On the non-embedding of ℓ 1 in the James Tree Space
aa r X i v : . [ m a t h . F A ] D ec ON THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE
IOAKEIM AMPATZOGLOU
Abstract.
James Tree Space (
J T ), introduced by R. James in [3], is the first Banachspace constructed having non-separable conjugate and not containing ℓ . James actuallyproved that every infinite dimensional subspace of J T contains a Hilbert space, whichimplies the ℓ non-embedding. In this expository article, we present a direct proof ofthe ℓ non-embedding, using Rosenthal’s ℓ - Theorem [5] and some measure theoreticarguments, namely Riesz’s Representation Theorem [6]. Introduction
For many years, the conjecture that a separable Banach space with non-separable con-jugate will contain ℓ , up to embedding, was an open question in Banach space theory. Itis well-known that ( ℓ ) ∗ coincides with ℓ ∞ , which is non-separable, so a natural questionis whether all separable Banach spaces with this property ”look like” ℓ , up to embedding.This conjecture was proved false by R. James [3] who made a ingenius construction call theJames Tree Space. The main idea relies on previous work of R. James [2], where a quasi-reflexive separable Banach space, isometric to its second conjugate and Hilbert-saturated,was constructed. By Hilbert-saturated we mean that each infinite dimensional subspacecontains a Hilbert space, up to embedding. This space, called the James space, clearly has aseparable conjugate though. R. James was able to preserve the Hilbert-saturation propertybut remove separability of the conjugate by creating a binary tree structure where intuitivelyeach infinite branch of the tree will be basis for a James space. The ℓ non-embedding thenfollows as an immediate consequence of the Hilbert-saturation property. In this work, weprovide a direct proof of the ℓ non-embedding, which was the main part of the conjecture,in a direct way i.e. without proving Hilbert-saturation property. For this purpose, we firstreview the Schauder bases theory, which is crucial for this construction and introduce theJames Tree Space. We then apply Riesz’s Representation Theorem [6] to an appropriate w ∗ -compact subset of the conjugate space and Rosenthal’s ℓ -Theorem [5] yields the claim.2. Preliminaries
In this preliminary section, we summarize some major results from Schauder bases inBanach spaces which will be useful throughout this paper. The majority of the proofs canbe found in most textbooks of Banach Space Theory, hence we provide proofs only forthe results which are not exactly stated in the bibliography as they are stated here. Ourapproach is based on [4].2.1.
Notation.
Let us clarify the notation used. Throughout this paper, all Banach spacesconsidered are assumed infinite dimensional unless stated. Given a Banach space X wedenote its unit ball by B X and its conjugate spaces by X ∗ , X ∗∗ , etc. Given a sequence { x n } n ∈ N in X we denote h x n : n ∈ N i to be the vector space spanned by this sequence and[ x n : n ∈ N ] to be the closure of the space spanned with respect to the norm. Finally. givena Banach space X we denote ∧ : X → X ∗∗ the canonical embedding of X in X ∗∗ given by b x ( x ∗ ) = x ∗ ( x ) , ∀ x ∈ X. Recall that the canonical embedding is a linear isometry and a Banach space is calledreflexive if the canonical embedding is surjective.
Definition of Schauder basis.
As known a Banach space necessarily has uncountablealgebraic basis. However, in most reasonable separable Banach spaces, we are able to finda countable topological basis i.e. each element can be expanded as a series with respect tothe norm. More precisely we give the following definition:
Definition 2.1.
Let X be a Banach space and { e n } n ∈ N be a sequence of distinct elementsof X . The sequence { e n } n ∈ N is called a Schauder basis, or just a basis, of X if for any x ∈ X there is a unique sequence { λ n } n ∈ N ⊆ R such that x = ∞ X n =1 λ n e n . The existence of Schauder basis is easily seen to be possible only in separable Banachspaces. However, the converse is not true, as shown by P. Enflo in [1].
Proposition 2.2.
Let X be a Banach space with a Schauder basis. Then X is separable. Consider X to be a Banach space with Schauder basis { e n } n ∈ N . For any n ∈ N we define e ∗ n : X → R by: e ∗ n ( x ) = e ∗ n ( ∞ X k =1 λ k e k ) = λ n . Clearly x = ∞ X n =1 e ∗ n ( x ) e n . One can easily prove the following: Proposition 2.3.
For any n ∈ N , e ∗ n ∈ X ∗ i.e. e ∗ n is a bounded linear functional. The functionals { e ∗ n } n ∈ N are called biorthogonal functionals of the basis { e n } n ∈ N .We now state an equivalent characterization of Schauder bases. In fact, this is how oneusually checks whether a given sequence in a Banach space is a Schauder basis. Proposition 2.4.
Let X be a Banach space and a sequence { e n } n ∈ N of pairwise distinct,non-zero elements of X . The following statements are equivalent:(i) { e n } n ∈ N is a Schauder basis of X (ii) The following hold: • X = [ e n : n ∈ N ] . • ∃ K > such that for any m > n ∈ N and λ , ..., λ m ∈ R , there holds || n X i =1 λ i e i || ≤ K || m X i =1 λ i e i || . (1) Remark 2.5.
The infimum number K in (1) is called constant of the basis. In the specialcase where this constant is unit, the basis is called monotone. It is clear that, for a basisto be monotone, condition (1) reduces to showing that for any n ∈ N and λ , ..., λ n +1 ∈ R ,there holds k n X i =1 λ i e i k ≤ k n +1 X i =1 λ i e i k . N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 3
Example 2.6.
