A new metric invariant for Banach spaces
aa r X i v : . [ m a t h . F A ] D ec A NEW METRIC INVARIANT FOR BANACH SPACES
F. BAUDIER, N. J. KALTON, AND G. LANCIEN
Abstract.
We show that if the Szlenk index of a Banach space X islarger than the first infinite ordinal ω or if the Szlenk index of its dual islarger than ω , then the tree of all finite sequences of integers equippedwith the hyperbolic distance metrically embeds into X . We show thatthe converse is true when X is assumed to be reflexive. As an application,we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms. Introduction
In 1976 Ribe proved in [22] that two uniformly homeomorphic Banachspaces are finitely representable in each other. This theorem gave birthto the “Ribe program” (see [4] or [17] for a detailed description). Localproperties of Banach spaces are properties which only involve finitely manyvectors. These are properties which are stable under finite representability.In view of Ribe’s result the “Ribe program” aims at looking for metricinvariants that characterize local properties of Banach spaces. The firstoccurence of the “Ribe program” is Bourgain’s metric characterization ofsuperreflexivity given in [4]. The metric invariant discovered by Bourgain isthe collection of the hyperbolic dyadic trees of arbitrarily large height N . Ifwe denote Ω = {∅} , the root of the tree. Let Ω i = {− , } i , B N = S Ni =0 Ω i .Thus B N endowed with its shortest path metric ρ is the hyperbolic dyadictree of height N .Let us recall some definitions. Let ( M, d ) and (
N, δ ) be two metric spacesand let f : M → N be an injective map. The distortion of f isdist( f ) := k f k Lip k f − k Lip = sup x = y ∈ M δ ( f ( x ) , f ( y )) d ( x, y ) . sup x = y ∈ M d ( x, y ) δ ( f ( x ) , f ( y )) . If dist( f ) is finite, we say that f is a Lipschitz or metric embedding of M into N . If there exists an embedding f from M into N , with dist( f ) ≤ C ,we use the notation M C ֒ → N .Bourgain’s characterization is the following: Theorem 1.1. (Bourgain 1986)
Let X be a Banach space. Then X isnot superreflexive if and only if there exists a universal constant C such thatfor all N ∈ N , ( B N , ρ ) C ֒ → X . Mathematics Subject Classification.
It has been proved in [1] that this is also equivalent to the metric embed-ding of the infinite hyperbolic dyadic tree ( B ∞ , ρ ) where B ∞ = S ∞ N =0 B N .We also recall that it follows from the Enflo-Pisier renorming theorem ([6]and [21]) that superreflexivity is equivalent to the existence of an equivalentuniformly convex and (or) uniformly smooth norm.In the series of papers [5], [16], [17] local properties such as linear type andlinear cotype are deeply studied and other occurrences of “Ribe’s program”are given.In a similar vein our paper is an attempt to investigate which asymptoticproperties admit a metrical characterization. Asymptotic properties havebeen intensively studied in [9], [7] and [20] and we refer to [11] for a precisedefinition of the asymptotic structure of a Banach space. The main result ofthis paper is an analogue of Bourgain’s theorem in the asymptotic setting.Let us first introduce a few notation and definitions. For a positive integer N , We denote T N = S Ni =0 N i , where N := {∅} . Then T ∞ = S ∞ N =1 T N isthe set of all finite sequences of positive integers. For s ∈ T ∞ , we denoteby | s | the length of s . There is a natural ordering on T ∞ defined by s ≤ t if t extends s . If s ≤ t , we will say that s is an ancestor of t . If s ≤ t and | t | = | s | + 1, we will say that s is the predecessor of t and t is a successor of s and we will denote s = t − . Then we equip T ∞ , and by restriction every T N , with the hyperbolic distance ρ , which is defined as follows. Let s and s ′ be two elements of T ∞ and let u ∈ T ∞ be their greatest common ancestor.We set ρ ( s, s ′ ) = | s | + | s ′ | − | u | = ρ ( s, u ) + ρ ( s ′ , u ) . We now define the asymptotic version of uniform convexity and uniformsmoothness that we will consider. Let ( X, k k ) be a Banach space and τ > B X its closed unit ball and by S X its unit sphere. For x ∈ S X and Y a closed linear subspace of X , we define ρ ( τ, x, Y ) = sup y ∈ S Y k x + τ y k − δ ( τ, x, Y ) = inf y ∈ S Y k x + τ y k − . Then ρ ( τ ) = sup x ∈ S X inf dim( X/Y ) < ∞ ρ ( τ, x, Y ) and δ ( τ ) = inf x ∈ S X sup dim( X/Y ) < ∞ δ ( τ, x, Y ) . The norm k k is said to be asymptotically uniformly smooth iflim τ → ρ ( τ ) τ = 0 . It is said to be asymptotically uniformly convex if ∀ τ > δ ( τ ) > . These moduli have been first introduced by Milman in [18]. new metric invariant for Banach spaces 3
We can now state the main result of our paper in a way that is clearlyan asymptotic analogue of Bourgain’s theorem.
Theorem 1.2.
