α -Large Families and Applications to Banach Space Theory
aa r X i v : . [ m a t h . F A ] N ov α -LARGE FAMILIES AND APPLICATIONS TO BANACHSPACE THEORY SPIROS A. ARGYROS AND PAVLOS MOTAKIS
Abstract.
The notion of α -large families of finite subsets of an infiniteset is defined for every countable ordinal number α , extending the knownnotion of large families. The definition of the α -large families is based onthe transfinite hierarchy of the Schreier families S α , α ă ω . We provethe existence of such families on the cardinal number 2 ℵ and we studytheir properties. As an application, based on those families we constructa reflexive space X α ℵ , α ă ω with density the continuum, such thatevery bounded non norm convergent sequence t x k u k has a subsequencegenerating ℓ α as a spreading model. Introduction
One of the most significant examples of Banach spaces is Tsirelson space(see [7], [14]), presented in the seventies. The main property of this space,is that it fails to contain a copy of c or ℓ p , answering in the negative aproblem posed by Banach. It is still an open problem whether there existTsirelson type spaces in the non-separable setting. A version of this problemhas recently been solved in the negative direction in [10], namely it is shownthat spaces spanned by an uncountable basic sequence such that their normsatisfies an implicit formula, similar to the one of Tsirelson space (see [7]),always contains a copy of c or ℓ p . To be more precise, if κ is an uncountableordinal number, B is a hereditary and compact family of finite subsets of κ ,0 ă θ ă } ¨ } θ, B is the unique norm defined on c p κ q satisfying the following implicit formula } x } θ, B “ max } x } , sup t θ n ÿ i “ } E i x } θ, B : t E i u di “ is B ´ admissible u ( then the completion of p c p κ q , } ¨ } θ, B q contains a copy of c or ℓ p . Primary 46B03, 46B06, 46B26, 03E05
Key words:
Spreading models, Reflexive spaces, Non separable Banach spaces, LargefamiliesResearch supported by APIΣTEIA program/1082.
As it seems not possible to have a non separable space, that stronglyresembles Tsirelson space, a natural question is which properties of this spacecan be transferred to the non separable setting. Besides being reflexive, oneof the main properties of Tsirelson space, is that it admits only ℓ as aspreading model, i.e. every bounded sequence without a norm convergentsubsequence has a subsequence that generates a spreading model equivalentto the usual basis of ℓ . The main goal of this paper is the constructionof a non separable reflexive Banach space X ℵ , with the aforementionedproperty. Theorem.
There exists a reflexive Banach space X ℵ generated by an un-conditional basic sequence t e ξ u ξ ă ℵ , admitting only ℓ as a spreading model.The construction of this space is based on the notion of α -large families,which is defined as follows. If A is an infinite set, B is a hereditary andcompact family of finite subsets of A and α is a countable ordinal number,we say that B is α -large, if its restriction on every infinite subset of A ,in a certain sense, contains a copy of S α , the Schreier family of order α .Equivalently, if its restriction on every infinite subset of A , has Cantor-Bendixson index, greater than or equal to ω α `
1. We prove the existenceof such families on the cardinal number 2 ℵ , by constructing for α ă ω , G α an α -large, hereditary and compact family of finite subsets of t , u N .We believe that these families are of independent interest, as they retainsome of the most important properties of the families S α , α ă ω . They aretherefore a generalization of the Schreier families, defined on the continuumand a study of them is included in the paper.In the first section of the paper, we define the notion of α -large familiesof finite subsets of an infinite set and a brief study of them is given.The second section is devoted to the construction of the families t G α u α ă ω .Initially, using the Schreier family S and diagonalization, we recursively de-fine some auxiliary families G α , α ă ω , which are subsets of rt , u N s ă ω ˆt , u N . The construction method used, imposes strong Schreier like prop-erties on the families G α , which are in fact the projection of G α , on thecomponent rt , u N s ă ω . Next, properties of these families, which are crucialfor the proof of the main result are included, among others, the fact that for α ă ω , G α is an α -large, compact and hereditary family of finite subsets of t , u N . Some additional results concerning the similarity of the G α to the S α , α ă ω are proven.The third section is concentrated on the construction of the space X ℵ .The first step is the definition of a sequence of spaces tp X n , }¨} n qu n , each onebased on the family G n . In particular, the norm of these spaces is definedon c p ℵ q in a similar manner as the norm of Schreier space is definedon c p N q (see [12]) and they all have the unit vector basis t e ξ u ξ ă ℵ as anunconditional Schauder basis. For n P N , the main two properties of thespace X n are the following. Firstly, every subsequence of the basis admitsonly ℓ as a spreading model and secondly the space X n is c saturated. -LARGE FAMILIES 3 Next, using the spaces X n , n P N and Tsirelson space T , a norm is definedon c p ℵ q , in the following manner. For x P c p ℵ q , set } x } “ } ÿ n “ n } x } n e n } T . The completion of c p ℵ q with this norm is the desired space X ℵ , whichhas the unit vector basis t e ξ u ξ ă ℵ as an unconditional Schauder basis. Theproof of the fact that this space admits only ℓ as a spreading model, relieson the study of the behavior of the } ¨ } n norms on a normalized weaklynull sequence t x k u k in X ℵ . Moreover, using the fact that the spaces X n are c saturated, we prove that every subspace of X ℵ contains a copy of asubspace of T , which yields that the space is reflexive.The fourth and final section concerns the construction, for α ă ω , ofreflexive spaces X α ℵ having an unconditional Schauder basis with size 2 ℵ ,admitting ℓ α as a unique spreading model. The construction method usedis a variation of the one used for the space X ℵ .1. α -large families We introduce the notion of α -large families which concerns the complex-ity of a family B of finite subsets of a given infinite set A . This notionextends the well known concept of large families and it is defined using thetransfinite hierarchy of the Schreier families t S α u α ă ω , first introduced in[1]. After providing the definition of α -large families we also give a usefulcharacterization linking this notion with the Cantor-Bendixson index of acompact and hereditary family of finite subsets of a given infinite set. Notation.
Let A be a set, B be a family of subsets of A , B be a subset of A and k be a natural number. We define r B s k “ t F Ă B : F “ k u and B æ B “ t F P B : F Ă B u . If F is a family of subsets of the natural numbers, L is an infinite subsetof N and φ : N Ñ L is the uniquely defined order preserving bijection, wedefine F r L s “ t φ p F q : F P F u . Definition 1.1.
Let A be an infinite set and B a family of finite subsets of A . (i) We say that B is large, if for every k P N , and B infinite subset of A , we have that r B s k X B ‰ ∅ .(ii) Given a countable ordinal number α , we say that B is α -large, if forevery B infinite subset of A , there exists a one to one map φ : N Ñ B ,such that φ p F q P B , for every F P S α . S. A. ARGYROS, P. MOTAKIS
Remark 1.2.
Using Ramsey theorem and a simple diagonalization argu-ment, it is easy to see that B is large, if and only if it is 1-large.The following lemma is an easy consequence of Theorem 1 from [8]. Lemma 1.3.
Let F , G be hereditary and compact families of finite subsetsof the natural numbers, such that for every L infinite subset of the naturalnumbers, the Cantor-Bendixson index of F æ L , is strictly smaller than theCantor-Bendixson index of G æ L . Then for every M infinite subset of thenatural numbers, there exists L a further infinite subset of M , such that F æ L Ă G æ L . Proposition 1.4.
Let A be an infinite set, B be a hereditary and compactfamily of finite subsets of A and α be a countable ordinal number. Then,the following assertions are equivalent.(i) B is α -large.(ii) For every B infinite subset of A , the Cantor-Bendixson index of B æ B is greater than or equal to ω α ` Proof.
