Large cardinals and basic sequences J. Lopez-Abad Instituto de - - PowerPoint PPT Presentation

Large cardinals and basic sequences

Large cardinals and basic sequences

• J. Lopez-Abad

Instituto de Ciencias Matem´

• aticas. CSIC. Madrid, Spain.

LC2011, July 2011

Large cardinals and basic sequences Introduction

The intend of this talk is to present a combinatorial approach to the study of the structure of a Banach space. In particular, we will focus on the existence of certain sequences, for example unconditional sequences.

Large cardinals and basic sequences Introduction

Problems:

(1) The separable quotient problem. (2) Subspaces. (3) Special sequences: basic, unconditional, subsymmetric...

Large cardinals and basic sequences Introduction

Tools:

(1) Structural Ramsey theorems. (2) (Not so) large cardinal numbers. (3) Forcing.

Large cardinals and basic sequences Basic definitions and results

Definition A Banach space (X, · ) is a vector space X (over the real numbers) endowed with a norm · (N.1) λx = |λ|x (N.2) x + y ≤ x + y (N.3) x = 0 iff x = 0 which is complete, i.e. Cauchy sequences are convergent.

Large cardinals and basic sequences Basic definitions and results

Examples:

• Rn with the euclidean norm (ai)i<n2 = (

i<n |ai|2)

1 2 ,

• the infinite dimensional separable Hilbert space

ℓ2 = {(ai)i∈N : (

i∈N |ai|2)

1 2 < ∞}, with the euclidean norm

(ai)i∈N2 = (

i∈N |ai|2)

1 2 ,

• the ℓp spaces, for p ≥ 1, ℓp = {(ai)i∈N : (

i∈N |ai|p)

1 p < ∞},

with the p-norm (ai)i∈Np = (

i∈N |ai|p)

1 p ,

Large cardinals and basic sequences Basic definitions and results

• c0 = {(ai)i∈N : limi→∞ ai = 0}, with the sup-norm

(ai)i∈N∞ = sup{|ai| : i ∈ N},

• ℓ∞ = {(ai)i∈N : supi→∞ |ai| < ∞}, with the sup-norm

(ai)i∈N∞ = sup{|ai| : i ∈ N},

• for a compact space K, the space C(K) of continuous

functions on K, endowed with the sup-norm, f = sup{|f(x)| : x ∈ K}. In particular, C([0, 1]).

Large cardinals and basic sequences Basic definitions and results

While (Rn, · p) are all isomorphic, the infinite dimensional versions are very much different:

Large cardinals and basic sequences Basic definitions and results

Basic notions

• A Banach space is infinite dimensional if it is not finite

dimensional.

• The density of a space X is the topological weight, i.e. the

smallest cardinality of a dense subset of X.

• A subspace Y of X will be understood as a linear subspace of

X, which is in addition closed. In particular Y with the norm · is also a Banach space.

• Given a subspace Y of X, the quotient space X/Y is the

Banach space over the linear quotient, endowed with the norm x + Y := d(x, Y).

Large cardinals and basic sequences Basic definitions and results

• An operator T : X → Y between two spaces X and Y is a

linear mapping which is continuous, or equivalently bounded, i.e., such that T := sup{Tx : x ∈ X, x ≤ 1} < ∞.

• An isomorphic embedding T : X → Y is a 1-1 operator such

that T(X) is a closed subspace of Y and the inverse U : T(X) → X is bounded.

• The dual X ∗ of a Banach space X the space of all operator

f : X → R. This is a Banach space with the norm f := sup{f(x) : x ≤ 1}. The elements of X ∗ are called functionals.

• A space X is called reflexive when X ∗∗ is canonically

identified with X.

Large cardinals and basic sequences Basic definitions and results

Special sequences

• A sequence (xγ)γ<κ in a Banach space (X, · ), indexed in

some cardinal number κ is called a biorthogonal sequence if for every α < κ there is a functional fα ∈ X ∗ such that fα(xβ) = δα,β. Notice that in particular biorthogonal sequences are linearly independent sequences.

• A normalized sequence (xγ)γ<κ in a Banach space (X, · ),

indexed in some cardinal number κ is called a (Schauder) basic sequence when there is a constant C ≥ 1 such that

• γ∈t

aγxγ ≤ C

• γ∈s

aγxγ for every sequence of scalars (aγ)γ∈s and every t ⊑ s ⊆ κ. It follows easily that (xα)α is a biorthogonal sequence.

Large cardinals and basic sequences Basic definitions and results

• A normalized sequence (xγ)γ<κ in X is a (Schauder) basis
• f X if it is basic, and the closed linear span of (xα)α is X.

Equivalently, every x ∈ X has a unique representation as x =

• γ<κ

aγxγ.