Let X be a reflexive Banach space. The following assertionsare equivalent.(i) There exists C ≥ such that T ∞ C ֒ → X .(ii) There exists C ≥ such that for any N in N , T N C ֒ → X .(iii) X does not admit any equivalent asymptotically uniformly smoothnorm or X does not admit any equivalent asymptotically uniformly convexnorm. The main tool for our proof will be the so-called
Szlenk index . We nowrecall the definition of the Szlenk derivation and the Szlenk index that havebeen first introduced in [24] and used there to show that there is no universalspace for the class of separable reflexive Banach spaces. So consider a realseparable Banach space X and K a weak ∗ -compact subset of X ∗ . For ε > V be the set of all relatively weak ∗ -open subsets V of K such thatthe norm diameter of V is less than ε and s ε K = K \ ∪{ V : V ∈ V} . We define inductively s αε K for any ordinal α , by s α +1 ε K = s ε ( s αε K ) and s αε K = ∩ β<α s βε K if α is a limit ordinal. Then we define Sz ( X, ε ) to be theleast ordinal α so that s αε B X ∗ = ∅ , if such an ordinal exists. Otherwise wewrite Sz ( X, ε ) = ∞ . The
Szlenk index of X is finally defined by Sz ( X ) =sup ε> Sz ( X, ε ) . We denote ω the first infinite ordinal and ω the first uncountable ordinal.Note that the dual of a separable Banach space X is separable if and onlyif Sz ( X ) < ω (this is a consequence of Baire’s theorem on the pointwiselimit of sequences of continuous functions). We will essentially deal withthe condition Sz ( X ) ≤ ω . The weak ∗ -compactness of B X ∗ implies thatthis is equivalent to the condition: Sz ( X, ε ) < ω , for all ε >
0. Besides,it follows from a theorem of Knaust, Odell and Schlumprecht ([11]) thata separable Banach space admits an equivalent asymptotically uniformlysmooth norm if and only if Sz ( X ) ≤ ω . Then it is easy to see that fora reflexive Banach space the condition Sz ( X ∗ ) ≤ ω is equivalent to theexistence of an equivalent asymptotically uniformly convex norm on X .Therefore condition ( iii ) in Theorem (1.2) is equivalent to( iv ) Sz ( X ) > ω or Sz ( X ∗ ) > ω. With this information at hand, we shall almost forget the formulationsin terms of renormings and work essentially with the notion of the Szlenkindex of a Banach space.
F. BAUDIER, N. J. KALTON AND G. LANCIEN
In order to have a complete view of the analogy between our result andBourgain’s theorem, it is worth noting at this point that the superreflexivitycan be similarly characterized by the behavior of an ordinal index. For agiven weak ∗ -compact convex subset C of X ∗ and a given ε >
0, let usdenote S be the set of all relatively weak ∗ -open slices S of C such thatthe norm diameter of S is less than ε and d ε C = C \ ∪{ S : S ∈ S} .We then define inductively d αε ( C ) for α ordinal as before and Dz ( X, ε )to be the least ordinal α so that d αε B X ∗ = ∅ , if such an ordinal exists.Otherwise we write Dz ( X, ε ) = ∞ . Finally, the weak ∗ -dentability index of X is Dz ( X ) = sup ε> Dz ( X, ε ) . Then it follows from [12] (see also thesurvey [13]) that the following conditions are equivalent:(i) X is super-reflexive.(ii) Dz ( X ) ≤ ω .(iii) Dz ( X ∗ ) ≤ ω .Let us now describe the organization of this article. In Section 2 we givethe construction of several embeddings and finally prove that T ∞ Lipschitz-embeds into X , whenever Sz ( X ) > ω or Sz ( X ∗ ) > ω . In Section 3 we showthe converse statement in the reflexive case. This will conclude the proof ofTheorem 1.2. In the last section we describe a few applications of our resultto the stability of certain classes of Banach spaces under coarse-Lipschitzembeddings or uniform homeomorphisms. The main consequence of ourwork is that the class of all separable reflexive spaces X so that Sz ( X ) ≤ ω and Sz ( X ∗ ) ≤ ω is stable under coarse-Lipschitz embeddings. It seems alsointeresting to us that a metric invariant (the embeddability of T ∞ in thiscase) is used to prove stability results, whereas the metric invariant is oftenlooked after, when the class is already known to be stable.2. Construction of the embeddings
Before to start, we need to introduce more notation concerning our trees.For s = ( s , . . . , s n ) and t = ( t , . . . , t m ) in T ∞ , we denote s ⌢ t = ( s , . . . , s n , t , . . . , t m ) and also ∅ ⌢ t = t ⌢ ∅ = t. For t ∈ T ∞ and k ≤ | t | , we denote t | k the ancestor of t of length k .For s ≤ t in T ∞ , we denote [ s, t ] = { u ∈ T ∞ , s ≤ u ≤ t } .For N in N and T ⊂ T N , we say that a map Φ : T N → T is a treeisomorphism if Φ( T N ) = T , Φ( ∅ ) = ∅ and for all s ∈ T N − and n ∈ N Φ( s ⌢ n ) = Φ( s ) ⌢ k s,n with k s,n ∈ N and k s,n < k s,m whenever n < m . Asubset T of T N is called a full subtree of T N if there exists a tree isomorphismfrom T N onto T or equivalently if ∅ ∈ T and for all s ∈ T ∩ T N − , the set ofsuccessors of s that also belong to T is infinite. new metric invariant for Banach spaces 5 We now begin with a very simple lemma.
Lemma 2.1.
Let ( x ∗ n ) ∞ n =0 be a weak*-null sequence in X ∗ such that k x ∗ n k ≥ for all n in N and let F be a finite dimensional subspace of X ∗ . Then thereexists a sequence ( x n ) n in B X such that for all y ∗ ∈ F , y ∗ ( x n ) = 0 and lim inf x ∗ n ( x n ) ≥ Proof.