Given that (i) holds, (ii) is an immediate consequence of the fact thatthe Candor-Bendixson index of S α is equal to ω α ` α (see Proposition 4.10 from [1]).For the converse, we may clearly assume that B is a hereditary and com-pact family of finite subsets of the natural numbers. For a given countableordinal α , if (ii) holds, we shall prove the following statement.For every infinite subset of the natural numbers M , there exists L aninfinite subset of M , such that S α r L s Ă B .The desired result evidently follows from the above. To prove this state-ment, we distinguish three cases. Case 1: α “
1. Assume that for every infinite subset of the natural numbers M , the Cantor-Bendixson index of B æ M is infinite. This means that everysuch M contains as subsets elements of B , of unbounded cardinality. Since B is hereditary, we conclude that it is large and therefore it also is 1-large. Case 2: α is a limit ordinal number. Then there is t β k u k a strictly increasingsequence of ordinal numbers with sup k β k “ α , such that S α “ Y k t F P S β k :min F ě k u .Using Lemma 1.3, choose L Ą ¨ ¨ ¨ Ą L k Ą ¨ ¨ ¨ infinite subsets of M , suchthat S β k æ L k Ă B , for all k .Choose L “ t ℓ ă ¨ ¨ ¨ ă ℓ k ă ¨ ¨ ¨ u an infinite subsets of M , with ℓ m P L k ,for every m ě k . It is not hard to check that S α r L s Ă B . Case 3: α is a successor ordinal number. If α “ β `
1, then the followingholds.For every M infinite subset of the naturals and n P N , there exists L afurther infinite subset of M , such that p S β ˚ A n q æ L Ă B , where S β ˚ A n “ tY ni “ F i : F i P S β , i “ , . . . , n u -LARGE FAMILIES 5 The above statement follows form Lemma 1.3 and the fact that theCantor-Bendixson index of S β ˚ A n is equal to ω β n ` ă ω α .Therefore, given M an infinite subset of the natural numbers, we maychoose L Ą ¨ ¨ ¨ Ą L n Ą . . . infinite subsets of M such that p S β ˚ A n q æ L n Ă B . Choose L “ t ℓ ă ¨ ¨ ¨ ă ℓ n ă ¨ ¨ ¨ u an infinite subsets of M , with ℓ m P L n ,for every m ě n . Once more, it is not hard to check that S α r L s Ă B . (cid:3) A transfinite sequence of compact and hereditary familiesof finite subsets of t , u N In this section we define a transfinite sequence G α , α ă ω of compact andhereditary families of finite subsets of t , u N with each G α being α -large for α ă ω . We shall first recursively define an auxiliary transfinite sequence t G α u α ă ω of subsets of rt , u N s ă ω ˆt , u N , which will then be used to definethe G α for α ă ω . We then prove the main properties of these families andwe conclude this section by showing the G α have some similar properties tothe Schreier families S α . Notation.
For σ “ t σ p i qu i “ and τ “ t τ p i qu i “ in t , u N , we define σ ^ τ and | σ ^ τ | as follows.(i) σ ^ τ “ σ and | σ ^ τ | “ 8 , if σ “ τ .(ii) σ ^ τ “ ∅ and | σ ^ τ | “
0, if σ p q ‰ τ p q .(iii) σ ^ τ “ t σ p i qu ℓi “ and | σ ^ τ | “ ℓ , if σ ‰ τ, σ p q “ τ p q and ℓ “ min t i P N : σ p i ` q ‰ τ p i ` qu For s “ t s p i qu ki “ and t “ t t p i qu ℓi “ finite sequences of 0’s and 1’s, we saythat s is an initial segment of t and write s Ď t , if k ď ℓ and s p i q “ t p i q for i “ , . . . , k . We say that s is a proper initial segment of t and write s Ł t ,if s Ď t and s ‰ t . Definition 2.1.
We define G to be all pairs p F, σ q , where F “ t τ i u di “ Prt , u N s ă ω , d P N and σ P t , u N , such that the following are satisfied.(i) σ ‰ τ i for i “ , . . . , d (ii) σ ^ τ ‰ ∅ and if d ą
1, then σ ^ τ Ł σ ^ τ Ł ¨ ¨ ¨ Ł σ ^ τ d (iii) d ď | σ ^ τ | Define Ą min p F, σ q “ | σ ^ τ | and Ą max p F, σ q “ | σ ^ τ d | .Assume that α is a countable ordinal number, G β have been defined for β ă α and that for p F, σ q P G β , Ą min p F, σ q and Ą max p F, σ q have also beendefined. Definition 2.2.
Let β ă α , p F i , σ i q di “ , d P N be a finite sequence of elementsof G β and σ P t , u N .We say that p F i , σ i q di “ is a skipped branching of σ in G β , if the followingare satisfied.(i) The F i , i “ , . . . , d are pairwise disjoint S. A. ARGYROS, P. MOTAKIS (ii) σ ‰ σ i for i “ , . . . , d (iii) σ ^ σ ‰ ∅ and if d ą
1, then σ ^ σ Ł σ ^ σ Ł ¨ ¨ ¨ Ł σ ^ σ d (iv) | σ ^ σ i | ă Ą min p F i , σ i q for i “ , . . . , d (v) d ď | σ ^ σ | Definition 2.3.
Let β ă α , σ P t , u N and p F i , σ q di “ , d P N be a finitesequence of elements of G β .We say that p F i , σ q di “ is an attached branching of σ in G β if the followingare satisfied.(i) The F i , i “ , . . . , d are pairwise disjoint(ii) If d ą
1, then Ą max p F i , σ q ă Ą min p F i ` , σ q , for i “ , . . . , d ´ d ď Ą min p F , σ q We are now ready to define G α , distinguishing two cases. Definition 2.4. If α is a successor ordinal number with α “ β `
1, we define G α to be all pairs p F, σ q , where F P rt , u N s ă ω and σ P t , u N , such thatone of the following is satisfied.(i) p F, σ q P G β .(ii) There is p F i , σ i q di “ a skipped branching of σ in G β such that F “Y di “ F i .In this case we say that p F, σ q is skipped . Moreover set Ą min p F, σ q “| σ ^ σ | and Ą max p F, σ q “ | σ ^ σ d | .(iii) There is p F i , σ q di “ an attached branching of σ in G β such that F “Y di “ F i .In this case we say that p F, σ q is attached . Moreover set Ą min p F, σ q “ Ą min p F , σ q and Ą max p F, σ q “ Ą max p F d , σ q .If α is a limit ordinal number, fix t β n u n a strictly increasing sequence ofordinal numbers with sup n β n “ α . We define G α “ ď n “ p F, σ q P G β n : Ą min p F, σ q ě n ( Remark 2.5. If α is a limit ordinal number, the sequence t β n u n may chosenin such a manner that both G α “ ď n “ p F, σ q P G β n : Ą min p F, σ q ě n ( and S α “ ď n “ F P S β n : min F ě n ( From now on, we shall assume that this is the case.
Remark 2.6.
Translating Definitions 2.1, 2.2, 2.3 and 2.4 one obtains thefollowing. -LARGE FAMILIES 7 (i) If p F, σ q P G , then F ď Ą min p F, σ q .(ii) If p F, σ q P G β ` and p F i , σ i q di “ is a skipped branching of σ in G β such that F “ Y di “ F i , then we have that d ď Ą min p F, σ q .(iii) If p F, σ q P G β ` and p F i , σ q di “ is an attached branching of σ in G β such that F “ Y di “ F i , then we have that d ď Ą min p F, σ q .We now proceed to prove some key properties of the families G α . Lemma 2.7.
