G Lancien
email: gilles.lancien@univ-fcomte.fr

Colin Petitjean
email: colin.petitjean@univ-fcomte.fr

A Proch Ázka

Antonin Procházka
email: antonin.prochazka@univ-fcomte.fr

ON THE COARSE GEOMETRY OF JAMES SPACES

Keywords: 2010 Mathematics Subject Classification. 46B20, 46B80, 46B85, 46T99

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space J nor into its dual J * . It is a particular case of a more general result on the non equicoarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property Q of Kalton. We conclude with a remark on the coarse geometry of the James tree space J T and of its predual.

Introduction

In a fundamental paper on the coarse geometry of Banach spaces ( [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF]), N. Kalton introduced a property of metric spaces that he named property Q. In particular, its absence served as an obstruction to coarse embeddability into reflexive Banach spaces. This property is related to the behavior of Lipschitz maps defined on a particular family of metric graphs that we shall denote ([N] k , d k K ) k∈N . We will recall the precise definitions of these graphs and of property Q in section 2.2. Let us just say, vaguely speaking for the moment, that a Banach space X has property Q if for every Lipschitz map f from ([N] k , d k K ) to X, there exists a full subgraph [M] k of [N] k , with M infinite subset of N, on which f satisfies a strong concentration phenomenon. It is then easy to see that if a Banach space X has property Q, then the family of graphs ([N] k , d k K ) k∈N does not equi-coarsely embed into X (see the definition in section 2.1). One of the main results in [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF] is that any reflexive Banach space has property Q. It then readily follows that a reflexive Banach space cannot contain a coarse copy of all separable metric spaces, or equivalently does not contain a coarse copy of the Banach space c 0 . In fact, with a sophistication of this argument, Kalton proved an even stronger result in [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF]: if a separable Banach space contains a coarse copy of c 0 , then there is an integer k such that the dual of order k of X is non separable. In particular, a quasi-reflexive Banach space does not contain a coarse copy of c 0 . However, Kalton proved that the most famous example of a quasireflexive space, namely the James space J , as well as its dual J * , fail property Q.

The main purpose of this paper is to show that, although they do not obey the concentration phenomenon described by property Q, neither J nor J * equi-coarsely contains the family of graphs ([N] k , d k K ) k∈N (Corollary 5.3). This provides a coarse invariant, namely "not containing equi-coarsely the Kalton graphs", that is very close to but different from property Q. This could allow to find obstructions to coarse embeddability between seemingly close Banach spaces. Our result is actually more general. We prove in Theorem 4.1 that a quasi-reflexive Banach space X such that both X and X * admit an equivalent p-asymptotically uniformly smooth norm (see the definition in section 3), for some p in (1, ∞), does not equi-coarsely contain the Kalton graphs. We conclude this note by showing that if the James tree space J T or its predual coarsely embeds into a separable Banach space X, then there exists k ∈ N so that the dual of order k of X is non separable. This extends slightly Theorem 3.5 in [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF].

Metric notions

2.1. Coarse embeddings. Let M , N be two metric spaces and f : M → N be a map. We define the compression modulus ρ f and the expansion modulus ω f as follows. For t ∈ [0, ∞), we set

ρ f (t) = inf{d N (f (x), f (y)) : d M (x, y) ≥ t}, ω f (t) = sup{d N (f (x), f (y)) : d M (x, y) ≤ t}.
We adopt the convention sup(∅) = 0 and inf(∅) = ∞. Note that for every

x, y ∈ M , ρ f (d M (x, y)) ≤ d N (f (x), f (y)) ≤ ω f (d M (x, y)).
We say that f is a coarse embedding if ω f (t) < ∞ for every t ∈ [0, +∞) and lim t→∞ ρ f (t) = ∞. Next, let (M i ) i∈I be a family of metric spaces. We say that the family (M i ) i∈I equi-coarsely embeds into a metric space N if there exist two maps ρ, ω : [0, +∞) → [0, +∞) and maps f i : M i → N for i ∈ I such that:

(i) lim t→∞ ρ(t) = ∞, (ii) ω(t) < ∞ for every t ∈ [0, +∞), (iii) ρ(t) ≤ ρ f i (t) and ω f i (t) ≤ ω(t) for every i ∈ I and t ∈ [0, ∞).