It is a classical consequence of Mazur’s technique for constructingbasic sequences (see for instance [14]), that lim inf d ( x ∗ n , F ) ≥ . Denote E = { x ∈ X ∀ x ∗ ∈ F x ∗ ( x ) = 0 } be the pre-orthogonal of F . Since F is finite dimensional, we have that F = E ⊥ . Therefore, for any x ∗ ∈ X ∗ , d ( x ∗ , F ) = k x ∗| E k E ∗ . This finishes the proof. (cid:3) Let now X be a separable Banach space. It follows from the metrizabilityof the weak ∗ topology on B X ∗ that if Sz( X, ε ) > ω then, for all N ∈ N there exists ( y ∗ s ) s ∈ T N in B ∗ X such that for all s ∈ T N − and all n ∈ N , k y ∗ s⌢n − y ∗ s k ≥ ε/ ε ′ and y ∗ s⌢n w ∗ → y ∗ s .It is an easy and well known fact that the map ε Sz(
X, ε ) is submulti-plicative (see for instance [13]). So, if Sz( X ) > ω , then Sz( X, ε ) > ω for any ε ∈ (0 , y ∗ s ) s ∈ T N we can take ε ′ = .By considering z ∗ s = y ∗ s − y ∗ s − for s = ∅ , z ∗∅ = y ∗∅ and re-scaling, this is clearlyequivalent to the existence, for all N ∈ N of ( z ∗ s ) s ∈ T N in X ∗ so that • ∀ s ∈ T N \ {∅} , k z ∗ s k ≥ • ∀ s ∈ T N − , z ∗ s⌢n w ∗ → • ∀ s ∈ T N , k P t ≤ s z ∗ t k ≤ z ∗ s ) s ∈ T N . Proposition 2.2.
Let X be a separable Banach space. If Sz ( X ) > ω , thenfor all N ∈ N and δ > there exist ( x ∗ s ) s ∈ T N in X ∗ and ( x s ) s ∈ T N in B X such that • ∀ s ∈ T N − , x ∗ s⌢n w ∗ → , • ∀ s ∈ T N \ {∅} , k x ∗ s k ≥ and ∀ s ∈ T N , k P t ≤ s x ∗ t k ≤ , • ∀ s ∈ T N , x ∗ s ( x s ) ≥ k x ∗ s k , • ∀ s = t, | x ∗ s ( x t ) | < δ .Proof. Let f : N → T N be a bijection such that ∀ s < t ∈ T N f − ( s ) < f − ( t )and ∀ s ∈ T N − ∀ n < m ∈ N f − ( s ⌢ n ) < f − ( s ⌢ m ) . Denote s i = f ( i ). In particular, ∅ = s . F. BAUDIER, N. J. KALTON AND G. LANCIEN
We now build inductively a tree isomorphism Φ : T N → Φ( T N ) ⊂ T N and afamily ( z Φ( s ) ) s ∈ T N in B X such that(2.1) z ∗ Φ( s ) ( z Φ( s ) ) ≥ , s ∈ T N and | z ∗ Φ( s ) ( z Φ( t ) ) | < δ, s = t ∈ T N . So set Φ( ∅ ) = ∅ , pick z Φ( ∅ ) in B X so that z ∗ Φ( ∅ ) ( z Φ( ∅ ) ) ≥ k z ∗ Φ( ∅ ) k and assumethat Φ( s ) , . . . , Φ( s k ) and z Φ( s ) , . . . , z Φ( s k ) have been constructed accord-ingdly to (2.1). Then, there exists i ∈ { , . . . , k } and p ∈ N such that s k +1 = s i ⌢ p . Since ( z ∗ Φ( s i ) ⌢n ) n ≥ is a weak ∗ -null sequence, Lemma 2.1 insures thatwe can pick n ∈ N and z Φ( s i ) ⌢n in B X such that | z ∗ Φ( s i ) ⌢n ( z Φ( s j ) ) | < δ forall j ≤ k , z ∗ Φ( s j ) ( z Φ( s i ) ⌢n ) = 0 for all j ≤ k and z ∗ Φ( s i ) ⌢n ( z Φ( s i ) ⌢n ) ≥ . Wenow set Φ( s k +1 ) = Φ( s i ) ⌢ n . If n is chosen large enough all the requiredproperties, including those needed for making Φ a tree isomorphism, aresatisfied.We conclude the proof by setting x ∗ s = z ∗ Φ( s ) and x s = z Φ( s ) , for s in T N . (cid:3) We shall improve progressively our embedding results and start with thefollowing.
Proposition 2.3.
There is a universal constant C ≥ such that, whenever X is a separable Banach space with Sz ( X ) > ω , we have that ∀ N ∈ N T N C ֒ → X and T
N C ֒ → X ∗ . Proof.
Let ( x ∗ s , x s ) s ∈ T N be the system given by Proposition 2.2. Our choiceof δ , will be specified later.We shall first embed the T N ’s into X . For that purpose, we mimic thenatural embedding of T N into ℓ ( T N ) (with ( x t ) t ∈ T N playing the role of thecanonical basis of ℓ ( T N )) and define F : T N → X by ∀ s ∈ T N F ( s ) = X t ≤ s x t . Since ( x t ) t ∈ T N ⊂ B X , we clearly have that F is 1-Lipschitz for the metric ρ on T N .Let now s = s ′ in T N and let u be their greatest common ancestor. Denote d = ρ ( u, s ) and d ′ = ρ ( u, s ′ ). Recall that ρ ( s, s ′ ) = d + d ′ and assume forinstance that d ≥ d ′ . Then h X t ≤ s x ∗ t , F ( s ) − F ( s ′ ) i ≥ d − δ | s | ( d + d ′ ) ≥ d − N δ ≥ d ≥ ρ ( s, s ′ ) , if δ was chosen less than N . new metric invariant for Banach spaces 7 Since k P t ≤ s x ∗ t k ≤
3, we obtain that for all s, s ′ in T N : k F ( s ) − F ( s ′ ) k ≥ ρ ( s, s ′ ) . This finishes the proof of our first embedding result.We now turn to the question of embedding the T N ’s into X ∗ .Our construction will copy the natural embedding of T N into c ( T N ), with( x ∗ t ) t ∈ T N replacing the canonical basis of c ( T N ). For s ∈ T N , we denote y ∗ s = P t ≤ s x ∗ t . Then we define G : T N → X ∗ by ∀ s ∈ T N G ( s ) = X t ≤ s y ∗ t . Since ( y ∗ t ) t ∈ T N is a subset of 3 B X ∗ , it is immediate that G is 3-Lipschitz.Let now s = s ′ in T N and denote again u their greatest common ancestor, d = ρ ( u, s ) and d ′ = ρ ( u, s ′ ). Assume for instance that d ≥ d ′ . Let us name v the unique successor of u such that v ≤ s and w the unique successor of u such that w ≤ s ′ if it exists. Then G ( s ) − G ( s ′ ) = X v ≤ t ≤ s y ∗ t − X w ≤ t ≤ s ′ y ∗ t . If s ′ ≤ s , [ w, s ′ ] is empty. Otherwise, ∀ t ∈ [ w, s ′ ] |h x v , y ∗ t i| ≤ δ | t | ≤ δN. On the other hand ∀ t ∈ [ v, s ] |h x v , y ∗ t i| ≥ − δ ( | t | − ≥ − δN. The two previous inequalities yield k G ( s ) − G ( s ′ ) k ≥ |h x v , G ( s ) − G ( s ′ ) i| ≥ d − δN ≥ d ≥ ρ ( s, s ′ ) , if δ was chosen in (0 , N ). This concludes our argument for the secondembedding. (cid:3) Remark . Let us just finally notice that in both cases we proved the state-ment for C = 24, but our argument allows us to get the result for anyconstant C > Remark . The end of this section will be devoted to various improvementsof Proposition 2.3, which are not fully needed in order to read the last twosections.We now turn to the problem of embedding T ∞ . We shall refine ourarguments in order to improve Proposition 2.3 and obtain: F. BAUDIER, N. J. KALTON AND G. LANCIEN
Theorem 2.4.
