Metrical characterization of super-reflexivity and linear type of Banach spaces

METRICAL CHARA CTERIZA TION OF SUPER-REFLEXIVITY AND LINEAR TYPE OF BANA CH SP A CES FLORENT BAUDIER † Abstract. W e prov e that a Banac h space X is not super-reflexive if an d only if the h yp erbol i c infinite tree embeds metrically into X . W e improv e one implication of J.Bourgain’s result who ga ve a metrical charact erization of sup er-reflexivity in Banac h spaces in terms of uniforms embeddings of the finite t rees. A characterization of the linear t ype for Banac h spaces is giv en using the em bedding of the infinite tree equipped with the metrics d p induced by the ℓ p norms. 1. Introduction and Not a tion W e fix some notation a nd reca ll basic results. Let ( M , d ) a nd ( N , δ ) b e tw o metric space s and an injective map f : M → N . F ollowing [11], we define the distortion of f to b e dist( f ) := k f k Lip k f − 1 k Lip = sup x 6 = y ∈ M δ ( f ( x ) , f ( y )) d ( x, y ) . sup x 6 = y ∈ M d ( x, y ) δ ( f ( x ) , f ( y )) . If dist( f ) is finite, we s ay that f is a metric embedding, or simply an em b edding o f M into N . And if there exists an em b edding f from M in to N , with dist( f ) ≤ C , we use the no tation M C ֒ → N . Denote Ω 0 = { ∅ } , the roo t of the tr ee. Let Ω n = {− 1 , 1 } n , T n = S n i =0 Ω i and T = S ∞ n =0 T n . Thus T n is the finite tree with n levels and T the infinite tree. F or ε , ε ′ ∈ T , we note ε ≤ ε ′ if ε ′ is an extension of ε . Denote | ε | the length of ε ; i.e the num b ers of no des of ε . W e define the h yp erb olic dis- tance betw een ε and ε ′ by ρ ( ε, ε ′ ) = | ε | + | ε ′ | − 2 | δ | , where δ is the g reatest common ancestor of ε and ε ′ . The metric on T n , is the restrictio n of ρ . F or a Banach space X , we deno te B X its closed unit ball, and X ∗ its dual space . T embeds isometrically into ℓ 1 ( N ) in a trivial wa y . Actually , let ( e ε ) ε ∈ T be the c anonical basis of ℓ 1 ( T ) ( T is countable), then the embedding is given by ε 7→ P s ≤ ε e s . Laboratoire de M ath ´ ematiques, UMR 6623 Unive rsit´ e de F r anc he-Comt ´ e, 25030 Besan¸ con, cedex - F rance † floren t.baudier@math.univ-fcomte .fr 2000 Mathematics Subje ct Classific ation . (46B20) (51F99) 1 2 FLORENT BA UDIE R † Aharoni prov ed in [1] that every separa ble metric space em b eds into c 0 , so T doe s . The main r esult o f this article is an impr ov emen t of Bourga in’s metrical ch ara cterization of s up er -reflexivity . Bourg ain proved in [2] t hat X is not sup er-r eflexive if and only if the finite trees T n uniformly embed into X (i.e with em b e dding constants independent of n ). Obviously if T embeds in to X then the T ′ n s em bed uniformly in to X and X is not sup er- reflexive, but if X is not sup er-refle x ive we did not k now w he ther