Let σ, σ , τ P t , u N , not all equal. The following are equiva-lent.(i) σ ^ τ Ł σ ^ σ (ii) σ ^ τ “ σ ^ τ . Proof.
Assume that (i) holds. We have that τ p j q “ σ p j q “ σ p j q , for j “ , . . . , | σ ^ τ | . Whereas, for j “ | σ ^ τ | `
1, we have that τ p j q ‰ σ p j q “ σ p j q .Therefore, | σ ^ τ | “ | σ ^ τ | , which means that σ ^ τ “ σ ^ τ .The inverse is proved similarly. (cid:3) Lemma 2.8.
Let α be a countable ordinal number and p F, σ q P G α . Thenthere exist τ m , τ M in F such that the following are satisfied.(i) Ą min p F, σ q “ | σ ^ τ m | and Ą max p F, σ q “ | σ ^ τ M | (ii) For τ P F we have that σ ^ τ m Ď σ ^ τ Ď σ ^ τ M Moreover, if α is a successor ordinal number with α “ β ` p F, σ q is skipped and p F i , σ i q di “ is a skipped branching of σ in G β such that F “ Y di “ F i , then for i “ , . . . , d and τ P F i , we have that σ ^ σ i “ σ ^ τ .(iv) If p F, σ q is attached and p F i , σ q di “ is an attached branching of σ in G β such that F “ Y di “ F i , then for 1 ď i ă j ď d and τ P F i , τ P F j ,we have that σ ^ τ Ł σ ^ τ . Proof.
We prove this lemma by transfinite induction. For α “ G . Assume now that α is acountable ordinal number and that the statement holds for every p F, σ q P G β ,for every β ă α . If α is a limit ordinal number, then the result followstrivially from the inductive assumption and the definition of G α . Assumetherefore that α “ β ` p F, σ q P G α .We treat first the case when p F, σ q is skipped. Let p F i , σ i q di “ be a skippedbranching of σ in G β , such that F “ Y di “ F i .We first prove part (iii), i.e. for τ P F i , we have that σ ^ σ i “ σ ^ τ , i “ , . . . , d .By the inductive assumption, there exist τ im P F i such that Ą min p F i , σ i q “| σ ^ τ im | and for every τ P F i we have that σ i ^ τ im Ď σ i ^ τ .Since, by definition, | σ ^ σ i | ă Ą min p F i , σ i q “ | σ i ^ τ im | ď | σ i ^ τ | , it followsthat σ ^ σ i Ł σ i ^ τ and by Lemma 2.7 σ ^ σ i “ σ ^ τ . S. A. ARGYROS, P. MOTAKIS
Choosing any τ m P F and τ M P F d , it is easy to see that (i) and (ii) aresatisfied.Assume now that p F, σ q is attached. Let p F i , σ q di “ be an attached branch-ing of σ in G β , such that F “ Y di “ F i .By the inductive assumption, there exist τ im , τ iM P F i such that Ą min p F i , σ q “| σ ^ τ im | , Ą max p F i , σ q “ | σ ^ τ iM | and for every τ P F i we have that σ ^ τ im Ď σ ^ τ Ď σ ^ τ iM .We will show that for 1 ď i ă j ď d , we have that σ ^ τ iM Ł σ ^ τ jm . Thisproves both (iv) and that τ m “ τ m , τ M “ τ dM have the desired properties.However, this follows immediately from the fact that | σ ^ τ iM | “ Ą max p F i , σ q ă Ą min p F j , σ q “ | σ ^ τ jm | . (cid:3) The following result is an immediate consequence of Lemma 2.8.
Corollary 2.9.
Let α be a countable ordinal number and p F, σ q P G α . Thenthe following hold.(i) Ą min p F, σ q “ min t| σ ^ τ | : τ P F u (ii) Ą max p F, σ q “ max t| σ ^ τ | : τ P F u Corollary 2.10.
Let α be a countable ordinal number and p F, σ q P G α ,such that F ě
2. Then Ą min p F, σ q ď min t| τ ^ τ | : τ , τ P F with τ ‰ τ u Proof.
Let τ ‰ τ be in F . By Lemma 2.8, there exists τ m P F , such that Ą min p F, σ q “ | σ ^ τ m | and σ ^ τ m Ď σ ^ τ as well as σ ^ τ m Ď σ ^ τ . Itfollows that σ ^ τ m Ď τ ^ τ . We conclude that Ą min p F, σ q ď | τ ^ τ | . (cid:3) Lemma 2.11.
Let α be a countable ordinal number and p F, σ q P G α , suchthat F ě
2. Then there exists σ P t , u N , such that p F, σ q P G α and Ą min p F, σ q “ min t| τ ^ τ | : τ , τ P F with τ ‰ τ u Proof.
We prove this lemma by transfinite induction on α . Assume that α “ p F, σ q P G , such that F ě F “ t τ i u di “ , d ě σ ^ τ Ł σ ^ τ and byLemma 2.7 we have that σ ^ τ “ τ ^ τ . We conclude that Ą min p F, σ q “| σ ^ τ | “ | τ ^ τ | . Corollary 2.10 yields that Ą min p F, σ q “ min t| τ ^ τ | : τ , τ P F with τ ‰ τ u and hence, the desired σ is σ itself.Assume now that α is a countable ordinal number and that the conclusionholds for every β ă α .If α is a limit ordinal number, choose t β n u n a strictly increasing sequenceof ordinal numbers with sup n β n “ α , such that the assumptions of Defini-tion 2.4 are satisfied. Let p F, σ q P G α with F ě
2. Then there is n P N such -LARGE FAMILIES 9 that p F, σ q P G β n and Ą min p F, σ q ě n . Corollary 2.10 yields the following.(1) min t| τ ^ τ | : τ , τ P F with τ ‰ τ u ě n By the inductive assumption, there exists σ P t , u N with p F, σ q P G β n and Ą min p F, σ q “ min t| τ ^ τ | : τ , τ P F with τ ‰ τ u . By (1) we have that Ą min p F, σ q P G α .Assume now that α is a successor ordinal number with α “ β ` p F, σ q P G α with F ě
2. If p F, σ q P G β , then the inductive assumptionyields the desired result. If this is not the case, then p F, σ q is either skipped,or attached. If it is attached, then there is p F i , σ i q di “ an attached branchingof σ , such that F “ Y di “ F i . If d “
1, then p F, σ q P G β and by the inductiveassumption we are done. Otherwise, choose τ P F , τ P F . Lemma 2.8 (iii)yields that σ ^ τ “ σ ^ σ Ł σ ^ σ “ σ ^ τ and by Lemma 2.7 we have that σ ^ τ “ τ ^ τ . We conclude that Ą min p F, σ q “ | σ ^ σ | “ | σ ^ τ | “ | τ ^ τ | and therefore, applying Corollary 2.10 we have that σ is the desired σ .If on the other hand p F, σ q is attached, using similar reasoning, Lemma2.8 (iv) and Corollary 2.9, we conclude the desired result. (cid:3) Corollary 2.12.
Let tp F k , σ k qu k be a sequence in Ť β ă ω G β such thatlim k Ą min p F k , σ k q “ 8 . Then, if F is an accumulation point of t F k u k , wehave that F ď Proof.
Let F be an accumulation point of t F k u k , and assume that thereare τ ‰ τ in F . Then there exists L an infinite subset of the naturalnumbers, such that τ , τ P F k , for every k P L . Corollary 2.10 yields that | τ ^ τ | ě Ą min p F k , σ k q , for all k P L . We conclude that | τ ^ τ | “ 8 , i.e. τ “ τ , a contradiction. (cid:3) The following two lemmas will both be useful in the sequel.
Lemma 2.13.
