On some universality questions in Banach space theory · Hierarchies Eﬀros Borel space Universality questions Amalgamations Conclusion On some universality questions in Banach space

Hierarchies Effros Borel space Universality questions Amalgamations Conclusion

On some universality questions in Banach

space theory

Ondrej Kurka

Off-site meeting of the Institute of Mathematics CAS,Ostrava, 20th-22nd September 2017


One of problems which a mathematician may deal with is thefollowing one:

Question

How much complicated a set/property/relation is?



Question


An approach to this general problem is proposed by descriptive settheory and its concepts of the Borel and the projective hierarchiesof sets.



Question



A particular problem is the following one:

Question

Is a set/property/relation as complicated as it seems to be?



Question



A particular problem is the following one:

Question

Is a set/property/relation as complicated as it seems to be?

If the studied object is less complicated than it seems to be, then ithas most likely a hidden simpler equivalent definition.


Definition

We say that a topological space X is a Polish space if it is separableand admits a compatible metric d such that (X , d) is complete.


Definition

We say that a topological space X is a Polish space if it is separableand admits a compatible metric d such that (X , d) is complete.

Example

The following spaces are Polish:

R, RN, NN, {0, 1}N, . . . ,

all compact metrizable spaces,

any Gδ subset of a Polish space,

all separable Banach spaces,

the hyperspace of all compact subsets of a Polish space(defined below).


Definition

Let X be a topological space. The hyperspace of compact subsetsof X is defined as

K(X ) = {K ⊆ X : K is compact}

equipped with the Vietoris topology, i.e., the topology generatedby the sets of the form

{K ∈ K(X ) : K ⊆ U},

{K ∈ K(X ) : K ∩ U 6= ∅},

where U varies over open subsets of X .


Definition

Let X be a topological space. The hyperspace of compact subsetsof X is defined as

K(X ) = {K ⊆ X : K is compact}

equipped with the Vietoris topology, i.e., the topology generatedby the sets of the form

{K ∈ K(X ) : K ⊆ U},

{K ∈ K(X ) : K ∩ U 6= ∅},

where U varies over open subsets of X .

Fact

If X is Polish, then so is K(X ).


G (or Σ01) ... open sets

F (or Π01) ... closed sets

Gδ (or Π02) ... countable intersections of open sets

Fσ (or Σ02) ... countable unions of closed sets

Gδσ (or Σ03) ... countable unions of Gδ sets

Fσδ (or Π03) ... countable intersections of Fσ sets

in this way, we can define classes Σ0ξ and Π0

ξ for every 1 ≤ ξ < ω1










G

F

Gδ ∩ Fσ

Gδ

Fσ

Gδσ ∩ Fσδ

Gδσ

Fσδ

. . .

⊆

⊆

⊆

⊆

⊆

⊆

⊆

⊆










G

F

Gδ ∩ Fσ

Gδ

Fσ

Gδσ ∩ Fσδ

Gδσ

Fσδ

. . .

⊆

⊆

⊆

⊆

⊆

⊆

⊆

⊆

All inclusions are proper in any uncountable Polish space.


Example

The set Q is an Fσ but not Gδ subset of R.


Example


Example

The set {x ∈ R : f ′(x) exists} is an Fσδ set for every continuousfunction f : R → R but there exists a Lipschitz f such that this setis not Gδσ.


Example


Example

The set {x ∈ R : f ′(x) exists} is an Fσδ set for every continuousfunction f : R → R but there exists a Lipschitz f such that this setis not Gδσ.

Example

The set{

x ∈ {0, 1}N×N : (∃∞n∃∞m)(x(m, n) = 0)}

is a Gδσδ but not Fσδσ subset of {0, 1}N×N.


Theorem (Suslin, 1917)

There exists a Gδ subset of R2 whose projection on the firstcoordinate is not Borel.




Thus, the class of Borel sets does not include all “reasonable” sets.




Thus, the class of Borel sets does not include all “reasonable” sets.

Definition

A subset A of a Polish space X is called an analytic set if thereexist a Polish space Y and a Borel subset B of X × Y such that Ais the projection of B on the first coordinate.

X

Y

B

A


There are many equivalent definitions of an analytic set:

Theorem

For a subset A of a Polish space X , the following assertions areequivalent:(i) A is analytic,(ii) A is the projection of a Gδ set B ⊆ X × R,(iii) A is the projection of a closed set B ⊆ X × NN,(iv) A is a continuous image of a Polish space,(v) A is empty or a continuous image of NN,(vi) A is the result of the Suslin operation applied on a system ofclosed sets, that is

A =⋃

(n1,n2,... )∈NN

∞⋂

k=1

Fn1,n2,...,nk,

where Fn1,n2,...,nk⊆ X are closed.