There is a constant C ≥ such that for any separableBanach space X satisfying Sz ( X ) > ω , we have T ∞ C ֒ → X and T ∞ C ֒ → X ∗ . Although this statement implies our previous results, we have chosen toseparate its proof in the hope of making it easier to read.
Proof.
So assume that Sz ( X ) > ω and fix a decreasing sequence ( δ i ) ∞ i =0 in(0 , S ∞ i =0 { i } × T i , one can actually build for every i ≥
0: ( x ∗ i,s ) s ∈ T i in X ∗ and ( x i,s ) s ∈ T i in B X such that(i) ∀ i ≥ , ∀ s ∈ T i − , x ∗ i,s⌢n w ∗ → ∀ i ≥ , ∀ s ∈ T i \ {∅} , k x ∗ i,s k ≥ ∀ s ∈ T i , k P t ≤ s x ∗ i,t k ≤ ∀ i ≥ , ∀ s ∈ T i , x ∗ i,s ( x i,s ) ≥ k x ∗ i,s k ,(iv) ∀ ( i, s ) = ( j, t ) , | x ∗ i,s ( x j,t ) | < δ i .Let us just emphasize the fact that the whole system ( x i,s , x ∗ i,s ) ( i,s ) is almostbiorthogonal. We wish also to note that the estimate given in (iv) dependsonly on i . This last fact relies on a careful application on Lemma 2.1.For i ≥
0, we denote F i a translate of the map defined on T i +1 in the proofof Proposition 2.3. So let F i ( ∅ ) = 0 and F i ( s ) = X ∅ 1) + (1 − λ s ) d | s | + λ s ′ d | s ′ | + (1 − λ s ′ ) d | s ′ | ) − δ l +2 ((1 − λ s ) d ( | s | − 1) + λ s d | s | + λ s ′ d | s ′ | + (1 − λ s ′ ) d | s ′ | ) ≥ d − d | s | ( δ l +1 + δ l +2 ) ≥ d − · l +2 ( δ l +1 + δ l +2 ) ≥ d ≥ ρ ( s, s ′ )8 , if the δ i ’s were chosen small enough.Since k P t ≤ s x ∗ i,t k ≤ i ≥ 0, we obtain the following lower bound k F ( s ) − F ( s ′ ) k ≥ ρ ( s, s ′ )96 . If s ′ = ∅ 6 = s ′ , the argument is similar but simpler. This concludes ourproof.In order to embed T ∞ into X ∗ , we use exactly the same technique. For i ≥ s ∈ T i denote y ∗ i,s = P t ≤ s x ∗ i,t and G i ( ∅ ) = 0 and G i ( s ) = X ∅ We will now study the condition “Sz( X ∗ ) > ω ”. We already know thatif Sz( X ∗ ) > ω , then T ∞ Lipschitz embeds into X ∗∗ and therefore, when X is reflexive, T ∞ Lipschitz embeds into X . We will show how to drop thereflexivity assumption in this statement. As before, we start with finitetrees. Proposition 2.5. There is a universal constant C ≥ such that, whenever X is a separable Banach space with Sz ( X ∗ ) > ω , we have that ∀ N ∈ N , T N C ֒ → X. Proof. If X ∗ is non separable, then Sz( X ) > ω and our problem is settledby Proposition 2.3. Thus we assume that X ∗ is separable. Then, for agiven positive integer N and a given δ > 0, Proposition 2.2 provides uswith ( x ∗ s ) s ∈ T N in B X ∗ and ( x ∗∗ s ) s ∈ T N in X ∗∗ such that • ∀ s ∈ T N − , x ∗∗ s⌢n w ∗ → • ∀ s ∈ T N \ {∅} , k x ∗∗ s k ≥ ∀ s ∈ T N , k P t ≤ s x ∗∗ t k ≤ • ∀ s ∈ T N , x ∗∗ s ( x ∗ s ) ≥ k x ∗∗ s k , • ∀ s = t, | x ∗∗ s ( x ∗ t ) | < δ .Let { s i , i ∈ N } be an enumeration of { s ∈ T N , | s | = N } and let B i = { t ∈ T N , t ≤ s i } be the corresponding branches of T N .For s ∈ T N denote y ∗∗ s = P t ≤ s x ∗∗ t .Let us now fix η > 0. For a given s ∈ T N , there is a unique i = i s ∈ N suchthat s ∈ B i s \ B i s − . Then, we can pick y s in X so that(2.2) k y s k ≤ ∀ t ∈ i s [ j =1 B j |h x ∗ t , y ∗∗ s − y s i| < η. In particular(2.3) ∀ t ≤ s |h x ∗ t , y ∗∗ s − y s i| < η. We now define G : T N → X by ∀ s ∈ T N G ( s ) = X t ≤ s y t . Since ( y t ) t ∈ T N is a subset of 3 B X , it is immediate that F is 3-Lipschitz.Let now s = s ′ in T N and denote again u their greatest common ancestor, d = ρ ( u, s ) and d ′ = ρ ( u, s ′ ), v the successor of u so that v ≤ s and w thesuccessor of u so that w ≤ s ′ , if they exist.Assume first that s and s ′ are comparable and for instance that s ′ ≤ s .Then u = s ′ , v exists, w does not and by (2.3)(2.4) h x ∗ v , F ( s ) − F ( s ′ ) i ≥ h x ∗ v , X v ≤ t ≤ s y ∗∗ t i − ηd ≥ d, for δ and η chosen small enough.Suppose now that s and s ′ are not comparable. Then v and w are definedand not comparable. Therefore i v = i w . For instance i v < i w . We will thenconsider two cases.