Let α be a countable ordinal number and p F, σ q P G α . Letalso σ P t , u N , such that σ ^ τ “ σ ^ τ for all τ P F . Then the followinghold.(i) p F, σ q P G α (ii) Ą min p F, σ q “ Ą min p F, σ q and Ą max p F, σ q “ Ą max p F, σ q Proof.
We prove this lemma by transfinite induction. The case α “ G . Assume now that the result holds for every β ă α . The case where α is a limit ordinal number is trivial, assume thereforethat α “ β ` p F, σ q P G α , σ P t , u N such that the assumptionsof the lemma are satisfied. Notice that it is enough to show that (i) is true,since part (ii) of the conclusion follows immediately from (i) and Corollary2.9. We treat first the case when p F, σ q is skipped, i.e. there exists p F i , σ i q di “ a skipped branching of σ in G β , with F “ Y di “ F i . To show that p F, σ q P G α ,it suffices to show that p F i , σ i q di “ is a skipped branching of σ .Notice that it is enough to show that σ ^ σ i “ σ ^ σ i for i “ , . . . , d ,which, by Lemma 2.7, is equivalent to σ ^ σ i Ł σ ^ σ for i “ , . . . , d .Fix 1 ď i ď d and chose τ P F i . Lemma 2.8 (iii) yields that σ ^ σ i “ σ ^ τ “ σ ^ τ . Once more, Lemma 2.7 yields that σ ^ σ i “ σ ^ τ Ł σ ^ σ .Assume now that p F, σ q is attached, i.e. there exists p F i , σ q di “ an attachedbranching of σ in G β , with F “ Y di “ F i . Since, by the inductive assumption,the conclusion holds for the p F i , σ q , i “ , . . . , d, σ , it is straightforwardto check that p F i , σ q di “ an attached branching of σ in G β and therefore p F, σ q P G α . (cid:3) Lemma 2.14.
Let p F, σ q P Ť β ă ω G β and σ P t , u N such that σ ^ τ Ł σ ^ τ for all τ P F . Then, if α “ min t β : p F, σ q P G β u , α is not a limitordinal number and the following hold.(i) If α “
1, then F “ α “ β `
1, then there exists σ P t , u N with p F, σ q P G β . Proof.
The fact that α is not a limit ordinal number follows trivially fromDefinition 2.4. The case α “ α “ β `
1. Since p F, σ q R G β , it is either skipped or attached.Assume first that there is p F i , σ i q di “ a skipped branching of σ in G β with F “ Y di “ F i . If d “
1, then σ “ σ is evidently the desired elementof t , u N . We will therefore prove that d “
1. Towards a contradiction,assume that d ě τ P F , τ P F .Lemma 2.8 (iii) yields that σ ^ τ “ σ ^ σ Ł σ ^ σ “ σ ^ τ . Bythe assumption, σ ^ τ Ł σ ^ τ and using Lemma 2.7 we conclude that σ ^ τ “ σ ^ σ . Similarly, we conclude that σ ^ τ “ σ ^ σ . We have shownthat σ ^ σ Ł σ ^ σ , which is absurd.If p F, σ q is attached, then using similar arguments and 2.8 (iv), one canprove the desired result. (cid:3) Proposition 2.15.
Let α be a countable ordinal number, p F, σ q P G α and G be a non empty subset of F . Then p G, σ q P G α . Proof.
We proceed by transfinite induction. For α “ G . Assume that the statement is true for every β ă α . The case when α is a limit ordinal number is an easy consequence ofthe inductive assumption and Corollary 2.9. Assume therefore that α “ β ` p F, σ q be in G α and G Ă F .Consider first the case, when p F, σ q is skipped and p F i q di “ be a skippedbranching of σ in G β , such that F “ Y di “ F i .Set t i ă ¨ ¨ ¨ ă i p u “ t i P t , . . . , d u : G X F i ‰ ∅ u and G j “ G X F i j for j “ , . . . , p . By the inductive assumption, p G j , σ i j q is in G β for j “ , . . . , p -LARGE FAMILIES 11 and, evidently, it is enough to show that p G j , σ i j q pj “ is a skipped branchingof σ .Obviously, assumptions (i), (ii) and (iii) from Definition 2.2 are satisfied.Corollary 2.9 yields that Ą min p F i j , σ i j q ď Ą min p G j , σ i j q and hence (iv) issatisfied. Moreover p ď d ď | σ ^ σ | ď | σ ^ σ i | , which means that (v) isalso satisfied.If on the other hand p F, σ q is attached, using similar reasoning and Corol-lary 2.9, the desired result can be easily proven. (cid:3) We are now ready to define the families G α , for α ă ω and prove theirmain properties. Definition 2.16.
For a countable ordinal number α we define G α “ t F Ă t , u N : there exists σ P t , u N with p F, σ q P G α u Y t ∅ u Remark 2.17.
It is clear that t G n u n ă ω is an increasing family of finitesubsets of t , u N . Proposition 2.15 also yields that G α is hereditary for all α ă ω . Proposition 2.18.
Let α be a countable ordinal number. Then G α is α -large. In particular, for every B infinite subset of t , u N there exists a oneto one map φ : N Ñ B with φ p F q P G α for every F P S α and α ă ω . Proof.
Let B be an infinite subset of t , u N . Choose t τ k u k pairwise disjointelements of B and σ P t , u N , with lim k τ k “ σ , such that σ ^ τ k Ł σ ^ τ k ` for all k P N . Define φ : N Ñ B , with φ p k q “ τ k .We shall inductively prove that for every α ă ω and F P S α , the followingholds.(i) p φ p F q , σ q P G α (ii) Ą min p φ p F q , σ q “ | σ ^ τ min F | and Ą max p φ p F q , σ q “ | σ ^ τ max F | The case α “ G . Assumenow that α is a countable ordinal number and that the statement is true forevery F P S β and β ă α .We treat first the case when α is a limit ordinal number. Choose t β n u n astrictly increasing sequence of ordinal numbers with sup n β n “ α , such that G α “ ď n “ p G, σ q P G β n : Ą min p G, σ q ě n ( as well as S α “ ď n “ F P S β n : min F ě n ( . Then, if F P S α , there exists n P N with F P S β n and min F ě n . Theinductive assumption yields that p φ p F q , σ q P G β n and Ą min p φ p F q , σ q “ | σ ^ τ min F | ě min F ě n . We conclude that p φ p F q , σ q P G α and, of course, Ą min p φ p F q , σ q “ | σ ^ τ min F | .Assume now that α “ β ` F P S α . Then there exist min F ď F ă ¨ ¨ ¨ ă F d in S β with F “ Y di “ F i .The inductive assumption yields that p φ p F i q , σ q di “ is an attached branch-ing of σ in G β and hence p φ p F q , σ q P G α .Moreover, Ą min p φ p F q , σ q “ Ą min p φ p F q , σ q “ | σ ^ τ min F | “ | σ ^ τ min F | .Similarly, we conclude that Ą max p φ p F q , σ q “ | σ ^ τ max F | . (cid:3) Remark 2.19.
With a little more effort, it can be proven that for α ă ω , G α is not α ` φ : N Ñ t , u N , such that φ p F q P G α , for every F P S α ` . To be even moreprecise, for every A infinite subset of t , u N , there exists B a countablesubset of A , such that the Cantor-Bendixson index of G α æ B is equal to ω α ` α ă ω . Since we do not make use of this fact, we omit theproof.The main result concerning the families G α , α ă ω is the following. Theorem 2.20.
Let α be a countable ordinal number. Then G α is an α -large, hereditary and compact family of finite subsets of t , u N . Proof.
