The stabilized set of p’s in Krivine’s theorem can be disconnected

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Pavlos Motakis

(joint work with Kevin Beanland and Daniel Freeman)

Department of Mathematics National Technical University of Athens

Brazilian Workshop in Geometry of Banach spaces 25 August 2014, Maresias

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Question: LetX be a Banach space with a basis, with a stabilized Krivine setF.

IsF necessarilyconnected?

During this lecture we shall demonstrate that this neednot always be the case.

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LetX be a Banach space with a Schauder basis(x_i)_i. Let also(e_j)j be a Schauder basic sequence, not necessarily inX.

We say that(e_j)j is finitely block represented in(x_i)i (or simply inX) if:

for every natural numbernandε >0 there exists a finite block sequence(y_j)ⁿ_j=1of(x_i)i

that is(1+ε)-equivalent to(e_j)ⁿ_j=1.

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LetX be a Banach space and(x_i)_i be a sequence inX. Let also(e_j)j be a sequence in some Banach space, which is not necessarilyX.

We say that(x_i)_i generates(e_j)_j as a spreading modelif:

for every natural numbernandε >0 there existsn₀∈N such that for every natural numbersn₀6i₁<· · ·<i_n we have that(x_i_j)ⁿ_j=1is(1+ε)-equivalent to(e_j)ⁿ_j=1.

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and(y_j)_j is a block sequence of(x_i)_ithat generates some sequence(e_j)_j as a spreading model,

then(e_j)_j is finitely block represented in(x_i)_i.

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Theorem (J. L. Krivine)

Let X be a Banach space with a Schauder basis. Then there exists a p∈[1,∞]such that the unit vector basis of`_pis finitely block represented in X (the case p=∞refers to the unit vector basis of c₀).

The set of allp’s that are finitely block represented inX is calledthe Krivine set of X and is denoted byK(X).

Remark: It follows that if for somep,X admits a spreading model equivalent to the unit vector basis of`p, thenpis in the Krivine set ofX.

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In his paper on Krivine’s theorem, H. P. Rosenthal observed the following:

On some block subspaceY ofX, the Krivine set is stabilized, i.e.

ifZ is a further block subspace ofY then the setsK(Y) andK(Z)coincide.

Rosenthal concluded his paper with the following question:

Is such a stabilized Krivine set necessarily always a singleton?

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E. Odell and Th. Schlumprecht constructed a spaceX with the property that every 1-unconditional basic sequence is finitely block represented in every block subspaceY ofX. This space has[1,∞]as its stabilized Krivine set and hence the answer the above question is negative.

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E. Odell and Th. Schlumprecht constructed a spaceX with the property that every 1-unconditional basic sequence is finitely block represented in every block subspaceY ofX. This space has[1,∞]as its stabilized Krivine set and hence the answer the above question is negative.

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Next Question: LetX be a Banach space with a basis, with a stabilized Krivine setF.

This question first appeared in a paper by P. Habala and N.

Tomczak-Jaegermann and was also later mentioned by Odell as one of 15 open problems in Banach spaces.

In their paper Habala and Tomczak-Jaegermann prove the following:

ifp<q are in the stabilized Krivine set ofX, thenX admits a block quotientZ such that everyr ∈[p,q]is finitely block represented inZ.

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Theorem

Let F ⊂[1,∞]be either afiniteset or a set consisting ofan increasing sequence and its limit. Then there exists a reflexive Banach space X with an unconditional basis such that for every infinite dimensional block subspace Y of X :

(i) For all16p6∞, the space`pis finitely block represented in Y if and only if p∈F .

(ii) If F is finite then the spreading models admitted by Y are exactly the spaces`_pfor p∈F .

(iii) If F is an increasing sequence with limit p_ωthen every spreading model admitted by Y is isomorphic to`_pfor some p∈F and for every p∈F\ {p_ω}`pis admitted as a spreading model by Y .

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Theorem

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Theorem

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Theorem

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In particular, the stabilized Krivine set ofX isF (which is either finite or consists of an increasing sequence and its limit) and hencenot connected.

This space also answers some questions concerning spreading models, which were asked by G. Androulakis, Odell, Schlumprecht and Tomczak-Jaegermann.

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In particular, the stabilized Krivine set ofX isF (which is either finite or consists of an increasing sequence and its limit) and hencenot connected.

This space also answers some questions concerning spreading models, which were asked by G. Androulakis, Odell, Schlumprecht and Tomczak-Jaegermann.

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Banach spaceX such that every subspace admitsn-many spreading models?

Answer: Yes, and they can be chosen to be`_p’s forp∈F for anyn-setF ⊂[1,∞].

Question: Does there exists a Banach spaceX such that every subspace admits countably infinite many spreading models?