...


analytic sets (or Σ11 sets) ... projections of Borel sets

coanalytic sets (or Π11 sets) ... complements of analytic sets

Σ12 sets ... projections of coanalytic sets

Π12 sets ... complements of Σ1

2 sets∆1

2 = Σ12 ∩ Π1

2

etc.


analytic sets (or Σ11 sets) ... projections of Borel sets

coanalytic sets (or Π11 sets) ... complements of analytic sets

Σ12 sets ... projections of coanalytic sets

Π12 sets ... complements of Σ1

2 sets∆1

2 = Σ12 ∩ Π1

2

etc.

Borel

analytic

coanalytic

∆12

Π12

Σ12

∆13

. . .⊆

⊆

⊆

⊆

⊆

⊆

⊆

⊆

All inclusions are proper in any uncountable Polish space.


Let us note that

{

f ∈ C ([0, 1]) : f (x) = 0 for some x ∈ [0, 1]}

is a closed subset of C ([0, 1]).


Let us note that

{

f ∈ C ([0, 1]) : f (x) = 0 for some x ∈ [0, 1]}


Example (Kaufman)

The set

{

f ∈ C ([0, 1]) : f (x) = 0 for some x ∈ [0, 1] \ Q}

is analytic but not Borel.


Let us note that

{

f ∈ C ([0, 1]) : f (x) = 0 for some x ∈ [0, 1]}


Example (Kaufman)

The set

{

f ∈ C ([0, 1]) : f (x) = 0 for some x ∈ [0, 1] \ Q}


Example (Mazurkiewicz)

The set{

f ∈ C ([0, 1]) : f is differentiable}

is coanalytic but not Borel.


Example (Hurewicz)

If X is an uncountable Polish space, then the set

{

K ∈ K(X ) : K is uncountable}



Example (Hurewicz)

If X is an uncountable Polish space, then the set

{

K ∈ K(X ) : K is uncountable}


Example (Ajtai, Becker)

For n ≥ 3, the set

{

K ∈ K(Rn) : K is path connected}

is Π12 but not Σ1

2.


Definition

Let X and Y be Banach spaces.We say that X and Y are isometric if there exists a surjectivelinear operator I : X → Y such that

‖Ix‖Y = ‖x‖X , x ∈ X .


Definition


‖Ix‖Y = ‖x‖X , x ∈ X .

We say that X and Y are linearly isomorphic (or just isomorphic) ifthere exist c > 0,C > 0 and a surjective linear operatorE : X → Y such that

c‖x‖X ≤ ‖Ex‖Y ≤ C‖x‖X , x ∈ X .


Definition


‖Ix‖Y = ‖x‖X , x ∈ X .


c‖x‖X ≤ ‖Ex‖Y ≤ C‖x‖X , x ∈ X .

Theorem (Banach, Mazur)

Every separable Banach space is isometric to a subspace ofC ([0, 1]).


Definition


‖Ix‖Y = ‖x‖X , x ∈ X .


c‖x‖X ≤ ‖Ex‖Y ≤ C‖x‖X , x ∈ X .

Theorem (Banach, Mazur)

Every separable Banach space is isometric to a subspace ofC ([0, 1]).

It turns out that the system of all subspaces of C ([0, 1]) can beequipped with a Borel σ-algebra.


Definition

The space of separable Banach spaces is defined as the set

SB ={

X ⊆ C ([0, 1]) : X is closed and linear}

equipped with the Effros Borel structure, defined as the σ-algebragenerated by the sets

{X ∈ SB : X ∩ U 6= ∅}

where U varies over open subsets of C ([0, 1]).


Definition


SB ={



{X ∈ SB : X ∩ U 6= ∅}


Fact (Bossard)

There is a Polish topology on SB whose Borel σ-algebra is theEffros Borel structure.


Definition


SB ={



{X ∈ SB : X ∩ U 6= ∅}


Fact (Bossard)

There is a Polish topology on SB whose Borel σ-algebra is theEffros Borel structure.

Therefore, it makes sense to say that a class of separable Banachspaces is Borel, analytic, coanalytic, Σ1

2, Π12, ...


Example (Rosendal)

For any separable Banach space Z , the class of all spaces isometricto Z is Borel.


Example (Rosendal)


Example (Godefroy, Bossard)

For 1 < p < ∞, the class of all spaces isomorphic to `p is Borel.


Example (Rosendal)


Example (Godefroy, Bossard)

For 1 < p < ∞, the class of all spaces isomorphic to `p is Borel.

Example (Bossard)

For 1 < p < ∞, p 6= 2, the class of all spaces isomorphic to Lp isanalytic but not Borel.