(a) If d ′ ≥ d . Then k G ( s ) − G ( s ′ ) k ≥ k P u Theorem 2.6. There is a universal constant C ≥ such that, whenever X is a separable Banach space with Sz ( X ∗ ) > ω , we have that T ∞ C ֒ → X. Proof. Again, we may directly assume that X ∗ is separable. The gluingargument that we used before to embed T ∞ does not seem to be efficient inthis case. We shall develop another technique. Fix first an integer K ≥ δ i ) i in (0 , X ∗ ) > ω we can build( x ∗∗ i,s ) s ∈ T Ki +1 in X ∗∗ and ( x ∗ i,s ) s ∈ T Ki +1 in B X ∗ such that • ∀ i ≥ , ∀ s ∈ T K i , x ∗∗ i,s⌢n w ∗ → • ∀ i ≥ , ∀ s ∈ T K i +1 \ {∅} , k x ∗∗ i,s k ≥ ∀ s ∈ T K i +1 , k P t ≤ s x ∗∗ i,t k ≤ • ∀ i ≥ , ∀ s ∈ T K i +1 , x ∗∗ i,s ( x ∗ i,s ) ≥ k x ∗∗ i,s k , • ∀ ( i, s ) = ( j, t ) , | x ∗∗ i,s ( x ∗ j,t ) | < δ i .For s in T K i +1 , we define y ∗∗ i,s = P t ≤ s x ∗∗ i,s .Let N i = P ik =0 K k , choose an enumeration { s ir , r ∈ N } of { s ∈ T N i , | s | = N i } and denote B ir = { t ∈ T N i , t ≤ s ir } the branch of T N i whose endpoint is s ir . We will also use an enumeration { t ir , r ∈ N } of the terminal nodes of T K i +1 and the corresponding branches C ir = { t ∈ T K i +1 , t ≤ t ir } .Let us first describe the general idea. We set G ( ∅ ) = 0. Consider now s ∈ T ∞ \ {∅} . Then, there exists n ∈ N and s , . . . , s n in T ∞ such that | s j | = K j for j ≤ n − 1, 1 ≤ | s n | ≤ K n and s = s ⌢ · · · ⌢ s n . For j ≤ n − s ⌢ · · · ⌢ s j is a terminal node of T N j that we denote s jr j . Weshall now define G ( s ) = X ∅ 1. Then, for any j ≤ i − s ⌢ · · · ⌢ s j is a terminal node of T N j that we denote s jr j . Besides, r j − ⌢ s j is a terminal node of T K j +1 that we denote t jk j . Let also k i ∈ N be such that s ∈ C ik i \ S k i − k =1 C ik .Then we pick y s,i in 3 B X satisfying the following conditions:(2.5) ∀ j ≤ i ∀ t ∈ k j [ k =1 C jk |h y ∗∗ s,i − y s,i , x ∗ t,j i| ≤ δ i Since any y s,i belongs to 3 B X , it is clear that G is 3-Lipschitz.We now start a discussion to prove that G − is Lipschitz. So let s = s ′ in T ∞ \ {∅} and n, m non negative integers so that N n − < | s | ≤ N n and N m − < | s ′ | ≤ N m (with the convention N − := 0). As usual, u is thegreatest common ancestor of s and s ′ and we denote p the integer suchthat N p − < | u | ≤ N p , d = ρ ( u, s ) and d ′ = ρ ( u, s ′ ). So we can write s = s ⌢ · · · ⌢ s n , s ′ = s ′ ⌢ · · · ⌢ s ′ m and u = u ⌢ · · · ⌢ u p ,with | s j | = K j for j ≤ n − 1, 0 < | s n | ≤ K n , | s ′ j | = K j for j ≤ m − < | s ′ m | ≤ K m , | u j | = K j for j ≤ p − < | u p | ≤ K p . Then, we havethat u j = s j = s ′ j for j ≤ p − u p is the greatest common ancestorof s p and s ′ p in T K p . Finally, if we denote s ⌢ · · · ⌢ s j = s jr j for j ≤ n − s ′ ⌢ · · · ⌢ s ′ j = s jr ′ j for j ≤ m − 1, we can write G ( s ) − G ( s ′ ) = X r p − ⌢u p 1) is a universal constant given by case (b). If d ′ ≥ M d , with M = 6 α , we obtain that k G ( s ) − G ( s ′ ) k ≥ α d ′ ≥ αρ ( s, s ′ )4 . new metric invariant for Banach spaces 15 So, we may as well assume that d ′ < M d . Now, with our usual carefulchoice of small δ n ’s we get h x ∗ , G ( s ) − G ( s ′ ) i ≥ a − b andeither h y ∗ , G ( s ) − G ( s ′ ) i ≥ b h z ∗ , G ( s ) − G ( s ′ ) i ≥ b . Then, using x ∗ + 25 y ∗ or x ∗ + 25 z ∗ , we obtain that k G ( s ) − G ( s ′ ) k ≥ d 104 = ( M + 1) d M + 1) ≥ d + d ′ M + 1) = ρ ( s, s ′ )104( M + 1) . All possible cases have been considered and our discussion is finished. (cid:3) On the non-embeddability of the hyperbolic trees Our aim is now to prove in the reflexive case the converse of the resultsgiven in the previous section. More precisely, the main result of this sectionis the following. Theorem 3.1. Assume that X is a separable reflexive Banach space andthat there exists C ≥ such that T N C ֒ → X for all N in N .Then either Sz ( X ) > ω or Sz ( X ∗ ) > ω . Before proceeding with the proof of this theorem, we need to recall twovery convenient renorming theorems essentially due to Odell and Schlumprecht.We refer to [19] and [20] for a complete exposition of the links between theSzlenk index of a Banach space and its embeddability into a Banach spacewith a finite dimensional decomposition with upper and lower estimates. Theorem 3.2. Let X be a separable reflexive Banach space. Then, thefollowing properties are equivalent.(i) Sz ( X ) ≤ ω .(ii) There exist < p < ∞ and an equivalent norm k · k on X such thatif U is a non-principal ultrafilter on N , x ∈ X and ( x n ) ∞ n =1 is any boundedsequence with lim n ∈U x n = 0 weakly (3.10) lim n ∈U k x + x n k ≤ lim n ∈U ( k x k p + k x n k p ) /p . This is contained in the proof of Theorem 3 of [20]. Theorem 3.3. Let X be a separable reflexive Banach space. Then, thefollowing properties are equivalent.