All we need to prove, is that G α is compact and we do so by transfiniteinduction. Let us first treat the case α “ F is in the closureof G .If F is finite, since G is hereditary, then F P G . It is therefore sufficientto show that F cannot be infinite. Since G is hereditary, we may assumethat F is countable and let t τ i : i P N u be an enumeration of F .We conclude, that setting F k “ t τ i : i “ , . . . , k u , then F k P G and F k “ k . Choose t σ k u k a sequence in t , u N such that p F k , σ k q P G for all k . Remark 2.6 yields that k ď Ą min p F k , σ k q for all k . On the other hand,by Corollary 2.10 we have that Ą min p F k , σ k q ď | τ ^ τ | . We conclude that k ď | τ ^ τ | for all k P N , which is obviously not possible.Assuming now that α is a countable ordinal number such that G β is com-pact for every β ă α , we will show that the same is true for G α .We treat first the case in which α is a limit ordinal number. Fix t β n u n astrictly increasing sequence of ordinal numbers with sup n β n “ α such that G α “ ď n “ p F, σ q P G β n : Ą min p F, σ q ě n ( Let F be in the closure of G α . As previously, if F is finite then it is in G α and it is therefore enough to show that F cannot be infinite. Once more, wemay assume that F “ t τ i : i P N u . Setting F k “ t τ , . . . , τ k u , we have that F k P G α , therefore there exist t σ k u k , with p F k , σ k q P G α . -LARGE FAMILIES 13 Using Corollary 2.10 we have that Ą min p F k , σ k q ď | τ ^ τ | “ d . In otherwords, p F k , σ k q P G β nk , with n k ď d for all k . Passing, if necessary, toa subsequence, we have that p F k , σ k q P G β n , for all k . We conclude that F P G β n , in other words G β n is not compact, which is absurd.Assume now that α “ β `
1. Let F be in the closure of G α . As previously,it is enough to show that F cannot be infinite. Once more, we may assumethat F “ t τ i : i P N u .Set F k “ t τ i : i “ , . . . , k u , for all k . Then F k P G α , i.e. there exists σ k such that p F k , σ k q P G α . Setting d “ | τ ^ τ | , Corollary 2.10, yields thefollowing.(2) Ą min p F k , σ k q ď d for all k By Definition 2.4, Remark 2.6 and (2), for every k P N , there exist t F kj u m k j “ pairwise disjoint sets in G β , with F k “ Y m k j “ F kj and m k ď d . Passing to asubsequence, we may assume that m k “ m , for all k .By the compactness of G β , we may pass to a further subsequence and find G , G , . . . , G m P G β , such that lim k F kj “ G j , for j “ , . . . , m .We conclude that F “ lim k F k “ lim k pY mj “ F kj q “ Y mj “ G j . Since Y mj “ G j is a finite set, this cannot be the case. (cid:3) Although the initial motivation behind the definition of the G α familieswas the construction of a non-separable reflexive space with ℓ as a uniquespreading model, we believe that they are of independent interest, as theyretain many of the properties of the families S α . They are therefore a versionof these families, defined on the Cantor set t , u N . We present a few moreproperties the G α have in common with the S α . Lemma 2.21.
Let α ă β be countable ordinal numbers. Then there exists n P N such that tp F, σ q P G α : Ą min p F, σ q ě n u Ă G β . Proof.
Fix α a countable ordinal number. We prove this proposition bymeans of transfinite induction, starting with β “ α `
1. In this case theresult follows from the definition of G β , for n “ β is a countable ordinal number with α ă β , such thatthe statement holds for every α ă γ ă β . If β “ γ `
1, by the inductiveassumption, there exists n P N , such that tp F, σ q P G α : Ą min p F, σ q ě n u Ă G γ . Evidently, we also have that tp F, σ q P G α : Ą min p F, σ q ě n u Ă G β .If β is a limit ordinal number, fix t β k u k a strictly increasing sequence ofordinal numbers, such that β “ lim k β k and G β “ ď k p F, σ q P G β k : Ą min p F, σ q ě k ( Choose k P N with α ă β k . By the inductive assumption, there exists m P N , such that tp F, σ q P G α : Ą min p F, σ q ě m u Ă G β k . Setting n “ max t k , m u , we have the desired result. (cid:3) Lemma 2.22.
Let α ă β be countable ordinal numbers. Then there exists n P N Y t u such that G α Ă G β ` n . Proof.
Fix β a countable ordinal number. We proceed by transfinite induc-tion on α . In the case α “
1, it is easily checked that G Ă G β . Assume nowthat α is a countable ordinal with α ă β , such that the statement holds forevery γ ă α . If α “ γ `
1, then by the inductive assumption there exists n P N Y t u with G γ Ă G β ` n . We conclude that G α Ă G β `p n ` q . If α isa limit ordinal, fix t α k u k a strictly increasing sequence of ordinal numbers,such that α “ lim k α k and G α “ ď k p F, σ q P G α k : Ą min p F, σ q ě k ( Lemma 2.21 yields that there exists m P N with tp F, σ q P G α : Ą min p F, σ q ě m u Ă G β . The inductive assumption, yields that for k “ , . . . , m ´
1, thereexists n k P N Y t u with G α k Ă G β ` n k . Setting n “ max t m, n , . . . , n m ´ u ,it can be easily checked that G α Ă G β ` n . (cid:3) Proposition 2.23.
Let α ă β be countable ordinal numbers. Then thereexists n P N such that F P G α : F ě t| τ ^ τ | : τ , τ P F, τ ‰ τ u ě n ( Ă G β . Proof.
Let α ă β be countable ordinal numbers. Choose n P N such thatthe conclusion of Lemma 2.21 is satisfied. We show that this n is the desirednatural number. Let F P G α with F ě t| τ ^ τ | : τ , τ P F, τ ‰ τ u ě n . Then there exists σ P t , u N with p F, σ q P G α . Lemma 2.11 yieldsthat there exists σ P t , u N such that p F, σ q P G α and Ą min p F, σ q ě n . Bythe choice of n , we have that p F, σ q P G β , i.e. F P G β . (cid:3) The following proposition is an obvious conclusion of Lemma 2.22
Proposition 2.24.
Let α ă β be countable ordinal numbers. Then thereexists n P N Y t u such that G α Ă G β ` n .The following fact is proven in [10], Proposition 7.4. If κ is an infinitecardinal number, then there exists a large, hereditary and compact familyof finite subsets of κ , if and only if κ is not ω -Erd˝os. The following questionarises naturally. Question.
Let κ be an infinite cardinal number which is not ω -Erd˝os and α be a countable ordinal number. Does there exist an α -large, hereditaryand compact family of finite subsets of κ ? -LARGE FAMILIES 15 The space X ℵ In this section we define the space X ℵ and prove that it is reflexive, hasan unconditional Schauder basis of length the continuum and that it admitsonly ℓ as a spreading model. In the beginning we define a sequence of nonseparable spaces X n , n P N . Each one is defined using the family G n in asimilar manner as the Schreier family S is used to define the space in [12].Then the construction of X ℵ is presented, which combines the spaces X n and Tsirelson space, using a method first appeared in [6]. In the end theproperties of the space X ℵ are deduced by directly using the structure ofthe families G n .Before proceeding to the definition of the spaces X n and X ℵ , let us firstrecall the notion of ℓ α spreading models. Definition 3.1.
Let t x k u k be a sequence in a Banach space and α be acountable ordinal number. We say that t x k u k generates an ℓ α spreadingmodel, if there exists a constant c ą F P S α and everyreal numbers t λ k u k P F the following holds: } ÿ k P F λ k x k } ě c ÿ k P F | λ k | . Let us from now on fix a one to one and onto map τ Ñ ξ τ from t , u N to the cardinal number 2 ℵ . Definition 3.2.