Example (Bossard)

The class of all separable reflexive spaces is coanalytic but notBorel.


Example (Bossard)


Example (K.)

The class of all separable spaces with the Schur property is Π12 but

not Σ12.

We say that a Banach space X has the Schur property if everyweakly convergent sequence in X is convergent in the norm.


Example (Bossard)


Example (K.)

The class of all separable spaces with the Schur property is Π12 but

not Σ12.

We say that a Banach space X has the Schur property if everyweakly convergent sequence in X is convergent in the norm.

Example

It is not known if the class of spaces with a Schauder basis is Borel.


Definition

Let C be a class of Banach spaces.

We say that a Banach space X is isometrically (isomorphically)universal for C if every space Y in C is isometric (isomorphic) to asubspace of X .


Definition

Let C be a class of Banach spaces.

We say that a Banach space X is isometrically (isomorphically)universal for C if every space Y in C is isometric (isomorphic) to asubspace of X .

The following general question can be considered.

Question

For a class of Banach spaces which share some property (P), doesthere exist an isomorphically/isometrically universal space thatsatisfies (P) as well?


Question (Banach, Mazur)

Does there exist an isomorphically universal space in the class ofseparable reflexive Banach spaces?

We say that a Banach space X is reflexive if the canonicalembedding J : X → X ∗∗ is surjective.Equivalently, X is reflexive if every bounded sequence has a weaklyconvergent subsequence.


The question has a negative answer.

Theorem (Szlenk, 1968)

If a Banach space E is isomorphically universal for all separablereflexive Banach spaces, then its dual E ∗ is not separable.





Sketch of the proof.

Let us assume that a separable Banach space E is isomorphicallyuniversal for all separable reflexive Banach spaces.






Let us assume that a separable Banach space E is isomorphicallyuniversal for all separable reflexive Banach spaces.For every separable Banach space X , it is possible to define anordinal index Sz(X ) ∈ [1, ω1] with properties

Sz(X ) < ω1 if and only if X ∗ separable,

Sz(X ) ≤ Sz(Y ) whenever X is isomorphic to a subspace of Y ,

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






Let us assume that a separable Banach space E is isomorphicallyuniversal for all separable reflexive Banach spaces.For every separable Banach space X , it is possible to define anordinal index Sz(X ) ∈ [1, ω1] with properties

Sz(X ) < ω1 if and only if X ∗ separable,

Sz(X ) ≤ Sz(Y ) whenever X is isomorphic to a subspace of Y ,

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

Since Sz(X ) ≤ Sz(E ) for every separable reflexive X , we obtainSz(E ) = ω1, and so E ∗ is not separable.


Question

Which classes C of separable reflexive spaces have the propertythat there is an isomorphically/isometrically universal separablereflexive space E for C?


Question


Fact (Bossard)

For a separable Banach space E, the set

A ={

X ∈ SB : X is isomorphic to a subspace of E}

is analytic.


Question


Fact (Bossard)

For a separable Banach space E, the set

A ={

X ∈ SB : X is isomorphic to a subspace of E}

is analytic.

Therefore, we obtain:

Necessary condition

There is an analytic set A ⊆ SB of reflexive spaces that containsall members of C up to isomorphism.


Some other results in this direction:

Theorem (Bourgain, 1980)

If a separable Banach space E is isomorphically universal for allseparable reflexive Banach spaces, then E is actually isomorphicallyuniversal for all separable Banach spaces.





Theorem (Bossard, 2002)

Every analytic subset A of SB containing all separable reflexivespaces up to isomorphism must also contain an element which isisomorphically universal for all separable Banach spaces.





Theorem (Bossard, 2002)

Every analytic subset A of SB containing all separable reflexivespaces up to isomorphism must also contain an element which isisomorphically universal for all separable Banach spaces.

Remark

Bossard ⇒ Bourgain, due to the fact from the previous slide.


It is possible to apply Bossard’s approach on questions concerningisometry as well.

Theorem (G. Godefroy, N. J. Kalton, 2007)

If a separable Banach space X is isometrically universal for allseparable strictly convex spaces, then X is actually isometricallyuniversal for all separable Banach spaces.


It is possible to apply Bossard’s approach on questions concerningisometry as well.

Theorem (G. Godefroy, N. J. Kalton, 2007)

If a separable Banach space X is isometrically universal for allseparable strictly convex spaces, then X is actually isometricallyuniversal for all separable Banach spaces.

Theorem (K., 2012)

If a separable Banach space X is isometrically universal for allseparable reflexive Frechet smooth Banach spaces, then X isactually isometrically universal for all separable Banach spaces.


Theorem

Let P be one of the following classes of Banach spaces:

the class of separable reflexive spaces,

the class of spaces with a separable dual,

the class of separable spaces which are not isomorphicallyuniversal for all separable Banach spaces.