Let ≤ p < ∞ . Recall by ℓ p we denote the Banach spaces ℓ p = ( x = { x n } n ∈ N : ∞ X n =1 | x n | p < ∞ ) , with norm k x k p = ( ∞ X n =1 | x n | p ) /p . Then the sequence { e n } n ∈ N , given by e n = (0 , , ..., , , ... ) (n-position) is a monotoneSchauder basis of ℓ p . We call it the natural basis of ℓ p .Proof. Let x = ( x , x ..., x n , ... ) ∈ ℓ p . Define s n = n X i =1 x i e i . Then we get || s n − x || = ( ∞ X i = n +1 | x i | p ) /p n →∞ −→ . Moreover, for all n ∈ N and λ , ..., λ n +1 ∈ R , we have || n X i =1 λ i e i || = ( n X i =1 | λ i | p ) /p ≤ ( n +1 X i =1 | λ i | p ) /p = || n +1 X i =1 λ i e i || , and the claim follows. (cid:3) We now deduce some useful criterias for w ∗ -convergence in Banach spaces with Schauderbases. Proposition 2.7.
Let X be a Banach space, x ∗ ∈ X ∗ \{ } and a bounded sequence { x ∗ n } n ∈ N in X ∗ . Assume there is norm-dense S ⊆ X such that x ∗ n ( s ) n →∞ −→ x ∗ ( s ) , ∀ s ∈ S. Then x ∗ n w ∗ → x ∗ .Proof. The result is trivial if x n = 0 , ∀ n ∈ N . Therefore, since { x ∗ n } n ∈ N is bounded too,we may well assume that 0 < M := sup n ∈ N k x ∗ n k < ∞ . It suffices to show that x ∗ ( x ) = lim n →∞ x ∗ n ( x ) , ∀ x ∈ X . Let x ∈ X . By density, there is asequence { s n } n ∈ N ⊆ S with x = lim n →∞ s n . Let ǫ > N, n ∈ N such that || s N − x || < ǫ { M, || x ∗ ||} and | x ∗ n ( s N ) − x ∗ ( s N ) | < ǫ , ∀ n ≥ n . Then for any n ≥ n , we have | x ∗ n ( x ) − x ∗ ( x ) | ≤ | x ∗ n ( x ) − x ∗ n ( s N ) | + | x ∗ n ( s N ) − x ∗ ( s N ) | + || x ∗ ( s N ) − x ∗ ( x ) | < M || s N − x || + ǫ k x ∗ kk s N − x k < ǫ ǫ ǫ ǫ. The result is proved. (cid:3)
IOAKEIM AMPATZOGLOU
Corollary 2.8.
Let X be a Banach space with basis { e n } n ∈ N . Consider x ∗ ∈ X ∗ and { x ∗ k } k ∈ N a bounded sequence in X ∗ If x ∗ k ( e n ) k →∞ −→ x ∗ ( e n ) ∀ n ∈ N , then x ∗ n w ∗ → x ∗ .Proof. Defining S = h e n : n ∈ N i . Then S is dense in X . Linearity of { x ∗ k } k ∈ N , x ∗ andProposition (2.7) yield the result. (cid:3) Basic sequences, blocks and equivalence.
In this section we introduce the notionbasic sequences in Banach spaces and some elementary type of basic sequences called blocks.
Definition 2.9.
Let X be a Banach space and { x n } n ∈ N a sequence of pairwise distinct,non-zero elements of X . The sequence { x n } n ∈ N is called basic sequence if it is Schauderbasis of the subspace [ x n : n ∈ N ] .We define the constant of the basic sequence { x n } n ∈ N as the constant of the Schauderbasis of [ x n : n ∈ N ] .Finally, we define its biorthogonal functionals as x ∗ n : [ x n : n ∈ N ] → R given by x ∗ n ( ∞ X k =1 λ k x k ) = λ n . Since { x n } n ∈ N is basic, Proposition (2.3) implies x ∗ n ∈ [ x k : k ∈ N ] ∗ , ∀ n ∈ N . Remark 2.10.
Hahn-Banach Theorem implies that for each n ∈ N , x ∗ n can be extended toan element of X ∗ . Therefore, without loss of generality, we may assume that x ∗ n ∈ X ∗ , foreach n ∈ N . We immediately get the following characterization:
Corollary 2.11.
Let X be a Banach space and a sequence { e n } n ∈ N of pairwise distinct,non-zero elements of X . The following statements are equivalent:(i) { e n } n ∈ N is basic sequence.(ii) ∃ K > such that for any m > n ∈ N and λ , ..., λ m ∈ R , there holds || n X i =1 λ i e i || ≤ K || m X i =1 λ i e i || . (2)We now define the notion of equivalence of two basic sequences. Definition 2.12.
Let
X, Y be Banach spaces. Consider the sequences { x n } n ∈ N ⊆ X and { y n } n ∈ N ⊆ Y . The sequences { x n } n ∈ N and { y n } n ∈ N are called equivalent if there exist c, C > such that for any n ∈ N and λ , ..., λ n ∈ R , there holds c || n X i =1 λ n x n || ≤ || n X i =1 λ n y n || ≤ C || n X i =1 λ n x n || . Remark 2.13.
It is clear that equivalence of sequences is an equivalence relation whichpreserves basic sequences.
Remark 2.14.
Let
X, Y be Banach spaces and { x n } n ∈ N ⊆ X, { y n } n ∈ N ⊆ Y equivalentsequences. Then the series P ∞ n =1 α n x n converges iff the series P ∞ n =1 α n y n converges. N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 5
Proof.
Let { α n } n ∈ N such that P ∞ n =1 α n x n converges. We will show that the sequence ofpartial sums { P ni =1 α i y i } n ∈ N is Cauchy. Indeed, for any ǫ > N ∈ N such that forall m > n > N , we have || m X i = n +1 α i x i || < ǫC . Then we get || m X i = n +1 α i y i || < ǫ. The other way is identical. (cid:3)
Equivalent sequences can be characterized in the following equivalent ways:
Proposition 2.15.