Answer: Yes, forF an increasing sequence the space constructed has this property.

Question: LetX be a Banach spaces such that every subspace admits both`1and`2spreading models. DoesX admit uncountably many spreading models?

Answer: No, forF ={1,2}the space constructed admits only`₁and`₂spreading models in every subspace.

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It is worth pointing out that the previously stated theorem is false if stated forF a decreasing sequence and its limit.

Indeed, as B. Sari has proved, if a Banach space admits a strictly increasing, with respect to domination, sequence of spreading models, then it admits uncountably many

spreading models.

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It is worth pointing out that the previously stated theorem is false if stated forF a decreasing sequence and its limit.

Indeed, as B. Sari has proved, if a Banach space admits a strictly increasing, with respect to domination, sequence of spreading models, then it admits uncountably many

spreading models.

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The definition of the norm uses the method ofsaturation under constraints, a method initialized by Odell and Schlumprecht to construct the earlier mentioned space with[1,∞]as its stabilized Krivine set.

The construction method used in the preset case is more related to the one used by S. Argyros, K. Beanland and P.

M. to construct Tsirelson like reflexive spaces. Among the properties of these spaces is that they admit only`₁andc₀ as a spreading model in every subspace.

Actually, the spaceX we construct forF ={1,∞}is a slight modification of the simplest case presented in that paper.

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From now on let us assume thatF consists of a strictly increasing sequence(p_k)^∞_k=1and its limitpω. (The case in whichF is finite is the same)

We fix a constant 0< θ61/4.

The normk · k_∗ of the spaceX satisfies an implicit formula which is based on countably infinite many layers.

Each layer also comes in various sizes.

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The base layer: form∈Nandx ∈c₀₀(N)define kxk_0,m=θsup 1

m^1/p⁰^ω

m

X

q=1

kE_qxk_∗

wherep_ω⁰ denotes the conjugate exponent ofp_ω and the supremum is taken over all successive subsets of the natural numberE₁<· · ·<E_m.

The index 0 states that this is the base layer of the norm, which comes in many sizes, indicated by the indexm.

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m^1/p⁰^ω

m

X

q=1

kE_qxk_∗

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m^1/p⁰^ω

m

X

q=1

kE_qxk_∗

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m^1/p⁰^ω

m

X

q=1

kE_qxk_∗

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We assume that for somek, the layers 0, . . . ,k−1 have been defined, i.e. for every layer 06i<k and every sizem∈N, the normk · k_i,m has been defined.

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Thek’th layer: form∈Nandx ∈c₀₀(N)define

kxk_k,m=θsup





d

X

q=1

kE_qxk^p_i^k

q,mq





1/pk

where the supremum is taken over alld ∈N, 06iq<k and alladmissible and very fast growing(E_q)^d_q=1,(m_q)^d_q=1, i.e. they satisfy

d 6E₁<· · ·<E_d, minE_i >(maxEi−1)²and m_i >maxEi−1andm_i >mfori=2, . . . ,d.

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kxk_k,m=θsup





d

X

q=1

kE_qxk^p_i^k

q,mq





1/pk

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kxk_k,m=θsup





d

X

q=1

kE_qxk^p_i^k

q,mq





1/pk

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kxk_k,m=θsup





d

X

q=1

kE_qxk^p_i^k

q,mq





1/pk

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kxk_k,m=θsup





d

X

q=1

kE_qxk^p_i^k

q,mq





1/pk

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Forx ∈c₀₀(N)we also define

kxk_ω =θsup





d

X

q=1

kE_qxk^p_∗^ω





1/pω

where the supremum is taken over alld ∈Nand all successive subsets of the natural numberE₁<· · ·<E_d.

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Forx ∈c₀₀(N)we also define

kxk_ω =θsup





d

X

q=1

kE_qxk^p_∗^ω





1/pω

where the supremum is taken over alld ∈Nand all successive subsets of the natural numberE₁<· · ·<E_d.

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The normk · k_∗ satisfies the following implicit formula for everyx ∈c₀₀(N):

kxk_∗=max (

kxk_∞, kxk_ω, sup

k,m

kxk_k,m )

.

For every block vectorsx₁<· · ·<x_nthe following estimate holds:

θ

m

X

i=1

kx_ik^p^ω

!1/pω

6

m

X

i=1

x_i

62

m

X

i=1

kx_ik^p¹

!1/p₁

.

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kxk_∗=max (

k,m

kxk_k,m )

.

θ

m

X

i=1

kx_ik^p^ω

!1/pω

6

m

X

i=1

x_i

62

m

X

i=1

kx_ik^p¹

!1/p₁

.

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kxk_∗=max (

k,m

kxk_k,m )

.