(i) Sz ( X ) ≤ ω and Sz ( X ∗ ) ≤ ω . (ii) There exist < p < q < ∞ and an equivalent norm k · k on X such that if U is a non-principal ultrafilter on N , x ∈ X and ( x n ) ∞ n =1 is anybounded sequence with lim n ∈U x n = 0 weakly (3.11) lim n ∈U ( k x k q + k x n k q ) /q ≤ lim n ∈U k x + x n k ≤ lim n ∈U ( k x k p + k x n k p ) /p . Let us remark that (ii) is equivalent to the statements that δ ( τ ) ≥ (1 + τ q ) /q − ρ ( τ ) ≤ (1 + τ p ) /p − . This result follows directly fromTheorem 7 of [20]. Proof of Theorem 3.1. Let X be a reflexive Banach space such that Sz( X ) ≤ ω and Sz( X ∗ ) ≤ ω . We will assume that the norm satisfies (3.11) and wemay assume for convenience that p and q are conjugate i.e. p + q = 1.Let us suppose that there is a constant C ≥ N ∈ N ,we have T N C ֒ → X . We will show that for large enough N this produces acontradiction. Let us pick a ∈ N such that a > (2 C ) q . We then pick m ∈ N with m > (2 C ) q and N = a m +1 . Suppose now that u : T N → X is a map such that u ( ∅ ) = 0 and:(3.12) ∀ s, s ′ ∈ T N ρ ( s, s ′ ) ≤ k u ( s ) − u ( s ′ ) k ≤ Cρ ( s, s ′ ) . We now consider an ultraproduct X of X modeled on the set N N ; this ideais inspired by similar considerations in [15]. Let U be a fixed non-principalultrafilter on N and define the seminorm on Z = ℓ ∞ ( N N , X ) by k x k X = lim n ∈U · · · lim n N ∈U k x ( n , . . . , n N ) k . If we factor out the set { x : k x k X = 0 } this induces an ultraproduct X . For x ∈ Z and 0 ≤ k ≤ N we define E k ( x )( n , . . . , n N ) = lim n k +1 ∈U · · · lim n N ∈U x ( n , . . . , n N )where each limit is with respect to the weak topology on X (recall that X is reflexive). For k < E k x = 0 . It will be useful tointroduce F k = I − E k for the complementary projections.We now use (3.11) to deduce that if F k x = 0 and E k y = 0 then( k x k q X + k y k q X ) /q ≤ k x + y k X ≤ ( k x k p X + k y k p X ) /p . From this it follows that the projections F k are contractive. Also if 0 = k < k < k < k r and x j ∈ Z with F k j x j = 0 and E k j − x j = 0 for 1 ≤ j ≤ r then(3.13) r X j =1 k x j k q X ! /q ≤ k r X j =1 x j k X ≤ r X j =1 k x j k p X ! /p . new metric invariant for Banach spaces 17 Let us now define z j ∈ Z for 1 ≤ j ≤ N by z j ( n , . . . , n N ) = u ( n , . . . , n j ) − u ( n , . . . , n j − ) . Here we understand that z ( n , . . . , n N ) = u ( n ) . We then define w j = z j − E j − z j and then w jk = E j − a k − z j − E j − a k z j , ≤ k < ∞ . Then z j = ∞ X k =0 w jk and by (3.13) m X k =1 k w jk k X ≤ m /p ( ∞ X k =0 k w jk k q X ) /q ≤ m /p k z j k X ≤ Cm /p . This implies that(3.14) N X j =1 m X k =1 k w jk k X ≤ Cm /p N. On the other hand if 0 ≤ r ≤ r + s ≤ N we note that by (3.12),lim n ′ r +1 ∈U lim n ′ r +2 ∈U · · · lim n ′ r + s ∈U k u ( n , . . . , n r , n ′ r +1 , . . . n ′ r + s ) − u ( n , . . . , n r + s ) k ≥ s. Hence if v ∈ ℓ ∞ ( N r , X ) we havelim n r +1 ∈U · · · lim n r + s ∈U k u ( n , . . . , n r + s ) − v ( n , . . . , n r ) k ≥ s. In particular if we let v ( n , . . . , n r ) = lim n r +1 ∈U · · · lim n r + s ∈U u ( n , . . . , n r + s )(with limits in the weak topology) we obtain kF r ( r + s X j = r +1 z j ) k X ≥ s. Now suppose s = a k where k ≥ 1. If r ≤ N − a k we have a k ≤ kF r ( r + a k X j = r +1 z j ) k X ≤ k r + a k X j = r +1 F j − a k z j k X . The last inequality follows from the fact that F k F l = F l F k = F l , whenever k ≤ l and from the contractivity of F r . On the other hand k r + a k X j = r +1 F j − a k − z j k X = k r + a k − X j = r +1 a − X i =0 F j +( i − a k − z j + ia k − k X ≤ r + a k − X j = r +1 a − X i =0 kF j +( i − a k − z j + ia k − k p X ! /p ≤ Ca k − a /p ≤ a k / . Combining these statements we have that if r = λa k with 1 ≤ k ≤ m and0 ≤ λ ≤ a m +1 − k − r ≤ N − a k = a m +1 − a k ) r + a k X j = r +1 k w jk k X ≥ a k / N X j =1 k w jk k X = a m +1 − k − X λ =0 λa k X j = λa k +1 k w jk k X ≥ N . This implies(3.15) N X j =1 m X k =1 k w jk k X ≥ mN . Now (3.14) and (3.15) give a contradiction since m > (2 C ) q . (cid:3) As an immediate consequence of Theorem 3.1 and section 2 we obtainthe following characterization, which yields Theorem 1.2 announced in ourintroduction. Corollary 3.4. Let X be a separable reflexive Banach space. The followingassertions are equivalent(i) Sz ( X ) > ω or Sz ( X ∗ ) > ω .(ii) There exists C ≥ such that T ∞ C ֒ → X .