Let
X, Y be Banach spaces. Consider a basic sequence { x n } n ∈ N ⊆ X and a sequence { y n } n ∈ N ⊆ Y . Then the following are equivalent:(i) { x n } n ∈ N and { y n } n ∈ N are equivalent.(ii) There is an isomorphism T : [ x n : n ∈ N ] → [ y n : n ∈ N ] , with T ( x n ) = y n , ∀ n ∈ N .Proof. ( i ) ⇒ ( ii ) We define the mapping T : [ x n : n ∈ N ] → [ y n : n ∈ N ] by T ( x ) = T ( ∞ X n =1 α n x n ) = ∞ X n =1 α n y n . The mapping T is well-defined and linear due to the fact that the sequence { x n } n ∈ N isbasic and Remark 2.14. We first show that T is bounded. For any n ∈ N , let us definethe linear operators T n : [ x n : n ∈ N ] → h y , ..., y n i , F n : h x , ..., x n i → h y , ..., y n i and P n : [ x n : n ∈ N ] → h x , ..., x n i , given respectively by T n ( ∞ X i =1 α i x i ) = n X i =1 α i y i ,F n ( n X i =1 α i x i ) = n X i =1 α i y i ,P n ( ∞ X i =1 α i x i ) = n X i =1 α i x i . Notice that F n is bounded since its domain is finite dimensional. Moreover, since { x n } n ∈ N is basic, Corollary (2.11) implies that for any x = P ∞ i =1 λ i x i ∈ X , we have P n ( x ) = k n X i =1 λ i x i k ≤ K k m X i =1 λ i x i k , ∀ m > n, where K is the constant of { x n } n ∈ N . Letting m → ∞ we get k P n ( x ) k ≤ K k x k , hence P n is bounded. But T n = F n ◦ P n , so T n is bounded for any n ∈ N . By definition of T n , thefollowing point-wise convergence holds: T ( x ) = lim n →∞ T n ( x ) , ∀ x ∈ X. Therefore, Banach-Steinhauss Theorem implies T is bounded. We will also show that T is a bijection and the result will come by the Open Mapping Theorem. Since { y n } n ∈ N is IOAKEIM AMPATZOGLOU equivalent to { x n } n ∈ N , it is basic as well. Therefore each y ∈ [ y n : n ∈ N ] can be uniquelywritten as y = P ∞ n =1 y n . This directly implies that T is a bijection, since T ( x n ) = y n , ∀ n ∈ N and T is bounded.( ii ) ⇒ ( i ) Comes immediately for c = 1 || T − || and C = || T || . (cid:3) Remark 2.16.
Let X be a Banach space with basis { x n } n ∈ N and Y a Banach space. Ifthere is a sequence { y n } n ∈ N in Y and c, C > such that for any n ∈ N and λ , ..., λ n ∈ R there holds c || n X i =1 λ n x n || ≤ || n X i =1 λ n y n || ≤ C || n X i =1 λ n x n || . Then X embeds isomorphically in Y .Proof. By Proposition 2.15, it is enough to show that the sequence { y n } n ∈ N is basic. Let K be the constant of { x n } n ∈ N . Consider m > n ∈ N and λ , ...λ m ∈ R . Then || n X i =1 λ i y i || ≤ CK || m X i =1 λ i x i || ≤ CKc || m X i =1 λ i y i || , and the claim is proved. (cid:3) We now restrict our attention to a specific class of basic sequences, called blocks. Blocksare much easier to handle than arbitrary basic sequences.
Definition 2.17.
Let X be a Banach space with basis { e n } n ∈ N . A sequence { u n } n ∈ N ofpairwise distinct non-zero elements of X will be called block of { e n } n ∈ N if there is a sequenceof real numbers { α i } i ∈ N and an increasing sequence { n i } i ∈ N of positive integers such thatfor any k ∈ N , there holds u k = n k +1 X i = n k +1 α i e i Remark 2.18.
It is worth mentioning that the notion of a block sequence is always definedwith respect to a given Schauder basis.
It is straightforward that blocks are basic sequences.
Proposition 2.19.
Let X be a Banach space with basis { e n } n ∈ N and constant K . Thenevery block sequence is basic of constant less or equal than K .Proof. For all m > k ∈ N and λ , ..., λ m ∈ R we have || k X j =1 λ j u j || = || k X j =1 λ j n j +1 X i = n j +1 α i e i || = || k X j =1 n j +1 X i = n j +1 λ j α i e i ||≤ K || m X j =1 n j +1 X i = n j +1 λ j α i e i || ≤ K || m X j =1 λ j u j || , so the sequence { u n } n ∈ N is basic with constant less or equal than K . (cid:3) N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 7
The following result gives some sufficient conditions for a basic sequence to be equivalentto a block up to subsequence. It is usually referred in literature as sliding hump argument.
Lemma 2.20. (Sliding Hump Argument) Let X be a Banach space with basis { e n } n ∈ N anda sequence { x n } n ∈ N in X such that • inf n ∈ N || x n || > . • lim n →∞ e ∗ k ( x n ) = 0 , ∀ k ∈ N . Then there is a subsequence { x ′ n } n ∈ N of { x n } n ∈ N which equivalent to a block of { e n } n ∈ N . We prove the following useful result about embeddings.
Proposition 2.21.