The Kalton interlaced graphs and property

Q. For k ∈ N and M an infinite subset of N, we put [M] ≤k = {S ⊂ M : |S| ≤ k}, [M] k = {S ⊂ M : |S| = k}, [M] ω = {S ⊂ M : S is infinite}, and [M] <ω = {S ⊂ M : S is finite}. We always list the elements of some m in [N] <ω or in [N] ω in increasing order, meaning that if we write m = (m 1 , m 2 , . . . , m l ) or m = (m 1 , m 2 , m 3 , . . .), we tacitly assume that m 1 < m 2 < • • • . For m = (m 1 , m 2 , . . . , m r ) ∈ [N] <ω and n = (n 1 , n 2 , . . . , n s ) ∈ [N] <ω , we write m ≺ n, if r < s ≤ k and m i = n i , for i = 1, 2, . . . ,

r, and we write

m n if m ≺ n or m = n. Thus m n if m is an initial segment of n.
Following Kalton [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF], for M ∈ [N] ω , we equip [M] k with a graph structure by declaring m = n ∈ [M] k adjacent if and only if

n 1 ≤ m 1 ≤ n 2 . . . ≤ n k ≤ m k or m 1 ≤ n 1 ≤ m 2 . . . ≤ m k ≤ n k . For any m, n ∈ [M] k , the distance d k K (m, n) is then defined as the shortest path distance in the graph [M] k . Remark 2.1. The distance d k K is independent of the set M and therefore [M 1 ] k is a metric subspace of [M 2 ] k whenever M 1 ∈ [M 2 ] ω .
This last claim is an immediate consequence of the following explicit formula for the distance.

Proposition 2.2. Let k ∈ N and M ∈ [N] ω . Then d k K (n, m) = d(n, m) for all n, m ∈ [M] k where d(n, m) = sup{ |n ∩ S| -|m ∩ S| : S segment of N}. Proof. It is easily seen that d is a metric on [M] k . Since d k K is a graph metric on [M] k , in order to show d k K = d it is enough to verify that d k K (n, m) = 1 if and only if d(n, m) = 1 and that d is a graph metric.
For A ⊂ N let us denote 1 A : N → {0, 1} the indicator function of A and let us first observe the following fact.

Fact: For every n, m ∈ [M] k , d(n, m) = max i F (i) -min i F (i)
where Without loss of generality we will assume that max F n,m > 0. Notice that the sets arg max(F ) and arg min(F ) are disjoint. We select inductively {a 1 < . . . < a p } ⊂ arg max(F ) and {b 1 < . . . < b q } ⊂ arg min(F ) (with p ≥ 1 and q ≥ 0) with the property that

F (i) = F n,m (i) = i j=1 1 n (j) -1 m (j) (and F (0) = 0). Indeed, we have for any segment S = [a, b] that |S ∩ n| -|S ∩ m| = j∈S 1 n (j) -1 m (j) = F (b) -F (a -1). In particular max S |S ∩ n| -|S ∩ m| ≤ max F -min F . On the other hand if S = [a, b] is such that {F (a -1), F (b)} = {max F, min F } then |S ∩ n| -|S ∩ m| ≥ max F -min F
• a 1 = min arg max(F ),

• For i ≥ 1, b i = min ({n > a i } ∩ arg min(F )), if this is not empty. • a i+1 = min ({n > b i } ∩ arg max(F ))
, if this set is not empty. Notice that {a 1 , . . . , a p } ⊂ n \ m and {b 1 , . . . , b q } ⊂ m \ n. Notice also that either p = q or p = q + 1. In the latter case we define b p := r for some r such that r > a p and F (r -1) > F (r). Such r must exist since F (max{n k , m k }) = 0. Also we have r ∈ m \ n. We will set