the infinite tree T embeds int o X . In this pape r , we prove that it is indeed the cas e : Theorem 1.1. L et X b e a non sup er-r eflexive Banach sp ac e, then ( T , ρ ) emb e ds into X . The pro o f of the direct part of Bourga in’s Theorem e s sentially uses James’ characteriza- tion o f sup er-reflexiv it y (see [7]) and a n enumeration of the finite trees T n . W e r ecall James’ Theorem : Theorem 1.2 (James) . L et 0 < θ < 1 and X a non su p er-r eflexive Banach sp ac e, then : ∀ n ∈ N , ∃ x 1 , x 2 , . . . , x n ∈ B X , ∃ x ∗ 1 , x ∗ 2 , . . . , x ∗ n ∈ B X ∗ s.t : x ∗ k ( x j ) = θ ∀ k < j x ∗ k ( x j ) = 0 ∀ k ≥ j 2. Metrical charac teriza tion of super-reflexivity The main obstruction to the e mbedding o f T in to any non-sup er- reflexive B anach space X is the finiteness of the sequences in James’ characteriz a tion. How, with a sequence of Bourga in’s type embedding, can we c o nstruct a s ingle embedding from T in to X ? In [13], Ribe shows in particular, that L 2 l p n and ( L 2 l p n ) L l 1 are uniformly homeo- morphic, where ( p n ) n is a sequence of nu mbers such that p n > 1, a nd p n tends to 1 . But T embeds into l 1 , hence via the uniform homeomorphism T em b eds into L 2 l p n . How ever T do es no t embed int o any l p n (they a re s uper -reflexive). The pro blem so lved in the next theo r em, inspir ed in part by Rib e’s pro of, is to c o nstruct a subspace with a Schauder decomp osition L F n where T 2 n +1 embeds into F n and to r e past prop erly the embeddings in order to o bta in the desired embedding. Pr o of of Th e or em 1.1 : Let ( ε i ) i ≥ 0 , a sequence of p ositive real num b ers s uch that Q i ≥ 0 (1 + ε i ) ≤ 2, and fix 0 < θ < 1. Let k n = 2 2 n +1 +1 − 1. First we constr uc t inductiv ely a s e quence ( F n ) n ≥ 0 of s ubs pa ces of X , whic h is a Sc hauder finite dimensional decomp ositio n of a subspace of X s.t the pro jection from L q i =0 F i onto L p i =0 F i , with kernel L q i = p +1 F i (with p < q ) is o f norm at most Q q − 1 i = p (1 + ε i ), and sequenc e s x n, 1 , x n, 2 , . . . , x n,k n ∈ B F n x ∗ n, 1 , x ∗ n, 2 , . . . , x ∗ n,k n ∈ B X ∗ s.t : x ∗ n,k ( x n,j ) = θ ∀ k < j x ∗ n,k ( x n,j ) = 0 ∀ k ≥ j. Metrical characteriz ation of sup er-refle xivity and line ar t yp e of Banach spaces 3 Denote Φ n : T n → { 1 , 2 , . . . , 2 n +1 − 1 } the enumeration of T n following the lex ic ographic order. It is an enumeration of T n such that a ny pa ir of segments in T n starting a t incompa- rable no des (with resp ect to the tre e or dering ≤ ) ar e mappe d inside disjoint interv als. Let Ψ n = Φ 2 n +1 and Γ n = T 2 n +1 . X is no n sup er-r eflexive, hence from James’ Theorem : ∃ x 0 , 1 , x 0 , 2 , . . . , x 0 , 7 ∈ B X , ∃ x ∗ 0 , 1 , x ∗ 0 , 2 , . . . , x ∗ 0 , 7 ∈ B X ∗ s.t : x ∗ 0 ,k ( x 0 ,j ) = θ ∀ k < j x ∗ 0 ,k ( x 