Let X be a Banach space with Schauder basis { e n } n ∈ N . Assume Y is aBanach space which embedds isomorphically in X , with Schauder basis { y n } n ∈ N , satisfying < m ≤ inf n ∈ N k y n k ≤ sup n ∈ N k y n k ≤ M < ∞ . Then there is a block of { e n } n ∈ N equivalent to { y n } n ∈ N .Proof. Since Y embedds in X , there is a sequence { x n } n ∈ N in X which is basic and equivalentto { y n } n ∈ N . Therefore, there are m ′ , M ′ > m ′ ≤ || x n || ≤ M ′ ∀ n ∈ N . For any k ∈ N , we have | e ∗ k ( x n ) | ≤ || e ∗ k || M, ∀ n ∈ N , so the real numbers sequence { e ∗ k ( x n ) } n ∈ N is bounded for all k ∈ N . Therefore, with a diagonal argument, we mayconstruct a subsequence { x ′ n } n ∈ N of { x n } n ∈ N such that the sequence { e ∗ k ( x ′ n ) } n ∈ N convergesfor all k ∈ N . Let us denote z n = x ′ n +1 − x ′ n . For any k ∈ N , we clearly havelim n →∞ e ∗ k ( z n ) = 0 . Using a sliding hump argument, we may find a subsequence { z ′ n } n ∈ N and a block { u n } n ∈ N of { x n } n ∈ N which are equivalent. Clearly { u n } n ∈ N is equivalent to { y n } n ∈ N . (cid:3) We finally mention, without proof, a very important Theorem which will turn out to beessential for our exposition. As known, ℓ cannot embedd in a space with separable dualsince its dual coincides with ℓ ∞ . H. Rosenthal proved a partial inverse of this, the famous ℓ -Theorem [5]. Theorem 2.22. ( ℓ -Theorem) Let X be a Banach space and { x n } n ∈ N a bounded sequencein X . Then there holds exclusively one of the following:i) The sequence { x n } n ∈ N has weak-Cauchy subsequence.ii) The sequence { x n } n ∈ N is basic and equivalent to the standard basis of ℓ . The one direction is immediate since the standard basis of ℓ cannot have a weak-Cauchysubsequence. The other direction is much more complicated though. Reader can find morein [5] . As an application of Rosenthal’s Theorem we prove the following useful Proposition: Proposition 2.23.
Let X a Banach space not containing ℓ . Then every infinite dimen-sional subspace of X has a weak-Cauchy unitary sequence. IOAKEIM AMPATZOGLOU
Proof.
Let Y be infinite dimensional subspace of X . Then B Y is not norm-compact. There-fore there is { s n } n ∈ N ⊆ B Y with non-convergent subsequence. But ℓ -Theorem and theassumption that ℓ does not embed in X , the sequence { s n } n ∈ N has a subsequence { s n k } k ∈ N which is weak-Cauchy. So the sequence (cid:8) s n k +1 − s n k (cid:9) k ∈ N is weakly null. Moreover, since { s n k } k ∈ N is not norm-convergent, there is θ > ∀ n ∈ N ∃ k, m ∈ N : n < k < m and || s k − s m || ≥ θ. (3)By (3) we determine an increasing sequence { p n } n ∈ N ⊆ N such that || s p n − s p n − || ≥ θ, ∀ n ∈ N . Defining u n = s p n − s p n − we get u n w −→ || u n || ≥ θ, ∀ n ∈ N . Hence, the sequence { z n } n ∈ N defined by z n = k u n k − u n is unitary and weakly null. (cid:3) Definition of
J T and non-separability of the conjugate
In this section, we define the
J T space and summarize some of its basic properties. Asmentioned in the introduction,
J T is the first example of a non-separable Banach spacewhich does not contain ℓ . In this section, we give the basic definitions about J T and showthat the conjugate space is non-separable.Let us begin with some basic definitions on the Cantor tree, which be the natural index setto define our basis. These definitions will turn out to be very important and will constantlybe used in the following. • We define the Cantor tree as follows:2 < N = { s = ( s , ..., s n ) : s i ∈ { , } , ∀ i = 1 , ..., n, n ∈ N } ∪ {∅} . The empty set ∅ is called root of the tree. Elements of 2 < N are called nodes of thetree. • We also define the set of sequences of 0 and 1 as follows:2 N = { σ = ( σ n ) n ∈ N : σ i ∈ { , } , ∀ i ∈ N } . • We define the level function | · | : 2 < N → N ∪ { } by | s | = ( , s = ∅ ,n, s = ( s , ..., s n ) . • We define the partial ordering ’ ⊑ ’ on 2 < N as follows: – ∅ ⊑ s, ∀ s ∈ < N – If s, u ∈ < N with s, u = ∅ , then s ⊑ t ⇔ | s | ≤ | t | and s i = t i , ∀ i = 1 , ..., | s | . • For σ ∈ N and n ∈ N , we will denote σ | n = ( σ , ..., σ n ) ∈ < N . If we considerpairwise distinct σ , ..., σ n ∈ N , then there is N ∈ N such that σ i | N = σ j | N ∀ i, j ∈{ , ..., n } with i = j . The minimum positive integer with this property is calledseparation level of σ , ..., σ n . • Let I ⊆ < N such that for any s, t ∈ I we have either s ⊑ t or t ⊑ s . If for any s, t ∈ I and w ∈ < N such that s ⊑ w ⊑ t , we have that w ∈ I , then I is called aninterval. Finite intervals are called segments and are typically denoted by F , whileinfinite intervals are called branches and are typically denoted by B . N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 9 • A segment F can be uniquely written as F = { s , s , ..., s n } where s i ∈ < N , ∀ i =1 , ..., n and s ⊑ s ⊑ , ..., ⊑ s n . Node s is called initial node of F and is denotedas in( F ), while s n is called ending node of F and is denoted as end( F ). The nodesin( F ) and end( F ) are called endpoints of F . It is clear that for any s, t ∈ < N with s ⊑ t , there is segment F with in( F ) = s and end( F ) = t . • For any branch B there is unique σ ( B ) ∈ N and unique n ∈ N such that B = { ( σ ( B ) n + k ) ∞ k =0 } . We define the initial node of B as in( B ) = σ ( B ) n . • Let B , ..., B n be pairwise distinct branches. Then σ ( B ) , ..., σ ( B n ) are clearly pair-wise distinct too. We define the separation level of B , ..., B n as the minimumpositive integer N such that σ ( B i ) | N = σ ( B j ) | N , ∀ i, j ∈ { , ..., n } with i = j and N ≥ in( B i ) , ∀ i = 1 , ..., n . • We define t = ∅ and t = (0). For n > t n = ( s , ..., s m ). If for all i = 1 , ...m we have s i = 1, we define t n +1 = ( s ′ , ..., s ′ m , s ′ m +1 ) where s ′ i = 0 for all i = 1 , ..., m + 1. Alternatively we consider i = max { i ∈ { , ..., m } : s i = 0 } anddefine t n +1 = ( s ′ , ..., s ′ m ) with s ′ i = s i , ∀ i = 1 , ..., i − s ′ i = 1 and s ′ i = 0 , ∀ i = i + 1 , ..., m . By this enumeration, it is clear that 2 < N = { t n : n ∈ N } and that 2 < N is countable. • We observe that for any n ∈ N , we have that | t n | = [log n ] where [ · ] denotes theinteger part of a positive number. • For s ∈ < N , we denote e s = X { s } . In particular we denote e n = X { t n } . Clearlythe sequences { e s } s ∈ < N and { e n } n ∈ N coincide and the represent the sequence of thecharacteristic functions of the nodes of the Cantor tree.We are now in the position to define the James Tree Space. Definition 3.1.