Let A ⊆ SB be an analytic set of spaces from P. Then there existsa separable Banach space E which belongs to P and which isisomorphically universal for of A.


Theorem




the class of separable spaces which are not isomorphicallyuniversal for all separable Banach spaces.

Let A ⊆ SB be an analytic set of spaces from P. Then there existsa separable Banach space E which belongs to P and which isisomorphically universal for of A.

The theorem is based on three results:

a version of this theorem for spaces with a Schauder basis(Argyros, Dodos, 2007),

a parameterized version of Zippin’s embedding theorem(Dodos, Ferenczi, 2007),

a parameterized version of the Bourgain-Pisier construction(Dodos, 2009).


Question (Godefroy)

Is there an isometric version of this theorem?


Question (Godefroy)


Theorem (K., 2015)




the class of separable spaces which are not isometricallyuniversal for all separable Banach spaces.

Let A ⊆ SB be an analytic set of spaces from P. Then there existsa separable Banach space E which belongs to P and which isisometrically universal for A.


Question (Godefroy)


Theorem (K., 2015)




the class of separable spaces which are not isometricallyuniversal for all separable Banach spaces.

Let A ⊆ SB be an analytic set of spaces from P. Then there existsa separable Banach space E which belongs to P and which isisometrically universal for A.

The theorem is based on three analogous ingredients.


Question (recalled)

Which classes C of separable reflexive spaces have the propertythat there is a universal separable reflexive space E for C?


Question (recalled)

Which classes C of separable reflexive spaces have the propertythat there is a universal separable reflexive space E for C?

Theorem

For a class C of separable reflexive Banach spaces, the followingassertions are equivalent:

1 There exists a separable reflexive Banach space,isomorphically universal for C.

2 There exists a separable reflexive Banach space,isometrically universal for C.

3 There is an analytic set A ⊆ SB of reflexive spaces thatcontains all members of C up to isomorphism.

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

(1) ⇔ (3) due to Dodos and Ferenczi (2007)(1) ⇔ (4) due to Odell, Schlumprecht and Zsak (2007)(2) ⇔ (3) is new


Theorem

For a class C of separable Banach spaces, the following assertionsare equivalent:

1 There exists a Banach space with a separable dual,isomorphically universal for C.

2 There exists a Banach space with a separable dual,isometrically universal for C.

3 There is an analytic set A ⊆ SB of spaces with a separabledual that contains all members of C up to isomorphism.

4 sup{Sz(X ) : X ∈ C} < ω1.

(1) ⇔ (3) due to Dodos and Ferenczi (2007)(1) ⇔ (4) due to Freeman, Odell, Schlumprecht and Zsak (2009)(2) ⇔ (3) is new


S. A. Argyros and P. Dodos, Genericity and amalgamation of classes of Banach spaces, Adv. Math. 209, no.

2 (2007), 666–748.

B. Bossard, A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fund.

Math. 172, no. 2 (2002), 117–152.

J. Bourgain, On separable Banach spaces, universal for all separable reflexive spaces, Proc. Amer. Math.

Soc. 79, no. 2 (1980), 241–246.

P. Dodos, On classes of Banach spaces admitting “small” universal spaces, Trans. Amer. Math. Soc. 361,

no. 12 (2009), 6407–6428.

P. Dodos, Banach spaces and descriptive set theory: selected topics, Lecture notes in mathematics 1993,

Springer, 2010.

P. Dodos and V. Ferenczi, Some strongly bounded classes of Banach spaces, Fund. Math. 193, no. 2

(2007), 171–179.

N. Ghoussoub, B. Maurey and W. Schachermayer, Slicings, selections and their applications, Canad. J.

Math. 44, no. 3 (1992), 483–504.

G. Godefroy, Analytic sets of Banach spaces, Rev. R. Acad. Cien. Serie A. Mat. 104, no. 2 (2010), 365–374.

O. Kurka, Amalgamations of classes of Banach spaces with a monotone basis, Studia Math. 234, no. 2

(2016), 121–148.

O. Kurka, Zippin’s embedding theorem and amalgamations of classes of Banach spaces, Proc. Amer. Math.

Soc. 144, no. 10 (2016), 4273–4277.

O. Kurka, Non-universal families of separable Banach spaces, Studia Math. 233, no. 2 (2016), 153–168.

E. Odell, Th. Schlumprecht, A. Zsak, Banach spaces of bounded Szlenk index, Studia Math. 183 (2007),

63–97.

Documents

On some universality questions in Banach space theory · Hierarchies Eﬀros Borel space Universality questions Amalgamations Conclusion On some universality questions in Banach space