ℓ = n ∪ {b 1 , . . . , b p } \ {a 1 , . . . , a p }.
It is clear that ℓ ∈ [M] k . We also have max F ℓ,m = max F n,m -1 and min F ℓ,m = min F n,m . Indeed, the point ℓ is constructed in such a way that when F n,m attains its maximum for the first time (going from the left), F ℓ,m is reduced by one and stays reduced by 1 until the next time the minimum of F n,m is attained (or until the point r) where this reduction is corrected back; and so on. Thus d(ℓ, m) = d(n, m) -1. Also, since the sets {a 1 , . . . , a p } and {b 1 , . . . , b p } are interlaced we have F n,m -1 ≤ F ℓ,m ≤ F n,m . Therefore, since F n,m = F n,ℓ + F ℓ,m , we have that 0 ≤ F n,ℓ ≤ 1 and so finally d(n, ℓ) = 1, since it is clear that n = ℓ.

Note that if X is a Banach space and f : ([M] k , d k K ) → X is a map with finite expansion modulus ω f , then ω f (1) is actually the Lipschitz constant of f as d k K is a graph distance on [M] k . In [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF] the property Q is defined in the setting of metric spaces. For homogeneity reasons, its definition can be simplified for Banach spaces. Let us recall it here. Definition 2.3. Let X be a Banach space. We say that X has property Q if there exists C ≥ 1 such that for every k ∈ N and every Lipschitz map

f : ([N] k , d k K ) → X, there exists an infinite subset M of N such that: ∀ n, m ∈ [M] k , f (n) -f (m) ≤ Cω f (1).
The following proposition should be clear from the definitions. We shall however include its short proof. Proposition 2.4. Let X be a Banach space. If X has property Q, then the family of graphs ([N] k , d k K ) k∈N does not equi-coarsely embed into X. Proof. Let C ≥ 1 be given by the definition of property Q. Aiming for a contradiction, assume that the family (

[N] k , d k K ) k∈N equi-coarsely embeds into X. That is, there are maps f k : ([N] k , d k K ) → X and two functions ρ, ω : [0, +∞) → [0, +∞) such that lim t→∞ ρ(t) = ∞ and ∀k ∈ N ∀t > 0 ρ(t) ≤ ρ f k (t) and ω f k (t) ≤ ω(t) < ∞.
Thus, for every k ∈ N, there exists an infinite subset [START_REF] Albiac | Topics in Banach Space Theory[END_REF]. This contradicts the fact that lim t→∞ ρ(t) = ∞.

M k of N such that diam (f ([M k ] k ))) ≤ Cω(1). Since diam ([M k ] k ) = k, this implies that for all k ∈ N, ρ(k) ≤ Cω
A concrete bi-Lipschitz copy of the metric spaces ([N] k , d k K ) in c 0 is given by the following proposition. Proposition 2.5. Let (s n ) ∞ n=1 be the summing basis of c 0 , that is

s n = n i=1 e i , where (e i ) ∞ i=1 is the canonical basis of c 0 . For k ∈ N, define f k : ([N] k , d k K ) → c 0 by f k (n) = k i=1 s n i . Then 1 2 d k K (n, m) ≤ f k (n) -f k (m) ∞ ≤ d k K (n, m) for all n, m ∈ [N] k . Proof. Since d k K = d, one can show (as in the Fact in the proof of Proposi- tion 2.2) that d k K (n, m) = max(f k (n) -f k (m)) -min(f k (n) -f k (m)). The result then follows easily since min(f k (n)-f k (m)) ≤ 0 ≤ max(f k (n)-f k (m)) for all n, m ∈ [N] k .
Remark 2.6. We already explained that c 0 cannot coarsely embed into any Banach space with property Q (in particular into any reflexive Banach space) and that Kalton even showed with additional arguments that if c 0 coarsely embeds into a separable Banach space X, then one of the iterated duals of X has to be non separable. An inspection of his proof shows that the uniformly discrete metric spaces

M k = k i=1 s n i × 1 A : (n 1 , . . . , n k ) ∈ [N] k , A ∈ [N] ω ⊂ c 0
do not equi-coarsely embed into any Banach space X such that X (r) is separable for all r. See Theorem 6.1 below for more on this subject.