We define
J T = ( x : 2 < N → R : sup ( m X i =1 | X s ∈ F i x s | ) < ∞ ) , where sup is taken over all finite families of pairwise disjoint segments { F i } mi =1 , m ∈ N . Itis immediate that J T is infinite dimensional vector space. For x ∈ J T , we define || x || = sup ( m X i =1 | X s ∈ F i x s | ) / , where sup is taken over all finite families of pairwise disjoint segments { F i } mi =1 . Remark 3.2.
Given x ∈ J T , we define its support as supp( x ) = (cid:8) s ∈ < N : x s = 0 (cid:9) . It is clear that the supremum taken for k x k can be restricted to all finite families of pairwisedisjoint segments contained in supp( x ) . Proposition 3.3. ( J T , || · || ) is a Banach space.Proof. We first show that || · || is a norm. The only non-trivial part is triangular inequality.Consider x, y ∈ J T and { F i } mi =1 pairwise disjoint segments. Then Minkowski’s inequality for sums implies( m X i =1 | X s ∈ F i ( x s + y s ) | ) / = [( X s ∈ F x s + X s ∈ F y s ) ++ ... + ( X s ∈ F m x s + X s ∈ F m y s ) ] / ≤ ( m X i =1 | X s ∈ F i x s | ) / + ( m X i =1 | X s ∈ F i y s | ) / ≤ || x || + || y || , so, taking supremum, since the intervals chosen are arbitrary, we obtain || x + y || ≤ || x || + || y || . Let us now show completeness. Let { x n } n ∈ N be a Cauchy sequence in J T . Then for any ǫ >
0, there is N such that for any k > n ≥ N , there holds || x k − x n || < ǫ . Considering s ∈ < N and the interval I s = { s } , the definition of the norm implies that | x k,s − x n,s | < ǫ, ∀ k > n > N. Therefore, the sequence { x n,s } n ∈ N converges for any s ∈ < N . Let us define the map x : 2 < N → R , given by x s = lim n →∞ x n,s , s ∈ < N . We will show x ∈ J T and that x = lim n →∞ x n . Indeed, consider ǫ > M ∈ N suchthat for any k > n ≥ M , we have || x n − x k || < ǫ { F i } mi =1 , we obtain( m X i =1 | X s ∈ F i | x m,s − x n,s ) | ) / < ǫ , ∀ m > n ≥ M m →∞ ⇒ (4)( m X i =1 | X s ∈ F i | x n,s − x s || ) / ≤ ǫ < ǫ, ∀ n ≥ M. (5)Fixing n = M in (4), we get that x M − x ∈ J T ⇒ x ∈ J T . Then (7) yields x = lim n →∞ x n . (cid:3) The following Lemma illustrates an interesting super-additive property of the norm whichwill yield that { e n } n ∈ N is a Schauder basis. Lemma 3.4.
Let x ∈ J T . Consider k ∈ N and s k = P ki =1 x i ( t i ) e i . Then the followingestimate holds: k s k k + k x − s k k ≤ k x k . N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 11
Proof.
Fix k ∈ N . Then it is clear that k s k k = sup ( m X i =1 | X s ∈ F i x s | ) , k x − s k k = sup l X j =1 | X s ∈ F ′ j x s | , (6)where the supremums are taken over all finite families of pairwise disjoint segments { F i } mi =1 , { F j } lj =1 contained in { t , ..., t k } and { t n : n ≥ k +1 } respectively. Considering such families,we define the family of disjoint intervals F = { F , ..., F m , F ′ , ..., F ′ l } . Then we clearly have m X i =1 | X s ∈ F i x s | + l X j =1 | X s ∈ F ′ j x s | = X F ∈F | X s ∈ F x s | ≤ k x k . Since the families are chosen arbitralily, we may take supremums and obtain the requiredestimate from (6). (cid:3)
Proposition 3.5.
The sequence { e n } n ∈ N is monotone and unitary Schauder basis of J T .Proof.
Clearly e n = 0 ∀ n ∈ N and || e n || = 1 ∀ n ∈ N . We first show that J T = [ e n : n ∈ N ]. Consider x ∈ J T . Let us define s n = n X i =1 x ( t i ) e i . We will show that x = lim n →∞ s n . Indeed, let ǫ >
0. The definition of the norm yiels that there are pairwise disjoint segments { F i } mi =1 such that, m X i =1 | X s ∈ F i x s | > || x || − ǫ . (7)For n = max ( | s | : s ∈ m [ i =1 F i ) , inequality (7) yields that || x || − || s n || < ǫ ∀ n ≥ n +1 . So for n ≥ n +1 , Lemma 3.4 implies that || x − s n || = || x − s n || + || s n || − || s n || = || x || − || s n || < ǫ . Therefore, x = lim n →∞ s n . Finally for n ∈ N and λ , ..., λ n , α n +1 ∈ R , we consider pairwisedisjoint segments { F i } mi =1 with t n +1 / ∈ m [ i =1 F i . Then, defining y = P ni =1 λ i e i , we obtain( m X i =1 | X s ∈ F i y s | )) / ≤ || n +1 X i =1 λ i e i || ⇒ || n X i =1 λ i e i || ≤ || n +1 X i =1 λ i e i || , and the claim is proved. (cid:3) Remark 3.6.