0 ,j ) = 0 ∀ k ≥ j. Γ 0 = T 2 embeds in to X via the embedding f 0 ( ε ) = P s ≤ ε x 0 , Ψ 0 ( s ) (see [2]). Let F 0 = Span { x 0 , 1 , . . . , x 0 , 7 } , then dim( F 0 ) < ∞ . Suppo se that F 0 , . . . , F p , and x p, 1 , x p, 2 , . . . , x p,k p ∈ B F p x ∗ p, 1 , x ∗ p, 2 , . . . , x ∗ p,k p ∈ B X ∗ verifying the req uir ed co nditio ns , ar e constructed for all p ≤ n . W e apply Mazur ’s Lemma (see [9] page 4) to the finite dimensio nal s ubs pace L n i =0 F i of X . Thu s there exists Y n ⊂ X s uch that dim( X/ Y n ) < ∞ and : k x k ≤ (1 + ε n ) k x + y k , ∀ ( x, y ) ∈ n M i =0 F i × Y n . But Y n is of finite co dimension in X , hence is not sup er-r eflexive. F ro m James’ Theorem and Hahn-B anach Theor em: ∃ x n +1 , 1 , x n +1 , 2 , . . . , x n +1 ,k n +1 ∈ B Y n , ∃ x ∗ n +1 , 1 , x ∗ n +1 , 2 , . . . , x ∗ n +1 ,k n +1 ∈ B X ∗ , s.t : x ∗ n +1 ,k ( x n +1 ,j ) = θ ∀ k < j x ∗ n +1 ,k ( x n +1 ,j ) = 0 ∀ k ≥ j. Γ n +1 embeds in to Y n via the em b edding f n +1 ( ε ) = P s ≤ ε x n +1 , Ψ n +1 ( s ) . Let F n +1 = Span { x n +1 ,j ; 1 ≤ j ≤ k n +1 } , then dim( F n +1 ) < ∞ , which achieves the induc- tion. Now define the following pro jections : Let, P n the pro jection from Span( S ∞ i =0 F i ) onto F 0 L · · · L F n with kernel Span( S ∞ i = n +1 F i ). It is easy to show that k P n k ≤ Q ∞ i = n (1 + ε i ) ≤ 2. W e denote now Π 0 = P 0 and Π n = P n − P n − 1 for n ≥ 1. W e have that k Π n k ≤ 4. 4 FLORENT BAUDIER † F ro m Bourg a in’s constructio n, for all n : (1) θ 3 ρ ( ε, ε ′ ) ≤ k f n ( ε ) − f n ( ε ′ ) k ≤ ρ ( ε, ε ′ ) , where f n denotes the Bourgain’s type e mbedding from Γ n in F n , i.e f n ( ε ) = P s ≤ ε x n, Ψ n ( s ) . Note that : ∀ n, ∀ ε ∈ Γ n k f n ( ε ) k ≤ | ε | . Now we define our embedding. Let f : T → Y = Span( S ∞ i =0 F i ) ⊂ X ε 7→ λf n ( ε ) + (1 − λ ) f n +1 ( ε ) , if 2 n ≤ | ε | ≤ 2 n +1 where, λ = 2 n +1 − | ε | 2 n And f ( ∅ ) = 0 . W e will pr ov e that : (2) ∀ ε, ε ′ ∈ T θ 24 ρ ( ε, ε ′ ) ≤ k f ( ε ) − f ( ε ′ ) k ≤ 9 ρ ( ε, ε ′ ) . Remark 2.1 W e hav e θ 24 | ε | ≤ k f ( ε ) k ≤ | ε | . First of all, we show that f is 9 − Lipschitz. W e can suppo se that 0 < | ε | ≤ | ε ′ | w.r.t rema rk 2.1. If | ε | ≤ 1 2 | ε ′ | then : ρ ( ε, ε ′ ) ≥ | ε ′ | − | ε | ≥ | ε | + | ε ′ | 3 Hence, k f ( ε ) − f ( ε ′ ) k ≤ 3 ρ ( ε, ε ′ ) . If 1 2 | ε ′ | < | ε | ≤ | ε ′ | , w e have t wo different ca ses to consider . 1) if 2 n ≤ | ε | ≤ | ε ′ | < 2 n +1 . Then, let λ = 2 n +1 − | ε | 2 n and λ ′ = 2 n +1 − | ε ′ | 2 n . k f ( ε ) − f ( ε ′ ) k = k λf n ( ε ) − λ ′ f n ( ε ′ ) + (1 − λ ) f n +1 ( ε ) − (1 − λ ′ ) f n +1 ( ε ′ ) k ≤ λ k f n ( ε ) − f n ( ε ′ ) k + | λ − λ ′ | ( k f n ( ε ′ ) k + k f n +1 ( ε ′ ) k ) + (1 − λ ) k f n +1 ( ε ) − f n +1 ( ε ′ ) k ≤ ρ ( ε, ε ′ ) + 2 ρ ( ε, ε ′ ) + 2 ρ ( ε, ε ′ ) ≤ 5 ρ ( ε, ε ′ ) , Metrical characteriz ation of sup er-refle xivity and line ar t yp e of Banach spaces 5 bec ause k f n ( ε ′ ) k < 2 n +1 , k f n +1 ( ε ′ ) k < 2 n +1 and, | λ − λ ′ | = | ε ′ | − | ε | 2 n ≤ ρ ( ε, ε ′ ) 2 n . 