θ

m

X

i=1

kx_ik^p^ω

!1/pω

6

m

X

i=1

x_i

62

m

X

i=1

kx_ik^p¹

!1/p₁

.

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kxk_∗=max (

k,m

kxk_k,m )

.

θ

m

X

i=1

kx_ik^p^ω

!1/pω

6

m

X

i=1

x_i

62

m

X

i=1

kx_ik^p¹

!1/p₁

.

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To show that the setF is contained in the Krivine set of every block subspace ofX, we show that for everyk,`_p_k is admitted as a spreading model by all subspaces ofX. In the present construction we use theα-indices of a block sequence.

Theα-index is a tool that has been used in earlier works

related to the method of constraints.

These indices determine the spreading models admitted by a block sequence.

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Theα-index is a tool that has been used in earlier works

related to the method of constraints.

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Theα-index is a tool that has been used in earlier works related to the method of constraints.

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Theα-index is a tool that has been used in earlier works related to the method of constraints.

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Letk be a natural number and(x_i)_i be a block sequence.

If for every layer 06k⁰ <k and strictly increasing

sequence of sizes(m_q)q, for every(x_i_q)qsubsequence of (x_i)_i we have that

limq kx_i_qk_k⁰_,m_q =0

then we say thattheα<k-index of(x_i)i is zero.

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Proposition

Let(x_i)i be a seminormalized block sequence in X generating a spreading model(y_j)_j.

The spreading model(y_j)_j of(x_i)_i is equivalent to the unit vector basis of`pω if and only if theα_<k index of(x_i)_i is zero for every k.

For every k ∈N, the spreading model(y_j)j of(x_i)i is equivalent to the unit vector basis of`p_k if and only if the α_<k index of(x_i)_i isnot zero, while theα_<k⁰ indexis zero for all k⁰ <k.

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Proposition

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Proposition

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We conclude that every spreading model admitted byX has to be`p, for somep ∈F.

It is also shown that allp’s inF, with the possible exception ofp_ω, occur as spreading models in every block subspace ofX.

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We conclude that every spreading model admitted byX has to be`p, for somep ∈F.

It is also shown that allp’s inF, with the possible exception ofp_ω, occur as spreading models in every block subspace ofX.

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Since for everyp∈F\ {p_ω}, every block subspace ofX admits an`pspreading model,

we conclude thatF \ {p_ω}and hence alsoF, is in the Krivine set of every block subspace ofX

We also show, by contradiction, that for everyp∈/F, the unit vector basis of`pis not finitely block represented inX. We derive that the Krivine set of every block subspace ofX is preciselyF.

Let us note that it is not known to us whether every block subspace ofX admits an`pω spreading model.

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Some words on how to obtain the desired spreading models in a a block subspace.

We use the following: if a block sequence generates an`p_k

spreading model, then an appropriate blocking of this sequence generates an`pk+1 spreading model. This blocking can be chosen to be increasingp_k-averages.

It is therefore sufficient to prove that every block subspace ofX admits an`p₁ spreading model.

We start with a block sequence(x_i)i in a block subspace of X and distinguish two cases:

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Case 1: The block sequence(x_i)_i admits an`pω spreading model.

In this case by appropriately blocking the sequence we pass to an other one generating an`p₁ spreading model.

This blocking can be chosen to be increasingp_ω-averages.

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Case 1: The block sequence(x_i)_i admits an`pω spreading model.

In this case by appropriately blocking the sequence we pass to an other one generating an`p₁ spreading model.

This blocking can be chosen to be increasingp_ω-averages.

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Case 2: The block sequence(x_i)i admits an`pk spreading model for somek.

We may then take block sequencences(x_i^m)_i,

m=k,k +1, . . .each one generating an`pm spreading model.

By carefully choosing block vectors, such that each one is an`_p_m average of elements of the sequence

we arrive at a sequence that generates an`p1 spreading model.

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Some words on how to prove that forp∈/F,`pis not finitely block represented inX.

Ifp∈/[p₁,p_ω]then the result follows easily from the fact that block vectors inX

satisfy a lower`_p_ω estimate with constantθand and upper`p1 estimate with constant 2.

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Roughly speaking, let us assume thatk is such that p_k <p<p_k+1,N is sufficiently large,εis sufficiently small and

(x_i)^N_i=1is a block sequence(1+ε)-equivalent to the unit vector basis of`^N_p,

Thek+1 layer of the norm provides`_p_k+1 structure to the space and hence

thek’th level is the one that has to be used to provide the

`_pestimate on some vectors.

It turns out however that the`pk structure imposed by the k’th level demolishes the`pone of the sequence.

We conclude that`pis not finitely block represented inX.

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