For n P N define a norm on c p ℵ q in the following man-ner. (i) For n P N , we may identify an F P G n with a linear functional F : c p ℵ q Ñ R in the following manner. For x “ ř ξ ă ℵ λ ξ e ξ P c p ℵ q F p x q “ ÿ τ P F λ ξ τ (ii) For x P c p ℵ q define } x } n “ sup t| F p x q| : F P G n u Set X n to be the completion of p c p ℵ q , } ¨ } n q . Proposition 3.3.
Let n P N . Then the following hold.(i) The space X n is c saturated.(ii) The unit vector basis t e ξ u ξ ă ℵ is a normalized, suppression uncondi-tional and weakly null basis of X n , with the length of the continuum.(iii) Any subsequence of the unit vector basis admits only ℓ as a spread-ing model. Proof.
To prove (i), notice that since G n is compact and contains only finitesets, it is scattered. The main Theorem from [11] yields that C p G n q is c saturated. Evidently, the map T : X n Ñ C p G n q with T x p F q “ F p x q is anisometric embedding and therefore, X n is c saturated. Property (ii) follows from the fact that G n is hereditary and property (iii)is a consequence of the fact that G n is 1-large. (cid:3) Remark 3.4.
For a cardinal number κ and B a compact, hereditary andlarge family of finite subsets of κ , one may define a c saturated space X B ,in the same manner as in Definition 3.2. Then any subsequence of the unitvector basis t e ξ u ξ ă κ admits only ℓ as a spreading model. Therefore theproblem of finding a basic sequence of length κ , admitting only ℓ as aspreading model, is reduced to the existence of such a family B . As it isproven in [10], Proposition 7.4, this is equivalent to κ not being ω -Erd˝os.It is also worth noting, that for a given cardinal number κ , it is easyto construct a reflexive space X with a basis t e ξ u ξ ă κ , having the propertythat every subspace has a sequence admitting ℓ as a spreading model. Asproven in [5], [9] and [13], any space X with an unconditional basis, embedsas a complemented subspace in a space with a symmetric basis D . As notedin [3], the construction in [5] has the following additional property. Everysubspace of D contains a copy of a subspace of X . One may therefore embedTsirelson space T into a space D with a symmetric basis t e n u n , saturatedwith subspaces of T . Since this basis is symmetric, it may naturally beextended to a basis t e ξ u ξ ă κ to define a space X having the desired property.However, this space also admits spreading models not equivalent to ℓ . Forinstance, the basis itself being symmetric and not equivalent to ℓ , fails thisproperty.By T we denote Tsirelson space as defined in [7] and by t e n u n we denoteits usual basis. We are now ready to define the space X ℵ , using the spaces X n , Tsirelson space T and a method first appeared in [6]. Definition 3.5.
Define the following norm on c p ℵ q . For x P c p ℵ q} x } “ ›› ÿ n “ n } x } n e n ›› T Set X ℵ to be the completion of p c p ℵ q , } ¨ }q .Set λ “ } ř n “ n e n } T and for ξ ă ℵ , ˜ e ξ “ λ e ξ . Since t e ξ u ξ ă ℵ is nor-malized and suppression unconditional in X n , and t e n u n is 1-unconditionalin T, we conclude that t ˜ e ξ u ξ ă ℵ is a normalized suppression unconditionalbasis of X ℵ .For n P N define P n : X ℵ Ñ X n with P n x “ n x . Evidently P n is welldefined and } P n } ď
1, for all n P N .The main result is the following, which is a combination of Proposition3.16 and Corollary 3.18, which will be presented in the sequel. Theorem 3.6.
The space X ℵ is a non separable reflexive space with asuppression unconditional Schauder basis with the length of the continuum,having the following property. Every normalized weakly null sequence in -LARGE FAMILIES 17 X ℵ has a subsequence that generates an ℓ n spreading model, for every n P N . Lemma 3.7.
Let t ˜ e ξ k u k be a subsequence of the basis t ˜ e ξ u ξ ă ℵ of X ℵ .Then it has a subsequence that generates an ℓ n spreading model for every n P N . Proof.
Set B “ t τ : ξ τ “ ξ k for some k P N u . By Proposition 2.18 thereexists a one to one map φ : N Ñ B such that φ p F q P G n for every F P S n and n P N .Pass to L an infinite subset of the natural numbers such that the map˜ φ : L Ñ ℵ with ˜ φ p j q “ ξ φ p j q is strictly increasing. We will show that t ˜ e ξ φ p j q u j P L admits an ℓ n spreading model for every n P N .By unconditionality, it is enough to show that there are positive constants c n such that for every n P N , F P S n , F Ă L and t t j u j P F positive realnumbers, we have that } ÿ j P F t j ˜ e ξ φ p j q } ě c n ÿ j P F t j By definition, we have that } ř j P F t j ˜ e ξ φ p j q } ě λ n } ř j P F t j e ξ φ p j q } n and by thechoice of φ , we have that φ p F q P G n . Hence, φ p F qp ř j P F t j e ξ φ p j q q “ ř j P F t j which yields that } ř j P F t j e ξ φ p j q } n “ ř j P F t j .We finally conclude that } ř j P F t j ˜ e ξ φ p j q } ě λ n ř j P F t j (cid:3) Proposition 3.8.
Let t x k u k be a normalized, disjointly supported blocksequence of t ˜ e ξ u ξ ă ℵ , such that lim sup k } x k } ą
0. Then t x k u k has asubsequence that generates an ℓ n spreading model for every n P N . Proof.
By unconditionality, it is quite clear, that by passing, if necessary,to a subsequence of t x k u k , there exist ε ą t ˜ e ξ k u k a subsequence of t ˜ e ξ u ξ ă ℵ , such that for any λ , . . . , λ m real numbers, one has that } m ÿ k “ λ k x k } ą ε } m ÿ k “ λ k ˜ e ξ k } Lemma 3.7 yields the desired result. (cid:3)
Proposition 3.9.
Let t x k u k be a normalized block sequence in X ℵ , suchthat lim k } P n x k } n “
0, for all n P N . Then t x k u k has a subsequence equiv-alent to a block sequence in T . In particular, t x k u k has a subsequence thatgenerates an ℓ n spreading model for every n P N . Proof.
Using a sliding hump argument, it is easy to see, that passing, ifnecessary, to a subsequence of t x k u k , there exist t I k u k increasing intervalsof the natural numbers, such that if we set y k “ ř n P I k n } x k } n e n , then t x k u k is equivalent to t y k u k . (cid:3) Lemma 3.10.
Let t x k u k be a normalized, disjointly supported block se-quence of t ˜ e ξ u ξ ă ℵ , such that the following holds. There exist c ą , n P N , p F k , σ k q P G n for k P N and σ P t , u N satisfying the following.(i) | F k p x k q| ą c for all k P N .(ii) The F k are pairwise disjoint.(iii) σ ‰ σ k for all k P N .(iv) σ ^ σ k Ł σ ^ σ k ` for all k P N .(v) | σ ^ σ k | ă Ą min p F k , σ k q for all k P N .Then t x k u k generates an ℓ n spreading model for every n P N . Proof.
By changing the signs of the x k , we may assume that F k p x k q ą c forall k P N .Arguing in a similar manner as in the proof of Proposition 2.18 one caninductively prove that for every n P N and G P S n the following hold.(a) pY k P G F k , σ q P G n ` n (b) Ą min pY k P G F k , σ q “ | σ ^ σ min G | and Ą max pY k P G F k , σ q “ | σ ^ σ max G | Since t x k u k is unconditional, it is enough find positive constants c n ą
0, such that fixing G P S n and t λ k u k P G non negative reals, we have thefollowing. } ÿ k P G λ k x k } ą c n ÿ k P G λ k Properties (a) and (b), yield that F “ Y k P G F k P G n ` n . This means thefollowing. } ÿ k P G λ k x k } ě } P n ` n p ÿ k P G λ k x k q} n ` n “ n ` n } ÿ k P G λ k x k } n ` n ą c n ` n ÿ k P G λ k (cid:3) Lemma 3.11.