2) if 2 n ≤ | ε | ≤ 2 n +1 ≤ | ε ′ | < 2 n +2 . Then, let λ = 2 n +1 − | ε | 2 n and λ ′ = 2 n +2 − | ε ′ | 2 n +1 . k f ( ε ) − f ( ε ′ ) k = k λf n ( ε ) + (1 − λ ) f n +1 ( ε ) − λ ′ f n +1 ( ε ′ ) − (1 − λ ′ ) f n +2 ( ε ′ ) k ≤ λ ( k f n ( ε ) k + k f n +1 ( ε ) k ) + (1 − λ ′ )( k f n +1 ( ε ′ ) k + k f n +2 ( ε ′ ) k ) + k f n +1 ( ε ) − f n +1 ( ε ′ ) k ≤ ρ ( ε, ε ′ ) + 2 λ | ε | + 2(1 − λ ′ ) | ε ′ | ≤ 9 ρ ( ε, ε ′ ) , bec ause, λ ≤ ρ ( ε, ε ′ ) 2 n , so λ | ε | ≤ 2 ρ ( ε, ε ′ ) . Similarly 1 − λ ′ = | ε ′ | − 2 n +1 2 n +1 ≤ ρ ( ε, ε ′ ) 2 n +1 and (1 − λ ′ ) | ε ′ | ≤ 2 ρ ( ε , ε ′ ) . Finally , f is 9-Lipschitz. Now we deal with the minoration. In our next discus s ion, whenever | ε | (resp ectively | ε ′ | ) will b elo ng to [2 n , 2 n +1 ), for so me int eger n , w e sha ll denote λ = 2 n +1 − | ε | 2 n (resp ectively λ ′ = 2 n +1 − | ε ′ | 2 n ) . W e ca n suppo s e that ε is smaller than ε ′ in the lexicog raphic order. Denote δ the gre atest common a ncestor of ε and ε ′ . And let d = | ε | − | δ | (resp ectively d ′ = | ε ′ | − | δ | ). 1) if 2 n ≤ | ε | , | ε ′ | ≤ 2 n +1 . W e hav e, x ∗ n, Ψ n ( δ ) Π n ( f ( ε ) − f ( ε ′ )) = θ ( λd − λ ′ d ′ ) x ∗ n +1 , Ψ n +1 ( δ ) Π n +1 ( f ( ε ) − f ( ε ′ )) = θ ((1 − λ ) d − (1 − λ ′ ) d ′ ) . Hence, k f ( ε ) − f ( ε ′ ) k ≥ θ ( d − d ′ ) 8 . And, − x ∗ n, Ψ n ( ε ) Π n ( f ( ε ) − f ( ε ′ )) = θ λ ′ d ′ − x ∗ n +1 , Ψ n +1 ( ε ) Π n +1 ( f ( ε ) − f ( ε ′ )) = θ (1 − λ ′ ) d ′ . 6 FLORENT BAUDIER † So, k f ( ε ) − f ( ε ′ ) k ≥ θd ′ 8 . Finally if we distinguish the cases d 2 ≤ d ′ , and d ′ < d 2 we o btain : k f ( ε ) − f ( ε ′ ) k ≥ θ ( d + d ′ ) 24 = θ 24 ρ ( ε, ε ′ ) . 2) if 2 n ≤ | ε | ≤ 2 n +1 ≤ 2 q +1 ≤ | ε ′ | ≤ 2 q +2 , or 2 n ≤ | ε ′ | ≤ 2 n +1 ≤ 2 q +1 ≤ | ε | ≤ 2 q +2 . If n < q , | x ∗ q +1 , Ψ q +1 ( δ ) Π q +1 ( f ( ε ) − f ( ε ′ )) + x ∗ q +2 , Ψ q +2 ( δ ) Π q +2 ( f ( ε ) − f ( ε ′ )) | = θ M ax ( d, d ′ ) Hence, k f ( ε ) − f ( ε ′ ) k ≥ θ 16 ρ ( ε, ε ′ ) . If n = q and | ε | ≤ | ε ′ | , | x ∗ n +1 , Ψ n +1 ( ε ) Π n +1 ( f ( ε ) − f ( ε ′ )) + x ∗ n +2 , Ψ n +2 ( δ ) Π n +2 ( f ( ε ) − f ( ε ′ )) | ≥ θ d ′ . So, k f ( ε ) − f ( ε ′ ) k ≥ θ 16 ρ ( ε, ε ′ ) . If n = q and | ε ′ | < | ε | , x ∗ n +1 , Ψ n +1 ( δ ) Π n +1 ( f ( ε ) − f ( ε ′ )) − x ∗ n +1 , Ψ n +1 ( ε ) Π n +1 ( f ( ε ) − f ( ε ′ ))+ x ∗ n +2 , Ψ n +2 ( δ ) Π n +2 ( f ( ε ) − f ( ε ′ )) = θ d. Hence, k f ( ε ) − f ( ε ′ ) k ≥ θ 24 ρ ( ε, ε ′ ) . Finally T 216 θ ֒ → X .  Corollary 2.2. X is n on sup er-r eflexive if and only if ( T , ρ ) emb e ds into X . Pr o of : It follows clearly fro m Bour g ain’s result [2] and The o rem 1 .1.  