(iii) There exists C ≥ such that for any N in N , T N C ֒ → X . Remark. Let us mention that we do not know if (iii) implies (i) for generalBanach spaces.4. Applications to coarse Lipschitz embeddings and uniformhomeomorphisms between Banach spaces We need to recall some definitions and notation. Let ( M, d ) and ( N, δ )be two unbounded metric spaces. We define ∀ t > ω f ( t ) = sup { δ ( f ( x ) , f ( y )) , x, y ∈ M, d ( x, y ) ≤ t } . new metric invariant for Banach spaces 19 We say that f is uniformly continuous if lim t → ω f ( t ) = 0. The map f issaid to be coarsely continuous if ω f ( t ) < ∞ for some t > L θ ( f ) = sup t ≥ θ ω f ( t ) t , for θ > L ( f ) = sup θ> L θ ( f ) , L ∞ ( f ) = inf θ> L θ ( f ) . A map is Lipschitz if and only if L ( f ) < ∞ . We will say that it is coarseLipschitz if L ∞ ( f ) < ∞ . Clearly, a coarse Lipschitz map is coarsely contin-uous. If f is bijective, we will say that f is a uniform homeomorphism (re-spectively, coarse homeomorphism, Lipschitz homeomorphism, coarse Lips-chitz homeomorphism ) if f and f − are uniformly continuous (respectively,coarsely continuous, Lipschitz, coarse Lipschitz). Finally we say that f is a coarse Lipschitz embedding if it is a coarse Lipschitz homeomorphism from X onto f ( X ).We conclude this brief introduction with the following easy and wellknown fact: if X and Y are Banach spaces, then for any map f : X → Y , ω f is a subadditive function. It follows that any coarsely continuous map f : X → Y is coarse Lipschitz. In particular, any uniform homeomorphismis a coarse Lipschitz homeomorphism. Theorem 4.1. Let X and Y be separable Banach spaces and suppose thatthere is a coarse Lipschitz embedding of X into Y . Suppose Y is reflexiveand Sz ( Y ) = ω . Then X is reflexive.Proof. We can assume by Theorem 3.2 that Y is normed to satisfy (3.10)for some 1 < p < ∞ . Now let f : X → Y be a coarse Lipschitz embedding. We may assumethat there exists C ≥ k x − x k − ≤ k f ( x ) − f ( x ) k ≤ C k x − x k + 1 x , x ∈ X. Suppose that X is a non reflexive Banach space and fix θ ∈ (0 , x n ) n in B X suchthat k y − z k ≥ θ , for all n ∈ N , all y in the convex hull of { x i } ni =1 and all z in the convex hull of { x i } i ≥ n +1 . In particular(4.16) k x n + .. + x n k − ( x m + .. + x m k ) k ≥ θk, n < .. < n k < m < .. < m k . For k ∈ N let N [ k ] denote the collection of all k -subsets of N (written inthe form ( n , . . . , n k ) where n < n < · · · < n k . We define h : N [ k ] → X by h ( n , . . . , n k ) = x n + · · · + x n k . On N [ k ] we define the distance d (( n , . . . , n k ) , ( m , . . . , m k )) = |{ j : n j = m j }| . Then h is Lipschitz with constant at most 2. Furthermore f ◦ h has Lipschitzconstant at most 2 C + 1. By Theorem 4.2 of [10] there is an infinite subset M of N so that diam f ◦ h ( M [ k ] ) ≤ C + 1) k /p . If n < n < · · · < n k The class of all reflexive Banach spaces with Szlenk indexequal to ω is stable under uniform homeomorphisms. As a final application we now state the main result of this section. Theorem 4.3. Let Y be a reflexive Banach space such that Sz ( Y ) ≤ ω andSz ( Y ∗ ) ≤ ω and assume that X is a Banach space which coarse Lipschitzembeds into Y . Then X is reflexive, Sz ( X ) ≤ ω and Sz ( X ∗ ) ≤ ω .Proof. First, it follows from Theorem 4.1 that X is reflexive. Assume nowthat Sz( X ) or Sz( X ∗ ) is greater than ω . Then, we know from Theorem2.4 that T ∞ Lipschitz embeds into X and therefore into Y . This is incontradiction with Theorem 3.1. (cid:3) Remark . Theorem 4.1, Corollary 4.2 and Theorem 4.3 should be comparedto the fact that in general reflexivity is not preserved under coarse Lipschitzembeddings or even uniform homeomorphisms. Indeed, Ribe proved in [23]that ℓ ⊕ ( P n ⊕ ℓ p n ) ℓ is uniformly homeomorphic to ( P n ⊕ ℓ p n ) ℓ , if ( p n ) n is strictly decreasing and tending to 1 (we also refer to Theorem 10.28 in [3]for a generalization of this result). The space X = ( P n ⊕ ℓ p n ) ℓ is of coursereflexive and standard computations yield that its Szlenk index is equal to ω . On the other hand, if the p n ’s are chosen in (1 , X ∗ is asymptotically uniformly smooth witha modulus of asymptotic smoothness ρ ( t ) = t . Thus, Sz( X ∗ ) = ω .So, in view of Corollary 4.2 and Theorem 4.3, Ribe’s example is optimal.Let us now recall that for a separable Banach