Studying further the property Q in [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF], Kalton exhibited non reflexive quasi-reflexive spaces with the property Q but showed that J and J * fail property Q. It is worth noticing that a theorem of Schoenberg [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF] implies that ℓ 1 coarsely embeds into ℓ 2 , and therefore ℓ 1 provides a simple example of a non-reflexive Banach space with property Q.

We conclude this section with two propositions that we state here for future reference. We start with a classical version of Ramsey's theorem.

Proposition 2.7 (Corollary 1.2 in [10]). Let (K, d) be a compact metric space, k ∈ N and f : [N] k → K. Then for every ε > 0, there exists an infinite subset M of N such that d(f (n), f (m)) < ε for every n, m ∈ [M] k . For a Banach space X, we call tree of height k in X any family (x(n)) n∈[N] ≤k , with x(n) ∈ X. Then, if M ∈ [N] ω , (x(n)) n∈[M] ≤k will be called a full subtree of (x(n)) n∈[N] ≤k . A tree (x * (n)) n∈[M] ≤k in X * is called weak * -null if for any n ∈ [M] ≤k-1 , the sequence (x * (n 1 , . . . , n k-1 , t)) t>n k-1 ,t∈M is weak * -null.
The next proposition is based on a weak * -compactness argument and will be crucial for our proofs. Although the distance considered on [N] k is different, the proof follows the same lines as Lemma 4.1 in [START_REF] Baudier | A new coarsely rigid class Banach spaces[END_REF]. We therefore state it now without further detail.

Proposition 2.8. Let X be a separable Banach space, k ∈ N, and f :

([N] k , d k K ) → X * a Lipschitz map. Then there exist M ∈ [N] ω and a weak * - null tree (x * (m)) m∈[M] ≤k in X * with x * m ≤ ω f (1) for all m ∈ [M] ≤k \ {∅} and so that ∀n ∈ [M] k , f (n) = k i=0 x * (n 1 , . . . , n i ) = m n
x * (m).

Uniform asymptotic properties of norms and related estimates

We recall the definitions that will be considered in this paper. For a Banach space (X, ) we denote by B X the closed unit ball of X and by S X its unit sphere. The following definitions are due to V. Milman [START_REF] Milman | Geometric theory of Banach spaces. II. Geometry of the unit ball (Russian), Uspehi Mat. Nauk[END_REF] and we adopt the notation from [START_REF] Johnson | Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces[END_REF]. For t ∈ [0, ∞) we define

ρ X (t) = sup x∈S X inf Y sup y∈S Y x + ty -1 ,
where Y runs through all closed subspaces of X of finite codimension. Then, the norm is said to be asymptotically uniformly smooth

(in short AUS) if lim t→0 ρ X (t) t = 0.
For p ∈ (1, ∞) it is said to be p-asymptotically uniformly smooth (in short p-AUS) if there exists c > 0 such that for all t ∈ [0, ∞), ρ X (t) ≤ ct p . We will also need the dual modulus defined by

δ * X (t) = inf x * ∈S X * sup E inf y * ∈S E x * + ty * -1 ,
where E runs through all finite-codimensional weak * -closed subspaces of X * . The norm of X * is said to be weak * asymptotically uniformly convex (in short AUC * ) if δ * X (t) > 0 for all t in (0, ∞). If there exists c > 0 and q ∈ [1, ∞) such that for all t ∈ [0, 1] δ * X (t) ≥ ct q , we say that the norm of X * is q-AUC * . The following proposition is elementary. Proposition 3.1. Let X be a Banach space. For any t ∈ (0, 1), any weakly null sequence (x n ) ∞ n=1 in B X and any x ∈ S X we have:

lim sup n→∞ x + tx n ≤ 1 + ρ X (t).
For any weak * -null sequence (x * n ) ∞ n=1 ⊂ X * and for any x * ∈ X * \ {0} we have

lim sup n→∞ x * + x * n ≥ x * 1 + δ * X lim sup x * n x * .
We will also need the following refinement (see Proposition 2.1 in [START_REF] Lancien | Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces[END_REF]).