Large cardinals and basic sequences Basic definitions and results

• (xγ)γ is called an unconditional basic sequence when there

is a constant C ≥ 1 such that C ≥ 1 such that

• γ∈t

aγxγ ≤ C

• γ∈s

aγxγ for every sequence of scalars (aγ)γ∈s and every t ⊆ s ⊆ κ.

• (xγ)γ is called subsymmetric when there is a constant C ≥ 1

such that

• k∈l

akxγk ≤ C

• k<l

akxξk for every sequence of scalars (ak)k<l and every γ0 < · · · < γl−1, ξ0 < · · · < ξl−1.

Large cardinals and basic sequences Basic definitions and results

• (xγ)γ is called weakly-null when the set

{γ < κ : |x∗(xγ)| ≥ ε} is finite for every x∗ ∈ X ∗ and every ε > 0.

Large cardinals and basic sequences Basic definitions and results

Examples:

The unit basis (un)n, un = (0, 0, ..., 0,

(n)

1 , 0, 0, ...), is a Schauder basis of each ℓp or c0. Indeed it is an unconditional and subsymmetric basis. For p > 1 it is also weakly-null.

Large cardinals and basic sequences Basic definitions and results

Basic results about special sequences

• (Banach-Mazur) Every infinite dimensional Banach space has

a basic sequence.

• (Mazur) Every normalized weakly-null sequence has a basic

subsequence.

• (Enflo) There are separable Banach spaces without bases.
• (Bessaga-Pelczynski) The structure of subspaces of a

Banach space with a basis (en)n is determined by block subsequences of the basis (en)n.

• (James) The reflexivity of a space with a basis is determined

by the basis.

• (James) A space with an unconditional basis is reflexive iff it

does not contain an isomorphic copy of c0 or ℓ1.

• (Krivine) c0 or ℓp are finitely block representable in any

Banach space.

Large cardinals and basic sequences Basic definitions and results

• (Tsirelson) There is a space T without isomorphic copies of

c0 or ℓp, p ≥ 1. Indeed T does not have subsymmetric basic sequences.

• The summing basis sn = (1, ..., 1,

(n+1)

0 , 0, . . . ) ∈ c0 does not have unconditional subsequences, and it is not weakly-null.

• (Rosenthal) Every norm-bounded sequence (xn)n has a

subsequence which is either equivalent to the unit basis of ℓ1 or a weakly-Cauchy, i.e., (f(xn))n is a numerical Cauchy sequence for each f ∈ X ∗.

• (Maurey-Rosenthal) There is a weakly-null basic sequence

without unconditional subsequences.

• (Gowers-Maurey) There is a reflexive Banach space without

unconditional basic sequences.

Large cardinals and basic sequences The problems: Uncountable sequences

Main goal: Existence of uncountable special sequences on a given Banach space. Theorem (Amir-Lindenstrauss) Every reflexive space of infinite density κ has a normalized weakly-null sequence of length κ. Question Is it true that a non-separable space has an uncountable biorthogonal sequence? Question Is it true that a non-separable space has an uncountable basic sequence?

Large cardinals and basic sequences The problems: Uncountable sequences

Theorem (Kunen) CH implies that there is a non-metrizable scattered compactum K such that C(K) does not have uncountable biorthogonal sequences. Theorem (Todorcevic) b = ω1 implies that there is a non-metrizable scattered compactum K such that C(K) does not have uncountable biorthogonal sequences.

Large cardinals and basic sequences The problems: Uncountable sequences

Theorem (Shelah) the ♦ principle implies that there is a non-separable Gurarij space without uncountable biorthogonal sequences. Theorem (Johnson-Lindenstrauss) There is a non-separable space without uncountable basic sequences.

Large cardinals and basic sequences The problems: Uncountable sequences

By the method of forcing it is possible to build generic spaces as direct limits of finite dimensional polyhedral spaces and isometries between them, providing a large variety of examples

• f spaces having certain nice uncountable sequences and not
• thers, as well as nice geometrical properties.

Applications:

1 Lindenstrauss spaces (preduals of L1) 2 Gurarij spaces 3 Distinction between different uncountable substructures of

the generic Banach space (e.g. biorthogonal-like systems).

Large cardinals and basic sequences The problems: Uncountable sequences

Examples of this are: Theorem (LA-Todorcevic 2010)

1 There are X ⊆ Y non-separable L∞,1+ such that: 1 X is Asplund and c0-sat., Y is Gurarij and Y/G ≡ X. 2 Both (X, w)n and (Y, w)n are HL for every integer n. 3 Both X and Y have no supported sets. 2 A pair X and Y related as above such that X have

uncountable fundamental ε-biorthogonal sequences for every ε > 0 but no uncountable biorthogonal sequences.