J T can be equivalently defined as the completion of < c (2 < N ) > under thenorm defined. Remark 3.7. If F is a segment, we define F ∗ = P s ∈ F e ∗ s . Clearly F ∈ J T ∗ since e s ∈J T ∗ , ∀ s ∈ < N .Consider now a branch B . The definition of the norm implies that for any x ∈ J T and ǫ > , there is n ∈ N such that for any m > n > n , there holds | m X i = n +1 e ∗ i ( x ) | < ǫ. Therefore, the series P s ∈ B e ∗ s ( x ) converges for any x ∈ J T . Defining B ∗ : J T → R by B ∗ ( x ) = X s ∈ B e ∗ s ( x ) , ∀ x ∈ J T , we obtain that B ∗ is linear and, by Banach-Steinhauss Theorem, we get B ∗ ∈ J T ∗ . It isclear that B ∗ w ∗ = X s ∈ B e ∗ s . Finally it is clear that || I ∗ || = 1 , for any interval I , and that for any x ∈ J T , we havethe following norm description: || x || = sup ( m X i =1 | I ∗ i ( x ) | ) / , where { I i } mi =1 are pairwise disjoint intervals, which can be either segments or branches. Let us introduce some notation. Recall that for σ ∈ N , we denote σ | n = ( σ , ..., σ n ) ∈ < N . We define σ ∗ w ∗ = X n =1 ∞ e ∗ σ | n ∈ J T ∗ , by Remark 3.7 . We can now easily see that the conjugate space is non-separable.
Proposition 3.8.
The conjugate space
J T ∗ is non-separable.Proof. The set (cid:8) σ ∗ : σ ∈ N (cid:9) is clearly uncountable and 1-separated i.e. there are x ∗ , y ∗ ∈J T ∗ with k x ∗ − y ∗ k ≥ . Indeed, considering σ = σ ∈ N , there is s ∈ < N with σ ∗ ( e s ) = 1 and σ ∗ ( e s ) = 0. Thus || σ ∗ − σ ∗ || ≥
1, so
J T ∗ is not separable. (cid:3) The ℓ non-embedding in J T
The goal of this section is to show that ℓ does not embed in J T . As mentioned before,we will use Riesz’s Representation Theorem to prove the non-embedding of ℓ .Let { I n } n ∈ N be pairwise disjoint intervals. For any x ∈ J T we have that ∞ X n =1 | I ∗ n ( x ) | ≤ || x || . N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 13
Moreover, for any sequence { α n } n ∈ N ∈ B ℓ , Cauchy-Schwartz inequality implies that ∞ X n =1 | α n || I ∗ n ( x ) | ≤ || x || , so P ∞ n =1 α n I ∗ n ( x ) converges absolutely for any x ∈ J T and w ∗ − P ∞ n =1 α n I ∗ n ∈ B J T ∗ . Letus note that by w ∗ − P ∞ n =1 α n I ∗ n we mean the series interpreted as a w ∗ -limit of partialsums.Let us define K ∗ = ( w ∗ − ∞ X n =1 α n I ∗ n : { α n } n ∈ N ∈ B ℓ and { I n } n ∈ N pairwise disjoint intervals ) . Clearly K ∗ ⊆ B J T ∗ . Proposition 4.1.
The set K ∗ is norming for J T i.e. || x || = sup { k ∗ ( x ) : k ∗ ∈ K ∗ } , ∀ x ∈ J T . We may also write || x || = sup {| k ∗ ( x ) | : k ∗ ∈ K ∗ } .Proof. Let x ∈ J T . Since K ∗ ⊆ B J T ∗ , we have thatsup { k ∗ ( x ) : k ∗ ∈ K ∗ } ≤ || x || . For the opposite direction, consider n ∈ N and define x n = n X i =1 e ∗ i ( x ) e i . Then there arepairwise disjoint intervals { I i } mi =1 such that || x n || − n < ( m X i =1 | I ∗ i ( x n ) | ) / . Defining λ i = ( m X i =1 | I ∗ i ( x n ) | ) − / I ∗ i ( x n ) , we have that k ∗ = m X i =1 λ i I ∗ i ∈ K ∗ and k ∗ ( x n ) = ( m X i =1 | I ∗ i ( x n ) | ) / , so || x n || − n < k ∗ ( x n ) ≤ sup { k ∗ ( x n ) : k ∗ ∈ K ∗ } n →∞ ⇒ || x || ≤ sup { k ∗ ( x ) : k ∗ ∈ K ∗ } , and the result follows. The second description is immediate. (cid:3) Proposition 4.2.