Proposition 3.2. Let X be a Banach space. Then the bidual norm on X * * has the following property. For any t ∈ (0, 1), any weak * -null sequence

(x * * n ) ∞ n=1 in B X * * and any x ∈ S X we have: lim sup n→∞ x + tx * * n ≤ 1 + ρ X (t).
Let us now recall the following classical duality result concerning these moduli (see for instance [START_REF] Dilworth | Equivalent norms with the property (β) of Rolewicz[END_REF] Corollary 2.3 for a precise statement). Proposition 3.3. Let X be a Banach space. Then X is AUS if and and only if

X * is AUC * . If p, q ∈ (1, ∞) are conjugate exponents, then X is p-AUS if

and and only if

X * is q-AUC * . We conclude this section with a list of a few classical properties of Orlicz functions and norms that are related to these moduli. A map ϕ : [0, ∞) → [0, ∞) is called an Orlicz function if it is continuous, non decreasing, convex and so that ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞. The Orlicz norm ℓϕ , associated with ϕ is defined on c 00 , the space of finitely supported sequences, as follows:

∀x = (x n ) ∞ n=1 ∈ c 00 , x ℓϕ = inf r > 0, ∞ n=1 ϕ(x n /r) ≤ 1 .
The following is immediate from the definition. (i) If there exists C > 0 such that ϕ(t) ≤ Ct p , for all t ∈ [0, 1], then there exists A > 0 such that x ℓϕ ≤ A x ℓp , for all x ∈ c 00 . (ii) If there exists c > 0 such that ϕ(t) ≥ ct p , for all t ∈ [0, 1], then there exists a > 0 such that x ℓϕ ≥ a x ℓp , for all x ∈ c 00 .

Assume now that ϕ : [0, ∞) → [0, ∞) is an Orlicz function which is 1-Lipschitz and such that lim t→∞ ϕ(t)/t = 1. Consider for (s, t) ∈ R 2 ,

N ϕ 2 (s, t) = |s| + |s|ϕ(|t|/|s|) if s = 0, |t| if s = 0.
Then define by induction for all n ≥ 3:

∀(s 1 , . . . , s n ) ∈ R n , N ϕ n (s 1 , . . . , s n ) = N ϕ 2 N ϕ n-1 (s 1 , . . . , s n-1
), s n . The following is proved in [START_REF] Kalton | Uniform homeomorphisms of Banach spaces and asymptotic structure[END_REF] (see Lemma 4.3 and its preparation). Lemma 3.5.

(i) For any n ≥ 2, the function

N ϕ n is an absolute (or lattice) norm on R n , meaning that N n (s 1 , . . . , s n ) ≤ N n (t 1 , . . . , t n ), whenever |s i | ≤ |t i | for all i ≤ n. (ii) For any n ∈ N and any x ∈ R n : 1 2 s ℓϕ ≤ N ϕ n (s) ≤ e s ℓϕ .
When X is a Banach space, it is easy to see that ρ X is a 1-Lipschitz Orlicz function such that lim t→∞ ρ(t)/t = 1. But due to its lack of convexity, δ * X

is not an Orlicz function and we need to modify it. Following [START_REF] Kalton | Uniform homeomorphisms of Banach spaces and asymptotic structure[END_REF], we define

δ(t) = t 0 δ * X (s) s ds.
It is easy to see that δ * X (t)/t is increasing and tends to 1 as t tends to ∞. Therefore, δ is an Orlicz function which is 1-Lipschitz, such that lim t→∞ δ(t)/t = 1 and satisfying:

∀t ∈ [0, ∞), δ * X (t/2) ≤ δ(t) ≤ δ * X (t).
The following statement is now a direct consequence of Lemmas 3.4 and 3.5.

Lemma 3.6. Let X be a Banach space and p ∈ [1, ∞).