Let t x k u k be a normalized, disjointly supported block se-quence of t ˜ e ξ u ξ ă ℵ , such that the following holds. There exist c ą , n P N , σ P t , u N , a sequence t F k u k in G n satisfying the following.(i) | F k p x k q| ą c for all k P N .(ii) The sets F k are pairwise disjoint(iii) p F k , σ q P G n for all k P N (iv) Ą max p F k , σ q ă Ą min p F k ` , σ q , for all k P N Then t x k u k generates an ℓ n spreading model for every n P N . Proof.
The proof is identical to the proof of Lemma 3.10. (cid:3)
Lemma 3.12.
Let t x k u k be a sequence in X ℵ and n P N such thatlim k } P n x k } n “
0. Then for every ε ą k P N such thatfor every k ě k the following holds. | F p x k q| ă ε for every F P G n -LARGE FAMILIES 19 Proof.
Fix ε ą
0. Choose k P N , such that } P n x k } n “ n } x k } n ă n ε , forevery k ě k . By definition of the norm } ¨ } n , this means the following. | F p x k q| ă ε for every F P G n (cid:3) Lemma 3.13.
Let t x k u k be a normalized, disjointly supported block se-quence of t ˜ e ξ u ξ ă ℵ , such that lim k } x k } “ n P N such that lim sup k } P n x k } n ą
0. Assume moreover, that if n “ min t n :lim sup k } P n x k } n ą u , there exists c ą , σ P t , u N and t F k u k a sequencein G n satisfying the following.(i) | F k p x k q| ą c for all k P N .(ii) The sets F k are pairwise disjoint(iii) p F k , σ q P G n for all k P N Then t x k u k has a subsequence that generates an ℓ n spreading model forevery n P N . Proof.
We shall prove that for every k , m natural numbers, there exist k ě k and G k Ă F k such that | G k p x k q| ą c { Ą min p G k , σ q ą m .If the above statement is true, we may clearly choose t G k u k in G n satis-fying the assumptions of Lemma 3.11, which will complete the proof.We assume that n ě
2, as the case n “ k } x k } “ k , m P N . By Lemma 3.12, choose k ě k , such that the followingholds.(3) | F p x k q| ă c m for every F P G n ´ We distinguish two cases.
Case 1:
There is p F ki , σ ki q di “ a skipped branching of σ in G n ´ with F k “Y di “ F ki . Case 2:
There is p F ki , σ q di “ an attached branching of σ in G n ´ with F k “Y di “ F ki .In either case, by Proposition 2.15 we have that if we set G k “ Y di “ m ` F ki ,then p G k , σ q P G n . Moreover, (3) yields that | G k p x k q| ą c { Ą min p G k , σ q ą m .If we are in case 1, then Ą min p G k , σ q “ | σ ^ σ km ` | . By Definition 2.2 wehave that | σ ^ σ ki | ă | σ ^ σ ki ` | for i “ , . . . , m , which of course yields that | σ ^ σ km ` | ą m .If, on the other hand, we are in case 2, then Ą min p G k , σ q “ Ą min p F km ` , σ q .By Definition 2.3 we have that Ą min p F ki , σ q ď Ą max p F ki , σ q ă Ą min p F ki ` , σ q for i “ , . . . , m , which yields that Ą min p F km ` , σ q ą m . (cid:3) Lemma 3.14.
Let t x k u k be a normalized, disjointly supported block se-quence of t ˜ e ξ u ξ ă ℵ , such that there exists n P N such that lim sup k } P n x k } n ą
0. Then, passing if necessary, to a subsequence, there exist c ą p F k , σ k q P G n satisfying the following.(i) The F k are pairwise disjoint.(ii) | F k p x k q| ą c for all k P N . Proof.
Pass to a subsequence of t x k u k and choose ε ą
0, such that thefollowing holds. } P n x k } n “ n } x k } n ą ε, for all k P N . By the definition of the norm } ¨ } n , there exist p F k , σ k q P G n with | F k p x k q| ą n ε , for all k P N . By virtue of Proposition 2.15 and the fact that t x k u k is disjointly supported, we may assume that the F k are pairwise disjoint.Setting c “ n ε finishes the proof. (cid:3) Proposition 3.15.
Let t x k u k be a normalized, disjointly supported blocksequence of t ˜ e ξ u ξ ă ℵ , such that lim k } x k } “ n P N suchthat lim sup k } P n x k } n ą
0. Then t x k u k has a subsequence that generates an ℓ n spreading model for every n P N . Proof.
Set n “ min t n : lim sup k } P n x k } n ą u and as in the proof of Lemma3.13 let us assume that n ě
2. Apply Lemmas 3.14 and 3.12, pass to asubsequence of t x k u k and find c ą p F k , σ k q P G n such that the followingare satisfied.(i) The F k are pairwise disjoint.(ii) | F k p x k q| ą c for all k P N .(iii) | F p x k q| ă c { k P N and F P G n ´ .Passing to a further subsequence, choose σ P t , u N such that lim k σ k “ σ . We distinguish two cases. Case 1: lim k max t| G p x k q| : G Ă F k with p G, σ q P G n u “ Case 2: lim sup k max t| G p x k q| : G Ă F k with p G, σ q P G n u ą t x k u k ,satisfying the following.(a) max t| G p x k q| : G Ă F k with p G, σ q P G n u ă c {
4, for all k P N .(b) σ ‰ σ k , for every k P N .(c) σ ^ σ k Ł σ ^ σ k ` for all k P N .We shall prove the following. For every k , there exists G k Ă F k , such thatthe following hold.(d) | G k p x k q| ą c { | σ ^ σ k | ă Ą min p G k , σ k q Combining (b), (c), (d) and (e), we conclude that the assumptions ofLemma 3.10 are satisfied, which proves the desired result, in case 1.Set G k “ t τ P F k : σ k ^ τ “ σ ^ τ u . Proposition 2.15 and Lemma2.13 yield that p G k , σ q P G n . Setting F k “ F k z G k , property (a) yields that | F k p x k q| ą c { -LARGE FAMILIES 21 Set G k “ t τ P F k : σ k ^ τ Ł σ ^ τ u . Once more, Proposition 2.15yields that p G k , σ k q P G n , however Lemma 2.14 yields that G k P G n ´ andtherefore, by (iii) we have that | G k p x k q| ă c { G k “ F k z G k . Then we have that | G k p x k q| ą c {
2, i.e. (d) holds.We will show that (e) also holds. By Corollary 2.9, there exists τ P G k ,with Ą min p G k , σ k q “ | σ k ^ τ | . Since τ R G k , we have that | σ k ^ τ | ‰ | σ ^ τ | .We will show that | σ ^ τ | ă | σ k ^ τ | . Assume that this is not the case,i.e. | σ k ^ τ | ă | σ ^ τ | . In other words, σ k ^ τ Ł σ ^ τ . This means that τ P G k , a contradiction.We conclude that σ ^ τ Ł σ k ^ τ . Lemma 2.7 yields that σ ^ τ “ σ k ^ σ .Applying Lemma 2.7 once more, we conclude that σ ^ σ k Ł σ k ^ τ , i.e. | σ ^ σ k | ă | σ k ^ τ | “ Ą min p G k , σ k q , which completes the proof for case 1.It only remains to treat case 2. Observe, that in this case, we may easilypass to a subsequence of t x k u k , satisfying the assumptions of Lemma 3.13.This completes the proof. (cid:3) Combining Propositions 3.8, 3.9 and 3.15, one obtains the following.