The set K ∗ is w ∗ -compact subset of J T ∗ .Proof. Since
J T is separable, the ball ( B X ∗ , w ∗ ) is a metric space, so it suffices to showthat K ∗ is sequentially compact. Let k ∗ n = w ∗ − ∞ X n =1 α i,n I ∗ i,n a sequence in K ∗ . Since thisseries converges absolutely, we may assume, after a possible re-ordering, that for any n ∈ N ,we have that | α i +1 ,n | ≤ | α i,n | ∀ i ∈ N . We first prove a claim. • Claim : Let { I n } n ∈ N a sequence of intervals. Then there is subsequence { I k n } n ∈ N andinterval I such that I ∗ k n w ∗ −→ I ∗ . Proof of the claim
Using a diagonal argument, we may find subsequence { I k n } n ∈ N suchthat (cid:8) I ∗ k n ( e s ) (cid:9) n ∈ N converges for any s ∈ < N . Let us define I = (cid:8) s ∈ < N : ∃ n s ∈ N with s ∈ I k n ∀ n ≥ n s (cid:9) . Then I is an interval. Indeed, consider s, t ∈ I . Then there is n ∈ N such that s, t ∈ I k n .Thus, either s ⊑ t or t ⊑ s . Let us consider s, t ∈ I and w ∈ < N such that s ⊑ w ⊑ t .From the definition of I , there is n ∈ N such that s, t ∈ I n , ∀ n ≥ n . Hence, we havethat w ∈ I n , ∀ n ≥ n ⇒ w ∈ I . We may easily show that lim n →∞ I ∗ k n ( s ) = I ∗ ( s ) , ∀ s ∈ < N and since || I ∗ n || = 1 , ∀ n ∈ N , Corollary 2.8, implies the claim. Main proof
Using a diagonal argument, we may find M ∈ [ N ] , sequence { α i } i ∈ N ∈ B ℓ such that α i,n n ∈ M −→ α i , ∀ i ∈ N and intervals { I i } i ∈ N such that I i = w ∗ − lim n ∈ M I ∗ i,n . Theintervals { I i } i ∈ N are clearly pairwise disjoint.We define k ∗ w ∗ = P ∞ n =1 α i I ∗ i ∈ K ∗ and we will show that k ∗ w ∗ = lim n ∈ M k ∗ n . Indeed, consider s ∈ < N and ǫ >
0. We pick N ∈ N such that( ∞ X i = N +1 α i ) / < ǫ . Then there is n ∈ N such that for any n ≥ n , there holds N X i =1 | α i,n I ∗ i,n ( e s ) − α i I ∗ i ( e s ) | < ǫ , and | α N +1 ,n − α N +1 | < ǫ . Then for any n ∈ M with n ≥ n , we have | ∞ X i =1 α i,n I ∗ i,n ( e s ) − ∞ X i =1 α i I ∗ i ( e s ) | ≤ N X i =1 | α i,n I ∗ i,n ( e s ) − α i I ∗ i ( e s ) | + | ∞ X i = N +1 α i,n I ∗ i,n ( e s ) | + | ∞ X i = N +1 α i I ∗ i ( e s ) | < ǫ ∞ X i = N +1 α i ) / ( ∞ X i = N +1 | I ∗ i ( e s ) | ) / + | α j,n | , for some j ≥ N + 1 < ǫ ǫ | α N +1 ,n |≤ ǫ | α N +1 ,n − α N +1 | + | α N +1 |≤ ǫ ǫ ǫ ǫ. Therefore k ∗ n ( e s ) n ∈ M −→ k ∗ ( e s ) ∀ s ∈ < N and the result is proved. (cid:3) N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 15
We will now use some measure theoretic arguments. Let ( X, A , µ ) a signed measure spaceand f ∈ L ( | µ | ), where | µ | is the total variation of µ . The integral of f with respect to µ isdefined as Z X f dµ = Z X f dµ + − Z X f dµ − . Dominated Convergence Theorem implies that if we consider a sequence of measurablefunctions f n : ( X, A ) → R with f = lim n →∞ f n a.e., such that there is g ∈ L ( | µ | ) with | f n | ≤ g ∀ n ∈ N a.e. in X, then lim n →∞ Z X | f n − f | dµ = 0 ⇒ lim n →∞ Z X f n dµ = Z X f dµ. We will now us a special form of Riesz’s Representation Theorem, whose proof can be foundin [6]. Let X be a topological space. We will write M f,r ( X ) for the set of all finite, regular,signed Borel measures of X and C ( X ) for the Banach space of continuous real functions on X , equipped with the k · k ∞ norm. Theorem 4.3. (Riesz’s Representation Theorem) Let X be a compact Hausdorff space.Then for any f ∗ ∈ C ∗ ( X ) there is unique µ f ∗ ∈ M f,r ( X ) such that f ∗ ( f ) = Z X f dµ f ∗ , ∀ f ∈ C ( X ) . Proposition 4.4.
Let { x n } n ∈ N be a bounded sequence in J T . If the sequence { I ∗ ( x n ) } n ∈ N converges for any interval I then { x n } n ∈ N is w -Cauchy.Proof. Let M = sup n ∈ N {|| x n ||} . We define T : J T → C ( K ) by T ( x ) = b x | K . Clearly T is awell-defined linear isometry. Indeed, || T ( x ) || ∞ = || b x | K || ∞ = sup {| b x ( x ∗ ) | : x ∗ ∈ K ∗ } = sup {| x ∗ ( x ) | : x ∗ ∈ K ∗ } = || x || , by Proposition 4.1. Therefore the conjugate map T ∗ : C ∗ ( K ) → J T ∗ is surjective. So, forany x ∗ ∈ J T ∗ , there is f ∗ ∈ C ∗ ( K ) such that x ∗ = f ∗ ◦ T . Riesz’s Representation Theoremimplies that for any x ∗ ∈ J T ∗ , there is unique µ x ∗ ∈ M f,r ( K ) such that for any n ∈ N , wehave x ∗ ( x n ) = Z K b x n | K , dµ x ∗ . Since || b x n | K || ≤ M for any n ∈ N and | µ x ∗ | ( K ) < ∞ for any x ∗ ∈ J T ∗ , DominatedConvergence Theorem yields it is enough to show that { x ∗ ( x n ) } n ∈ N converges for all x ∗ ∈ K ∗ . Indeed, let x ∗ w ∗ = P ∞ i =1 λ i I ∗ i ∈ K ∗ . By assumption we may write α i = lim n →∞ I ∗ i ( x n ) . Then ( P ∞ i =1 α i ) / ≤ M and P ∞ i =1 | λ i α i | ≤ M , by Cauchy-Schwartz inequality. Hence, theseries P ∞ i =1 λ i α i is absolutely convegent. Consider ǫ > N ∈ N such that( ∞ X i = N +1 λ i ) / < ǫ M , and | ∞ X i = N +1 λ i α i | < ǫ . Fixing n ∈ N such that for any n ≥ n there holds N X i =1 | λ i I ∗ i ( x n ) − λ i α i | < ǫ , ∀ n ≥ n , we obtain | x ∗ ( x n ) − ∞ X i =1 λ i α i | ≤ N X i =1 | λ i I ∗ i ( x n ) − λ i α i | + | ∞ X i = N +1 λ i I ∗ i ( x n ) | + | ∞ X i = N +1 λ i α i | < ǫ ∞ X i = N +1 λ i ) / ( ∞ X i = N +1 | I ∗ i ( x n ) | ) / + ǫ < ǫ ǫ ǫ ǫ. The proof is complete. (cid:3)
We will use the following notation. Given a countable set M , we will write [ M ] to denotethe set of infinite subsets of M . Let us first prove the following Lemma. Lemma 4.5.
