W is reflexive and Sz(X^∗) =Sz(W^∗), c) if X^∗ has thew^∗-Unconditional Tree Property then (w_i) is

unconditional, and

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Main Result

Theorem

a) Sz(W) =Sz(X), where Sz(X) is the Szlenk index ofX ,

b) if X is reflexive then W is reflexive and Sz(X^∗) =Sz(W^∗), c) if X^∗ has thew^∗-Unconditional Tree Property then (w_i) is

unconditional, and

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Main Result

Theorem

a) Sz(W) =Sz(X), where Sz(X) is the Szlenk index ofX , b) if X is reflexive then W is reflexive and Sz(X^∗) =Sz(W^∗),

c) if X^∗ has thew^∗-Unconditional Tree Property then (w_i) is unconditional, and

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Main Result

Theorem

unconditional, and

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Main Result

Theorem

unconditional, and

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Main Result

Theorem

unconditional, and

w^∗-Unconditional Tree Property(Johnson-Zheng):

Everyw^∗-null tree inSX^∗, (inf. countably branching, inf.

countable height) has a branch which is unconditional.

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FDD version of the Main result

We first prove the following FDD version of our main result, and then apply a construction of Lindenstrauss and Tzafriri, in order to get from FDD’s to bases.

Theorem

Assume that X is a Banach space with separable dual. Then X embeds into a space Z with a shrinking FDD(Z_i) so that

a) Sz(Z) =Sz(X),

b) if X is reflexive then Z is reflexive and Sz(X^∗) =Sz(Z^∗), and c) if X^∗ has an skipped unconditional FMD then(Z_i)is

unconditional.

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FDD version of the Main result

Theorem

a) Sz(Z) =Sz(X),

b) if X is reflexive then Z is reflexive and Sz(X^∗) =Sz(Z^∗), and c) if X^∗ has an skipped unconditional FMD then(Z_i) is

unconditional.

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FDD version of the Main result

Theorem

a) Sz(Z) =Sz(X),

b) if X is reflexive then Z is reflexive and Sz(X^∗) =Sz(Z^∗), and c) if X^∗ has an skipped unconditional FMD then(Z_i) is

unconditional.

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FDD version of the Main result

Theorem

a) Sz(Z) =Sz(X),

b) if X is reflexive then Z is reflexive and Sz(X^∗) =Sz(Z^∗), and

c) if X^∗ has an skipped unconditional FMD then(Z_i) is unconditional.

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FDD version of the Main result

Theorem

a) Sz(Z) =Sz(X),

b) if X is reflexive then Z is reflexive and Sz(X^∗) =Sz(Z^∗), and c) if X^∗ has an skipped unconditional FMD then(Zi) is

unconditional.

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Finite Dimensional Markushevich Decompositions

AssumeX is a separable Banach space. A sequence (E_n) of finite dimensional subspaces ofX is called Finite Dimensional

Markushevich Decomposition (FMD) ofX if

1 (E_k) is fundamental: X = span(Ej :j ∈N),

2 (E_k) is minimal: E_k∩span(E_j :j ∈N\ {k}) ={0},k ∈N. In that case we call (Fj), with

F_k = span(E_j :j ∈N\ {k})^⊥={x^∗ ∈X^∗:x^∗|_span(E

j:j∈N\{k})≡0} the biorthogonal sequenceof (E_k), and

3 (E_k) is total:

∀x ∈X (∀k ∈N,x^∗∈Fk x^∗(x) = 0)⇒x = 0 (i.e. span(F_k :k∈N) is w^∗-dense inX^∗).

If dim(Ek) = 1, for allk ∈N, say Ek = span(ek), then (ek) is called a Markushevich basis.

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Finite Dimensional Markushevich Decompositions

F_k = span(E_j :j ∈N\ {k})^⊥={x^∗ ∈X^∗:x^∗|_span(E

j:j∈N\{k})≡0} the biorthogonal sequenceof (E_k), and

3 (E_k) is total:

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Finite Dimensional Markushevich Decompositions

Fk = span(Ej :j ∈N\ {k})^⊥={x^∗ ∈X^∗ :x^∗|_span(E

j:j∈N\{k})≡0}

the biorthogonal sequenceof (Ek), and

3 (E_k) is total:

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Finite Dimensional Markushevich Decompositions

j:j∈N\{k})≡0}

3 (E_k) is total:

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Finite Dimensional Markushevich Decompositions

j:j∈N\{k})≡0}

3 (E_k) is total:

If dim(E_k) = 1, for allk ∈N, say E_k = span(e_k), then (e_k) is called a Markushevich basis.

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A finite dimensional Markushevich decomposition is called

1 c- norming,for 0<c ≤1, if sup

x^∗∈span(Fk:k∈N),kx^∗k≤1

|x^∗(x)| ≥ckxk.

2 shrinking if span(F_k :k∈N) is norm dense in X^∗ and, thus (F_k) is an FMD for X^∗ (The sequence (F_k) is always an FMD of its closed linear span with (EK) being its biorthogonals). Theorem

Markushevich, 1943: Every separable Banach space has a 1-norming Markushevich basis (ej), which can be chosen to be shrinking if X^∗ is separable.

Ovsepian & Pe lczy´nski, 1975: (e_j) can be chosen to be bounded, i.e. sup_j₌₁ke_jk · ke_j^∗k<c, c universal.

Pe lczy´nski, 1976: For ε >0,(e_j) can be chosen so that, sup_j=1ke_jk · ke_j^∗k<1 +ε.

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|x^∗(x)| ≥ckxk.

2 shrinking if span(F_k :k∈N) is norm dense in X^∗ and, thus (F_k) is an FMD for X^∗ (The sequence (F_k) is always an FMD of its closed linear span with (EK) being its biorthogonals). Theorem

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|x^∗(x)| ≥ckxk.

2 shrinking if span(F_k :k∈N) is norm dense inX^∗ and, thus (F_k) is an FMD for X^∗ (The sequence (F_k) is always an FMD of its closed linear span with (EK) being its biorthogonals).

Theorem

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|x^∗(x)| ≥ckxk.

Theorem

Pe lczy´nski, 1976: Forε >0,(e_j) can be chosen so that, sup_j=1ke_jk · ke_j^∗k<1 +ε.

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|x^∗(x)| ≥ckxk.

Theorem

Pe lczy´nski, 1976: Forε >0,(e_j) can be chosen so that, sup_j=1ke_jk · ke_j^∗k<1 +ε.