(i) If there exists C > 0 such that ρ X (x) ≤ Ct p , for all t ∈ [0, 1], then there exists A > 0 such that

∀n ∈ N ∀x ∈ R n , N ρ X n (x) ≤ A x ℓ n p . (ii) If there exists c > 0 such that δ * X (t) ≥ ct p , for all t ∈ [0, 1], then there exists a > 0 such that ∀n ∈ N ∀x ∈ R n , N δ n (x) ≥ a x ℓ n p .
We will also use the following reformulation of Propositions 3.1 and 3.2 in terms of the norms N δ 2 and N ρ X 2 . Lemma 3.7. Let X be a Banach space.

(i) Let (x * n ) ⊂ X * be weak * -null. Then for any x * ∈ X * we have lim sup

n→∞ x * + x * n ≥ N δ 2 ( x * , lim sup x * n ). (ii) Similarly, if (x * * n ) ⊂ X * * is weak * -null and x ∈ X, then lim inf n→∞ x + x * * n ≤ N ρ X 2 ( x , lim inf x * * n ).
Proof. If x * = 0 there is nothing to do, so we may assume that x * = 0. By application of Proposition 3.1 we see that

lim sup n→∞ x * + x * n ≥ x * 1 + δ * X lim sup x * n x * ≥ x * 1 + δ lim sup x * n x * = N δ 2 ( x * , lim sup x * n )
The proof of the second claim is even simpler so we leave it to the reader.

The general result

Let us first recall that a Banach space is said to be quasi-reflexive if the image of its canonical embedding into its bidual is of finite codimension in its bidual. We can now state our main result.

Theorem 4.1. Let X be a quasi-reflexive Banach space, let p ∈ (1, ∞) and denote q its conjugate exponent. Assume that X admits an equivalent p-AUS norm and that X * admits an equivalent q-AUS norm. Then the family ([N] k , d k K ) k∈N does not equi-coarsely embed into X * * . We immediately deduce the following. Corollary 4.2. Let X be a quasi-reflexive Banach space, let p ∈ (1, ∞) and denote q its conjugate exponent. Assume that X admits an equivalent p-AUS norm and that X * admits an equivalent q-AUS norm. Then the family ([N] k , d k K ) k∈N does not equi-coarsely embed into X, nor does it equi-coarsely embed into any iterated dual X (r) (r ≥ 0) of X.

Proof. Since X is quasi reflexive we infer that X (r) admits an equivalent p-AUS norm when r is even and it admits an equivalent q-AUS norm when r is odd. Indeed, note that when r is even X (r) is isomorphic to X ⊕ p F where F is finite-dimensional (resp. X (r) ≃ X * ⊕ q F when r is odd). Now it is obvious from Theorem 4.1 that ([N] k ) k∈N do not equi-coarsely embed into X (r) when r is even. When r is odd, we just exchange the roles of p and q.

Before going into the detailed proof of Theorem 4.1 let us briefly indicate the main idea. We assume that there is an equi-coarse family of embeddings (f k ) of [N] k into X * * with moduli ρ and ω. We fix k sufficiently large and observe that, up to passing to a subgraph, f k can be represented as the sum along the branches of a weak * -null countably branching tree of height k, say (z n ) n∈[N ] ≤k . Moreover the norms of the elements of this tree stabilize on each level towards values (K i ) k i=1 ⊂ [0, ω(1)]. Applying the existence of a q-AU S norm on X * one can show that k i=1 K p i ≤ c p ω(1) p where c is a constant depending only on X. The benefit of this observation is twofold. On one hand we will be able to construct two elements n 0 , m 0 ∈ [N] l (with l ≤ k) such that l i=1 z (n 1 ,...,n i )z (m 1 ,...,m i ) is small in norm (say less than 2cω(1)) while d l K (n 0 , m 0 ) is large (say ρ(d l K (n 0 , m 0 )) > 3cω(1)). On the other hand the p -AU S renormability of X together with the quasi-reflexivity allows to extend these elements to elements n, m ∈