3 A non-separable space X with uncountable

1 + ε-Schauder basic sequences for every ε > 0 but no uncountable monotone basic sequences.

Large cardinals and basic sequences The problems: Uncountable sequences

4 All of the spaces above have few operators. 5 A non-metrizable Poulsen simplex and a non-metrizable

Bauer simplex such that the corresponding space of probability measures is hereditarily separable in all finite powers. All these examples are built in a unified way, and they are consequence of the diamond principle ♦.

Large cardinals and basic sequences The problems: Uncountable sequences

Question 1: Is it possible to find a non-separable space without uncountable biorthogonal sequences and without isomorphic copies of c0? Question 2: Is it possible to find a non-separable space hereditarily Lindel¨

• f

with respect to its weak topology and with the Radon Nikodym property?

Large cardinals and basic sequences The problems: countable sequences

Separable Quotient Problem

Question Does every infinite dimensional Banach space X has a non-trivial quotient with a Schauder basis? Theorem (Johnson-Rosenthal) Every separable space has a non-trivial quotient with a Schauder basis.

Large cardinals and basic sequences The problems: countable sequences

Theorem (Todorcevic) It follows from Martin’s Maximum Axiom (MM) that every Banach space of density ℵ1 has a quotient with a Schauder basis of length ω1, and therefore a quotient with a Schauder basis. Theorem (Hagler-Johnson) Suppose that X is such that X ∗ has an unconditional sequence. Then X has a non-trivial quotient with an unconditional basis.

Large cardinals and basic sequences The problems: countable sequences

Question 1 What is the minimal cardinal κ, denoted by nc if exists, such that every Banach space of density at least κ has an unconditional basic sequence? Question 2 What is the minimal cardinal κ, denoted by ncrefl if exists, such that every reflexive Banach space of density at least κ has an unconditional basic sequence? Question 3 What is the minimal cardinal κ, denoted by nc0 if exists, such that every normalized weakly-null sequence (xγ)γ<κ has an unconditional basic sequence?

Large cardinals and basic sequences The problems: countable sequences

Question 4 What is the minimal cardinal κ, denoted by ns, such that every space of density κ has a subsymmetric basic sequence? Question 5 What is the minimal cardinal κ, denoted by nsseq, such that every sequence of length κ has a subsymmetric subsequence?

Large cardinals and basic sequences The problems: countable sequences

Proposition ncrefl ≤ nc, nc0. The first inequality is trivial. The second follows from the Amir-Lindenstrauss Theorem.

Large cardinals and basic sequences The problems: countable sequences

Proposition nc ≤ ns ≤ nsseq. For the second inequality: Let X be a space of density κ := nsseq. Let (xα)α<κ be a normalized 1-separated sequence in X, i.e. xβ − xα ≥ 1 for all α < β < κ. Let (yn)n be a subsymmetric subsequence of (xα)α. By Rosenthal’s ℓ1-theorem, there is a subsequence (zn)n∈N of (yn)n which is either equivalent to the unit basis of ℓ1, and therefore it is a basic sequence,

• r else weakly-Cauchy. Since (zn)n is separated,

((z2n+1 − z2n)/z2n+1 − z2n)n is a normalized weakly-null sequence, hence by Mazur’s theorem, it has a basic subsequence, which is in addition subsymmetric.

Large cardinals and basic sequences The problems: countable sequences

Now suppose that X has density ns. Then X has a subsymmetric basic sequence. Therefore it has a subsequence equivalent to the unit basis of ℓ1, hence unconditional, or else a subsequence (yn)n which is weakly-Cauchy. Now the normalized difference sequence ((y2n+1 − y2n)/y2n+1 − y2n)n is a normalized weakly-null subsymmetric basic sequence. Now it readily follows from Schreier unconditionality that that sequence has an unconditional basic subsequence.

Large cardinals and basic sequences The problems: countable sequences

Lower Bounds

nc > ω: Gowers-Maurey HI space, 1993.

Large cardinals and basic sequences The problems: countable sequences

Lower Bounds

nc > 2ω: Argyros-Tolias non-separable HI space, 2004. ncrefl > ω1: Argyros-LA-Todorcevic, 2005. ns > 2ω: Odell, 1985.

Large cardinals and basic sequences The problems: countable sequences

Lower Bounds

nc0 ≥ ωω: LA-Todorcevic, 2011. nsseq = first ω-Erd¨

• s cardinal number: LA-Todorcevic, 2011.

Definition κ is ω-Erd¨

• s when κ → (ω)<ω

2 , i.e. for every coloring

c : [κ]<ω → 2 there is an infinite set X ⊆ κ such that for every n ∈ N the restriction c ↾ [X]n is constant.