space the condition “Sz( X ) ≤ ω ” is equivalent to the existence of an equivalent asymptotically uniformlysmooth norm on X and that for a reflexive separable Banach space thecondition “Sz( X ∗ ) ≤ ω ” is equivalent to the existence of an equivalent new metric invariant for Banach spaces 21 asymptotically uniformly convex norm on X (see [20] for a survey on theseresults and proper references). Let us now denote as in [20]: C auc = { Y : Y is separable ref lexive and has an equivalent a.u.c. norm } and C aus = { Y : Y is separable ref lexive and has an equivalent a.u.s. norm } . Then, we can restate Corollary 4.2 and Theorem 4.3 as follows Theorem 4.4. The class C aus is stable under uniform homeomorphisms andthe class C auc ∩ C aus is stable under coarse Lipschitz embeddings. References [1] F. Baudier, Metrical characterization of super-reflexivity and linear type of Banachspaces , Arch. Math. (2007), 419–429.[2] F. Baudier and G. Lancien, Embeddings of locally finite metric spaces into Banachspaces , Proc. Amer. Math. Soc. (2008), 1029–1033.[3] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1 ,American Mathematical Society Colloquium Publications, vol. 48, American Math-ematical Society, Providence, RI, 2000.[4] J. Bourgain, The metrical interpretation of super-reflexivity in Banach spaces , IsraelJ. Math. (1986), 221–230.[5] J. Bourgain, V. Milman, and H. Wolfson, On type of metric spaces , Trans. Amer.Math. Soc. (1986), 295–317.[6] P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm ,Israel J. Math. (1972), 281–288 (1973).[7] G. Godefroy, N. J. Kalton, and G. Lancien, Szlenk indices and uniform homeomor-phisms , Trans. Amer. Math. Soc. (2001), 3895–3918 (electronic).[8] R. C. James, Uniformly non-square Banach spaces , Ann. of Math. (2) (1964),542–550.[9] W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Almost Fr´echetdifferentiability of Lipschitz mappings between infinite-dimensional Banach spaces ,Proc. London Math. Soc. (3) (2002), 711–746.[10] N. J. Kalton and N. L. Randrianarivony, The coarse Lipschitz structure of ℓ p ⊕ ℓ q ,Math. Ann. (2008), 223–237.[11] H. Knaust, E. Odell, and T. Schlumprecht, On asymptotic structure, the Szlenk indexand UKK properties in Banach spaces , Positivity (1999), 173–199.[12] G. Lancien, On uniformly convex and uniformly Kadec-Klee renormings , SerdicaMath. J. (1995), 1–18.[13] , A survey on the Szlenk index and some of its applications , Revista RealAcad. Cienc. Serie A Mat. (2006), 209–235.[14] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, I, Sequence spaces ,Springer-Verlag, Berlin, 1977.[15] B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann, Asymptotic infinite-dimensional theory of Banach spaces , Geometric aspects of functional analysis(Israel, 1992), Oper. Theory Adv. Appl., vol. 77, Birkh¨auser, Basel, 1995, pp. 149–175.[16] M. Mendel and A. Naor, Scaled Enflo type is equivalent to Rademacher type , Bull.Lond. Math. Soc. (2007), 493–498.[17] , Metric cotype , Ann. of Math.(2) (2008), 247–298.[18] V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball ,Uspehi Mat. Nauk (1971), 73–149 (Russian). English translation: Russian Math.Surveys (1971), 79–163. [19] E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces , Trans. Amer.Math. Soc. (2002), 4085–4108 (electronic).[20] E. Odell and T. Schlumprecht, Embeddings into Banach spaces with finite dimen-sional decompositions , Revista Real Acad. Cienc. Serie A Mat. (2006), 295–323.[21] G. Pisier, Martingales with values in uniformly convex spaces , Israel J. Math. (1975), 326–350.[22] M. Ribe, On uniformly homeomorphic normed spaces , Ark. Mat. (1976), 237–244.[23] , Existence of separable uniformly homeomorphic nonisomorphic Banachspaces , Israel J. Math. (1984), 139–147.[24] W. Szlenk, The non existence of a separable reflexive Banach space universal for allseparable reflexive Banach spaces , Studia Math. (1968), 53–61. Universit´e de Franche-Comt´e, Laboratoire de Math´ematiques UMR 6623,16 route de Gray, 25030 Besanc¸on Cedex, FRANCE. E-mail address : [email protected] Department of Mathematics, University of Missouri-Columbia, Columbia,MO 65211 E-mail address : [email protected] Universit´e de Franche-Comt´e, Laboratoire de Math´ematiques UMR 6623,16 route de Gray, 25030 Besanc¸on Cedex, FRANCE. E-mail address ::