Let X = ∅ and f n : X → R sequence of functions such that sup n ∈ N {| f n ( x ) |} < ∞ ∀ x ∈ X. If for any ǫ > and M ∈ [ N ] there is L ∈ [ M ] with lim sup n ∈ L f n ( x ) − lim inf n ∈ L f n ( x ) < ǫ, ∀ x ∈ X, the sequence { f n } n ∈ N has pointwise subsequence.Proof. By induction we will construct a decreasing sequence { L k } k ∈ N of infinite subsets of N such that for any k ∈ N we havelim sup n ∈ L k f n ( x ) − lim inf n ∈ L k f n ( x ) < k , ∀ x ∈ X. Consider a strictly increasing sequence { n k } k ∈ N such that n k ∈ L k ∀ k ∈ N and the set L ∞ = { n k : k ∈ N } . Clearly L ∞ is infinite and for any k ∈ N we have L ∞ ⊆ L k . Then forall x ∈ X we have thatlim sup n ∈ L ∞ f n ( x ) − lim inf n ∈ L ∞ f n ( x ) ≤ lim sup n ∈ L k f n ( x ) − lim inf n ∈ L k f n ( x ) < k , ∀ k ∈ N , so letting k → ∞ , we obtain lim sup n ∈ L ∞ f n ( x ) = lim inf n ∈ L ∞ f n ( x ) , thus the sequence ( f n ) n ∈ L ∞ is pointwise convergent. (cid:3) We are now able to prove the non-embedding of ℓ in J T . Theorem 4.6. ℓ does not embed in J T . N THE ℓ NON-EMBEDDING IN THE JAMES TREE SPACE 17
Proof.
Assume ℓ embeds in J T . Then Proposition 2.21 implies there is block of the basisof
J T equivalent to the standard basis of ℓ . We will show that any block has w-Cauchysubsequence, which contradicts the ℓ -Theorem. Let { u n } n ∈ N a block and let us write M =sup n ∈ N {|| u n ||} . We will show there is subsequence { u n k } k ∈ N such that { I ∗ ( u n k } k ∈ N convergesfor any interval I and the contradiction will come by Proposition 4.4. For segments, using adiagonal argument, we may find K ∈ [ N ] such that { S ∗ ( u n ) } n ∈ K converges for any segment S . Therefore, using Lemma 4.5, it suffices to show that for any ǫ > M ∈ [ K ], thereis L ∈ [ M ] such that lim sup n ∈ L σ ∗ ( u n ) − lim inf n ∈ L σ ∗ ( u n ) ≤ ǫ, ∀ σ ∈ N . Arguing by contradiction, assume there is ǫ > M ∈ [ K ], such that for any L ∈ [ M ],there is σ L ∈ N withlim sup n ∈ L σ ∗ L ( u n ) − lim inf n ∈ L σ ∗ L ( u n ) > ǫ ⇒ lim sup n ∈ L σ ∗ L ( u n ) + lim inf n ∈ L σ ∗ L ( u n ) > ǫ . (8)We pick k ∈ N such that k ǫ > M . Consider L ∈ [ M ] and σ ∈ N such that (8)holds. Then at least one of lim sup n ∈ L σ ∗ ( u n ) and lim inf n ∈ L σ ∗ ( u n ) is greater than ǫ L ∈ [ L ] such that lim n ∈ L σ ∗ ( u n ) > ǫ σ ∈ N which satisfies (8). It isclear that σ = σ . Continuing inductively we may find L k ⊆ L k − ⊆ , ..., ⊆ L ∈ N and σ , ..., σ k ∈ N pairwise distinct such that lim n ∈ L i σ ∗ i ( u n ) > ǫ i = 1 , ..., k , thuslim n ∈ L k σ ∗ i ( u n ) > ǫ , ∀ i = 1 , ..., k . Therefore there is N ∈ L k such that for any i = 1 , ..., k ,we have σ ∗ i ( u n ) > ǫ , ∀ n ∈ L k : n ≥ N. But σ , ..., σ k finally separate, since they are pairwise distinct and the sequence { u n } n ∈ N isblock, so we may find n ∈ L k with n ≥ N such that || u n || ≥ k X i =1 σ ∗ i ( u n ) > k ǫ > M . But this contradicts the boundness assumption on { u n } n ∈ N . The proof is complete. (cid:3) Corollary 4.7.
Every bounded sequence in
J T has w − Cauchy subsequence.Proof.
It comes immediately by the fact that ℓ does not embed in J T and the ℓ -Theorem. (cid:3) References [1] P. Enflo,
A counterexample to the approximation property in Banach spaces , Acta Mathematica, vol.130, pp. 309-317, 1973.[2] R. James,
A non-reflexive space isometric with its second conjugate , Proc. Nat. Acad. Sci. U.S.A., pp.174-177, 1951.[3] R. James,
A separable somewhat reflexive Banach space with non-separable dual , Bull. Amer. Math.Soc., vol. 80, pp. 738-743, 1974. [4] J. Lindenstrauss and L. Tzafriri,
Classical Banach spaces I and II , Springer, 1996.[5] H.P. Rosenthal,
A characterization of Banach spaces containing ℓ , Proc. Nat. Acad. Sci. 71 (1974),241, 2413.[6] W. Rudin, Real and Complex Analysis , McGraw-Hill, New York, 1966.
Ioakeim Ampatzoglou, Department of Mathematics, The University of Texas at Austin.
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