Large cardinals and basic sequences The problems: countable sequences

Upper Bounds

nsseq is smaller or equal than the first ω-Erd¨

• s cardinal.

Consequently, nc, too. Ketonen 1974. It is consistent relative to the existence of a measurable cardinal number that nc0 ≤ ωω. Dodos-LA-Todorcevic, 2011. Definition κ is said to be measurable if there exists a κ-complete normal ultrafilter U on κ.

Large cardinals and basic sequences The problems: countable sequences

It is consistent relative to the existence of infinitely many strongly compact cardinals that nc ≤ ωω. Dodos-LA-Todorcevic, 2011. Definition κ is said to be strongly compact if every κ-complete filter can be extended to a κ-complete ultrafilter.

Large cardinals and basic sequences The problems: countable sequences

Hints on proofs. nc0 ≥ ωω

Proposition There is a graph Gnc0(κ) whose set of vertexes is a dense family of finite sequences of finite sets of κ such that if the chromatic number χ(Gnc0(κ)) is countable, then there is a normalized weakly null sequence (xα)α<κ without unconditional sequences. Theorem The graph Gnc0(ωn) is countably chromatic for every integer n.

Large cardinals and basic sequences The problems: countable sequences

Hints on proofs. nc0 ≥ ωω

Definition Given s, t ∈ [κ]<ω with the same cardinality, let θs,t : s → t denote the unique order-preserving bijection between s and t. For such s, t, let M(s, t) := {α ∈ s ∩ t : θs,t(α) = α}.

Large cardinals and basic sequences The problems: countable sequences

Proof of nsseq = 1st ω-Erd¨

• s cardinal number

Theorem nsseq > κ iff κ is not ω-Erd¨

• s, i.e. κ → (ω)ω

2.

Proof: Suppose first that κ is ω-Erd¨

• s, and let (xγ)γ<κ be a

normalized sequence in a Banach space. For each n-set s ∈ [κ]n, color s by the type of the sequence (xγ)γ∈s. Since there are only 2ω many types, by the Erd¨

• s-property of κ, there is some infinite set X ⊆ κ such that if

s, t ⊆ X have the same cardinality, then they have the same type. This means that (xγ)γ∈X is subsymmetric.

Large cardinals and basic sequences The problems: countable sequences

Now suppose that κ → (ω)<ω

2 . Fix a bad coloring f : [κ]<ω → 2

without infinite f-homogeneous sets. Let B be the family of all f-homogeneous subsets of κ. By the hypothesis on κ, B consists on finite subsets of κ. Moreover, B is a compact family, and hereditary under inclusion. Define the following Schreier norm on c00(κ), the vector space consisting on all eventually zero sequences x : κ → R: For each x ∈ c00(κ), let xB := max{x, χs : s ∈ B}. Let XB be the completion of (c00(κ), · B). Since B is hereditary under inclusion, the unit sequence (uγ)γ<κ is a unconditional basis of XB.

Large cardinals and basic sequences The problems: countable sequences

Claim (xγ)γ does not have subsymmetric subsequences. This is proved using the finite Ramsey theorem to prove that every infinite subset of κ has arbitrarily large finite f-homogeneous subsets, and then Ptak’s Lemma.

Large cardinals and basic sequences The problems: countable sequences

nc0 ≤ ωω

Definition Let κ be a cardinal number, d ∈ N. By Pld(κ) we shall denote the combinatorial principle asserting that for every coloring c :

• [κ]d <ω → ω there exists a sequence

(Mn) of infinite disjoint subsets of κ such that for every m ∈ ω the restriction c ↾ m

n=0[Mn]d is constant.

Theorem nc0 ≤ κ if Pl1(κ). Theorem (Di Prisco-Todorcevic) It is consistent relative to the existence of a measurable cardinal that Pl1(ℵω).

Large cardinals and basic sequences The problems: countable sequences

nc ≤ ωω

Theorem nc ≤ κ if Pl2(κ). Theorem (Shelah) It is consistent relative to the existence of infinitely many strongly compact cardinals that Pl2(ℵω).

Large cardinals and basic sequences The problems: countable sequences

nc0 ≤ 2ℵ0

Definition Let κ, λ be two cardinal numbers. We say that κ has the λ-real measure extension property, ME(κ, λ) in short, if for every family F of subsets of κ of cardinality ≤ λ there is a σ-additive real measure µ on κ such that every element of F is µ-measurable. Theorem nc0 ≤ κ provided that ME(κ, κ). Theorem (Solovay) It is consistent relative to the existence of a weakly-compact cardinal κ (i.e. κ → (κ)2

2) that ME(2ℵ0, 2ℵ0).

Large cardinals and basic sequences The problems: countable sequences

Conjectures

C1 It is consistent with CH that nc = ω2. C2 It is consistently with CH that ns = ω2.