Proposition 3.16.
Let t x k u k be a normalized weakly null sequence in X ℵ .Then t x k u k has a subsequence that generates an ℓ n spreading model for every n P N . Proposition 3.17.
The space X ℵ is saturated with subspaces of Tsirelsonspace. Proof.
It is an immediate consequence of Proposition 3.16 that X ℵ doesnot contain a copy of c . By Proposition 3.3, the spaces X n are c saturatedand therefore, the operators P n : X ℵ Ñ X n , are strictly singular.We conclude, that in any infinite dimensional subspace Y of X ℵ , n P N and ε ą
0, there exists x P Y with } x } “ } P n x } n ă ε for n “ , . . . , n . One may easily construct a normalized sequence in Y , satisfyingthe assumption of Proposition 3.9, which completes the proof. (cid:3) In particular, the previous result yields that neither c nor ℓ embed into X ℵ . Using James’ well known theorem for spaces with an unconditionalbasis, we conclude the following. Corollary 3.18.
The space X ℵ is reflexive. Remark 3.19.
As is well known, (see [2], Lemma 37), if t x k u k is a nor-malized weakly null sequence in a Banach space X and x P X , then t x k u k admits an ℓ spreading model, if and only if t x k ´ x u k admits an ℓ spread-ing model as well. Since X ℵ is reflexive and every normalized weaklynull sequence admits an ℓ spreading model, we conclude that any boundedsequence in X ℵ , without a norm convergent subsequence, admits an ℓ spreading model. In other words, every spreading model admitted by X ℵ ,is either trivial or equivalent to the usual basis of ℓ . Spaces admitting ℓ α as a unique spreading model The goal of the present section, is to give an outline of the construction,for a given countable ordinal number α , of a non separable reflexive space X α ℵ , having the following property. Every normalized weakly null sequencein X α ℵ has a subsequence that generates an ℓ α spreading model. Definition 4.1.
Let α be a countable ordinal number. Define } ¨ } T α to bethe unique norm on c p N q that satisfies the following implicit formula, forevery x P c p N q . } x } T α “ max } x } ,
12 sup d ÿ i “ } E i x } T α ( where the supremum is taken over all E ă ¨ ¨ ¨ ă E d subsets of the naturalnumbers with t min E i : i “ , . . . , d u P S α .Define the Tsirelson space of order α , denoted by T α , to be the completionof c p N q with the aforementioned norm.The space T α is reflexive and the unit vector basis t e n u n , forms a 1-unconditional basis for T α . Moreover, every normalized weakly null sequencein T α , has a subsequence that generates an ℓ α spreading model. For moredetails see [4].Given a countable ordinal number α , we shall construct t G αn u n an increas-ing sequence of families of finite subsets of t , u N , strongly related to t G n u n .As before, we first define some auxiliary families G αn , n P N . Definition 4.2.
We define G α to be all pairs p F, σ q , where F “ t τ i u di “ Prt , u N s ă ω , d P N and σ P t , u N , such that the following are satisfied.(i) σ ‰ τ i for i “ , . . . , d (ii) σ ^ τ ‰ ∅ and if d ą
1, then σ ^ τ Ł σ ^ τ Ł ¨ ¨ ¨ Ł σ ^ τ d (iii) t| σ ^ τ i | : i “ , . . . , d u P S α Define Ą min p F, σ q “ | σ ^ τ | and Ą max p F, σ q “ | σ ^ τ d | .Assume that n P N , G αk have been defined for k ď n and that for p F, σ q P G αk , Ą min p F, σ q and Ą max p F, σ q have also been defined. Definition 4.3.
Let p F i , σ i q di “ , d P N be a finite sequence of elements of G αn and σ P t , u N .We say that p F i , σ i q di “ is a skipped branching of σ in G αn , if the followingare satisfied.(i) The F i , i “ , . . . , d are pairwise disjoint(ii) σ ‰ σ i for i “ , . . . , d (iii) σ ^ σ ‰ ∅ and if d ą
1, then σ ^ σ Ł σ ^ σ Ł ¨ ¨ ¨ Ł σ ^ σ d (iv) | σ ^ σ i | ă Ą min p F i , σ i q for i “ , . . . , d (v) t| σ ^ σ i | : i “ , . . . , d u P S α -LARGE FAMILIES 23 Definition 4.4.
Let σ P t , u N and p F i , σ q di “ , d P N be a finite sequenceof elements of G αn .We say that p F i , σ q di “ is an attached branching of σ in G αn if the followingare satisfied.(i) The F i , i “ , . . . , d are pairwise disjoint(ii) If d ą
1, then Ą max p F i , σ q ă Ą min p F i ` , σ q , for i “ , . . . , d ´ t Ą min p F i , σ q : i “ , . . . , d u P S α We are now ready to define G αn ` . Definition 4.5.
We define G αn ` to be all pairs p F, σ q , where F P rt , u N s ă ω and σ P t , u N , such that one of the following is satisfied.(i) p F, σ q P G αn .(ii) There is p F i , σ i q di “ a skipped branching of σ in G αn such that F “Y di “ F i .In this case we say that p F, σ q is skipped . Moreover set Ą min p F, σ q “| σ ^ σ | and Ą max p F, σ q “ | σ ^ σ d | .(iii) There is p F i , σ q di “ an attached branching of σ in G αn such that F “Y di “ F i .In this case we say that p F, σ q is attached . Moreover set Ą min p F, σ q “ Ą min p F , σ q and Ą max p F, σ q “ Ą max p F d , σ q . Definition 4.6.
For a countable ordinal number α and n P N we define G αn “ t F Ă t , u N : there exists σ P t , u N with p F, σ q P G αn u Y t ∅ u Remark 4.7.
It is clear that for α “ G αn “ G n , for every n P N . More-over, for a countable ordinal number α , every result stated for G n , up toProposition 2.15, holds also for G αn and the proofs are identical. On theother hand, if for n P N we denote by S nα the convolution of S α with itself n times, Proposition 2.18 can be restated as follows. Proposition 4.8.
Let α be a countable ordinal number. Then for every B infinite subset of t , u N there exists a one to one map φ : N Ñ B with φ p F q P G αn for every F P S nα and n P N .Theorem 2.20 takes the following form and the proof uses the compactnessof S α and Corollary 2.12. Theorem 4.9.
Let α be a countable ordinal number and n P N . Then G αn is an α -large, hereditary and compact family of finite subsets of t , u N .In order to define the desired space X α ℵ , one takes the same steps as inthe previous section. All proofs are identical. Definition 4.10.
For α a countable ordinal number and n P N define anorm on c p ℵ q in the following manner. (i) For n P N , we may identify an F P G αn with a linear functional F : c p ℵ q Ñ R in the following manner. For x “ ř ξ ă ℵ λ ξ e ξ P c p ℵ q F p x q “ ÿ τ P F λ ξ τ (ii) For x P c p ℵ q define } x } αn “ sup t| F p x q| : F P G αn u Set X αn to be the completion of p c p ℵ q , } ¨ } αn q . Definition 4.11.
Define the following norm on c p ℵ q . For x P c p ℵ q} x } “ ›› ÿ n “ n } x } αn e n ›› T α Set X α ℵ to be the completion of p c p ℵ q , } ¨ }q . Theorem 4.12.
The space X α ℵ is a non separable reflexive space with asuppression unconditional Schauder basis with the length of the continuum,having the following property. Every normalized weakly null sequence in X α ℵ has a subsequence that generates an ℓ α spreading model. References [1] D. E. Alspach and S. A. Argyros,
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