Metrical characteriz ation of sup er-refle xivity and line ar t yp e of Banach spaces 7 3. Metric characteriza tion of the linear type First we identify ca nonicaly {− 1 , 1 } n with K n = {− 1 , 1 } n × Q k>n { 0 } . Let p ∈ [1 , ∞ ). Then we define an other metric on T = S K n as fo llows : ∀ ε, ε ′ ∈ T , d p ( ε, ε ′ ) = ( ∞ X i =0 | ε i − ε ′ i | p ) 1 p . The length of ε ∈ T can b e viewed as | ε | = ( d p ( ε, 0)) p . The nor m k . k p on ℓ p coincides with d p for the elements of T . W e recall now tw o c la ssical definitions : Let X and Y be tw o Banac h spaces. If X and Y are linearly isomorphic, the Ba nach- Mazur distanc e b etw een X and Y , deno ted by d B M ( X, Y ), is the infimum of k T k k T − 1 k , ov er a ll linear is o morphisms T from X o nto Y . F or p ∈ [1 , ∞ ], we say that a Ba nach space X uniformly contains the ℓ n p ’s if there is a constant C ≥ 1 such that for every in teger n , X admits a n n - dimens io nal subspace Y so that d B M ( ℓ n p , Y ) ≤ C . W e state and prov e now the following result. Theorem 3.1. L et p ∈ [1 , ∞ ) . If X uniformly c ontains the ℓ n p ’s then ( T , d p ) emb e ds into X . Pr o of : W e first r ecall a fundamental r esult due to Kriv ine (for 1 < p < ∞ in [8 ]) and James (for p = 1 a nd ∞ in [7]). Theorem 3.2 (J ames-Kriv ine) . L et p ∈ [1 , ∞ ] and X b e a Banach sp ac e uniformly c on- taining the ℓ n p ’s. Then, for any fi n ite c o dimensional su bsp ac e Y of X , any ǫ > 0 and any n ∈ N , ther e exists a subsp ac e F of Y su ch that d B M ( ℓ n p , F ) < 1 + ǫ . Using Theore m 3.2 together with the fac t that each ℓ n p is finite dimens io nal, we c a n build inductively finite dimensional subspaces ( F n ) ∞ n =0 of X and ( R n ) ∞ n =0 so that for every n ≥ 0, R n is a linear isomorphism from ℓ n p onto F n satisfying ∀ u ∈ ℓ n p 1 2 k u k ≤ k R n u k ≤ k u k and also s uch that ( F n ) ∞ n =0 is a Sc hauder finite dimensio na l decomp osition o f its close d linear span Z . More pr e cisely , if P n is the pro jection fro m Z o nt o F 0 ⊕ ... ⊕ F n with kernel Span ( S ∞ i = n +1 F i ), we will as sume as we may , that k P n k ≤ 2 . W e denote now Π 0 = P 0 and Π n = P n − P n − 1 for n ≥ 1. W e have that k Π n k ≤ 4. W e now consider ϕ n : T n → ℓ n p defined b y ∀ ε ∈ T n , ϕ n ( ε ) = | ε | X i =1 ε i e i , where ( e i ) is the canonica l basis of ℓ n p . The map ϕ n is clea rly an isometr ic embedding of T n int o ℓ n p . 8 FLORENT BAUDIER † Then we set : ∀ ε ∈ T n , f n ( ε ) = R n ( ϕ n ( ε )) ∈ F n . Finally we constr uct a map f : T → X as follows : f : T → X ε 7→ λf m ( ε ) + (1 − λ ) f m +1 ( ε ) , if 2 m ≤ | ε | < 2 m +1 , where, λ = 2 m +1 − | ε | 2 m . Remark 3.3 W e hav e 1 16 | ε | 1 p ≤ k f ( ε ) k ≤ | ε | 1 p . Like in th e pro of of Theorem 1.1 ,we prov e that f is 9-Lipschitz using exa ctly the sa me computations. W e shall now pr ove that f − 1 is Lips chit z. W e co nsider ε, ε ′ ∈ T and assume a gain that 0 < | ε | ≤ | ε ′ | . W e need to study t wo different cases. Again, whenever | ε | (resp ectively | ε ′ | ) will b elong to [2 m , 2 m +1 ), fo r some integer m , we shall deno te λ = 2 m +1 − | ε | 2 m (resp ectively λ ′ = 2 m +1 − | ε ′ | 2 m ) . 