[N] k such that d k K (n, m) is still large and k i=l+1 z (n 1 ,...,n i ) -z (m 1 ,...,m i ) ∼ k i=l+1 z (n 1 ,...,n i ) -z (m 1 ,...,m i ) p 1/p ∼ ( k i=l+1 K p i ) 1/p ≤ cω(1).
Eventually, summing the tree from 1 to k over the branches ending by n and m we get the desired contradiction

3cω(1) < ρ(d k K (n, m)) ≤ f k (n) -f k (m) ≤ 3cω(1)
. Proof of Theorem 4.1. Let us assume that there are two maps ρ, ω : [0, +∞) → [0, +∞) and maps

f k ([N] k , d k K ) : → (X * * , ) for k ∈ N such that: (i) lim t→∞ ρ(t) = ∞, (ii) ω(t) < ∞ for every t ∈ (0, +∞), (iii) ρ(t) ≤ ρ f k (t) and ω f k (t) ≤ ω(t) for every k ∈ N and t ∈ (0, ∞).
Note that all f k 's are ω(1)-Lipschitz for and so ω(1) > 0. Since all the sets [N] k are countable, we may and will assume that X and therefore, by the quasi-reflexivity of X, that all its iterated duals are separable. Let us fix N ∈ N. Pick α ∈ N such that α ≥ p q and set k = N 1+α ∈ N. We also fix η > 0. We shall provide at the end of our proof a contradiction if N is chosen large enough and η small enough. We denote the original norm on X, as well as its dual and bidual norms. Let us assume, as we may, that is p-AUS on X. We denote its modulus of asymptotic uniform smoothness ρ or simply ρ X .

For the first step of the proof we shall exploit the existence of an equivalent q-AUS norm | | on X * (we also denote | | its dual norm on X * * ). It is worth mentioning that if X is not reflexive, | | cannot be the dual norm of an equivalent norm on X (see for instance Proposition 2.6 in [START_REF] Causey | Prescribed Szlenk index of separable Banach spaces[END_REF]). Assume also that there exists b > 0 such that

(4.1) ∀z ∈ X * * b z ≤ |z| ≤ z .
Then we have that all f k 's are also ω(1)-Lipschitz for | |. By Proposition 3.3, we have that there exists c > 0 such that for all t ∈ [0, 1],

δ * | | (t) ≥ ct p . We denote again δ(t) = t 0 δ * | | (s) s ds.
Recall that Lemma 3.6 ensures the existence of a > 0 such that for all n ∈ N, N δ n ≥ 2a ℓ n p . First, using the separability of X * and Proposition 2.8, we may assume by passing to a full subtree, that there exist a weak * -null tree (z(m))

m∈[N] ≤k in X * * with |z m | ≤ ω(1) for all m ∈ [N] ≤k \ {∅} and so that ∀n ∈ [N] k , f k (n) = k i=0 z(n 1 , . . . , n i ) = m n z(m). For r ∈ N we denote E r = {m = (m 1 , . . . , m j ) ∈ [N] ≤k \ {∅}, m j = r} and F r = r u=1 E u . Fix a sequence (λ r ) ∞ r=1 in (0, 1) such that ∞ r=1 λ r > 1 2
. We now use Lemma 3.7 (i) and the fact that (z(m)) m∈[N] ≤k is a weak * -null tree to build inductively n 1 < . . . < n r so that for all n 1 , . . . , n L ∈ F nr-1 , for all ε 1 , . . . , ε L ∈ {-1, 1} and all n ∈ E nr , we have

z(n) + L l=1 ε l z(n l ) ≥ λ r N δ 2 L l=1 ε l z(n l ) , z(n) .
Therefore, using the fact that N δ 2 is an absolute norm and after passing to a full subtree, we may assume that for all

r 1 < • • • < r L in N, all ε 1 , . . . , ε L ∈ {-1, 1} and all n 1 , . . . , n L so that n l ∈ E r l for 1 ≤ l ≤ L, we have (4.2) L l=1 ε l z(n l ) ≥ 1 2 N δ L |z(n 1 )|, . . . , |z(n L )| ≥ a L i=1 z(n l ) p 1/p . Assume now that n = (n 1 , . . . , n k ) ∈ N k is such that n 1 < • • • < n k are even and choose m = (m 1 , . . . , m k ) so that n 1 < m 1 < • • • < n k < m k . It follows from (4.2) that f (n) -f (m)| = k i=1 z(n 1 , . . . , n i ) -z(m 1 , . . . , m i ) ≥ a k i=1 z(n 1 , . . . , n i ) p + z(m 1 , . . . , m i ) p 1/p
.