1) if 2 m ≤ | ε | , | ε ′ | < 2 m +1 . d p ( ε, ε ′ ) ≤ k λ P | ε | i =1 ε i e i − λ ′ P | ε ′ | i =1 ε ′ i e i k p + k (1 − λ ) P | ε | i =1 ε i e i − (1 − λ ′ ) P | ε ′ | i =1 ε ′ i e i k p ≤ 2 k Π m ( f ( ε ) − f ( ε ′ )) k + 2 k Π m +1 ( f ( ε ) − f ( ε ′ )) k ≤ 16 k f ( ε ) − f ( ε ′ ) k . 2) if 2 m ≤ | ε | ≤ 2 m +1 ≤ 2 q +1 ≤ | ε ′ | < 2 q +2 . if m < q , d p ( ε, ε ′ ) ≤ 2 d p ( ε ′ , 0) ≤ 2((1 − λ ′ ) d p ( ε ′ , 0) + λ ′ d p ( ε ′ , 0)) ≤ 2(2 k Π q +2 ( f ( ε ) − f ( ε ′ )) k + 2 k Π m +1 ( f ( ε ) − f ( ε ′ )) k ) ≤ 32 k f ( ε ) − f ( ε ′ ) k . Metrical characteriz ation of sup er-refle xivity and line ar t yp e of Banach spaces 9 if m = q , d p ( ε, ε ′ ) ≤ λd p ( ε, 0) + k (1 − λ ) P | ε | i =1 ε i e i − λ ′ P | ε ′ | i =1 ε ′ i e i k p + (1 − λ ′ ) d p ( ε ′ , 0) ≤ 2 k Π m ( f ( ε ) − f ( ε ′ )) k + 2 k Π m +1 ( f ( ε ) − f ( ε ′ )) k + 2 k Π m +2 ( f ( ε ) − f ( ε ′ )) k ≤ 24 k f ( ε ) − f ( ε ′ ) k . Finally we obtain that f − 1 is 32 -Lipschitz, and T 288 ֒ → X .  In the s equel a Banach s pace X is said to hav e typ e p > 0 if ther e exists a constant T < ∞ such that for every n and every x 1 , . . . , x n ∈ X , E ε k n X j =1 ε j x j k p X ≤ T p n X j =1 k x j k p X , where the exp ectation E ε is with res pe c t to a unifor m choice of s ig ns ε 1 , . . . , ε n ∈ {− 1 , 1 } n . The set of p ’s for which X contains ℓ n p ’s unifor mly is closely rela ted to the type of X according to the follo wing result due to Maurey , Pisier [10] and Krivine [8 ], which clarifies the mea ning o f these notions. Theorem 3.4 (Maurey-Pisier -Krivine) . L et X b e an infinite-dimensional Banach sp ac e. L et p X = sup { p ; X is of type p } , Then X c ont ains ℓ n p ’s un iformly for p = p X . Equivalently, we ha ve p X = inf { p ; X co ntains ℓ n p ’s uniformly } . W e deduce from Theorem 3 .1 tw o corollar ies. Corollary 3.5. L et X a Banach sp ac e and 1 ≤ p < 2 . The fol lowing assertions ar e e quivalent : i) p X ≤ p . ii) X uniformly c ontains the ℓ n p ’s. iii) the ( T n , d p ) ’s uniformly emb e d into X . iv) ( T , d p ) emb e ds into X . Pr o of : ii ) implies i ) is obvious. i ) implies ii ) is due to Theor e m 3.2 and the work o f Bretag nolle, Dacunha-Castelle and Krivine [4]. F or the equiv a lence be t ween ii ) and ii i ) see the work of Bourg ain, Milman and W o lfson [3] a nd Krivine [8]. iv ) implies iii ) is obvious. And i i ) implies iv ) is Theor em 3.1. 10 FLORENT BAUDIER †  Corollary 3.6. 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