We now use the fact that d k K (n, m) = 1 and f is ω(1)-Lipschitz, to deduce

k i=1 z(n 1 , . . . , n i ) p 1/p ≤ 1 a ω(1).
So replacing N with 2N and setting A = 1/a, we may assume that

(4.3) ∀n ∈ [N] k , k i=1 z(n 1 , . . . , n i ) p 1/p ≤ Aω(1)
.

By Ramsey's theorem (Proposition 2.7), we may also assume, after passing again to a full subtree, that for all i ∈ {1, . . . , k} there exists

K i ∈ [0, ω(1)] such that ∀(n 1 , . . . , n i ) ∈ [N] i , K i ≤ |z(n 1 , . . . , n i )| ≤ K i + η.
The estimate (4.3) yields (4.4)

k i=1 K p i ≤ A p ω(1) p .
Therefore, since k = N 1+α , there exists j ∈ {0, N, . . . , N (N α -1)} such that

j+N i=j+1 K p i ≤ A p ω(1) p N α .
Then we deduce from Hölder's inequality that (4.5)

j+N i=j+1 K i ≤ N 1/q Aω(1) N α/p ≤ Aω(1)
.

We now use the assumption that X is quasi-reflexive, so that X * * = X⊕F , where F is of finite dimension. Thus, for each (n 1 , . . . , n i ) ∈ [N] ≤k , we can decompose z(n 1 , . . . , n i ) = x(n 1 , . . . , n i )+e(n 1 , . . . , n i ), with x(n 1 , . . . , n i ) ∈ X and e(n 1 , . . . , n i ) ∈ F . Then, the compactness of bounded sets in F and another application of Proposition 2.7 allows us to assume, after passing to a full subtree, that ∀i ∈ {1, . . . , k} ∀n, v ∈ [N] i , e(n)e(v) < η, Which implies that for all i ∈ {1, . . . , k} and all n, v ∈ [N] i we have

(4.6) z(n) -z(v) -x(n) -x(v) < η.
We are now ready for the last step of the proof, where we shall build m and u in

[N] k so that d k K (m, u) = N , but |f (m) -f (u)
| is bounded by a constant depending only on ω(1) and on X. This will yield a contradiction with the fact lim N →∞ ρ(N ) = ∞.

First, we set m i = u i = i, for all 1 ≤ i ≤ j. Then, for j + 1 ≤ i ≤ j + N , we set m i = i and u i = i + N . Finally, we shall build m i = u i inductively, for j + N < i ≤ k. Note, that when this will be done, we will indeed have

d k K (m, u) = N .
First, we obviously have (4.7)

j i=1 z(m 1 , . . . , m i ) -z(u 1 , . . . , u i ) = 0.
The next estimate follows from (4.5).

(4.8)

j+N i=j+1 z(m 1 , . . . , m i ) -z(u 1 , . . . , u i ) ≤ j+N i=j+1 2(K i + η) ≤ 3Aω(1),
if η was initially chosen small enough. We now select the remaining coordinates of m and u inductively using the fact that is p-AUS. To shorten the notation for the end of the proof, we shall now denote

x i = x(m 1 , . . . , m i ), z i = z(m 1 , . . . , m i ), x ′ i = x(u 1 , . . . , u i ) and z ′ i = z(u 1 , . . . , u i ).
First, we simply set m j+N +1 = u j+N +1 = j + 2N + 1. We now use the fact that the tree (z(m)) m∈[N] ≤k is weak * -null and Lemma 3.7 (ii) to find m j+N +2 = u j+N +2 > j + 2N + 1 such that

x j+N +1 -x ′ j+N +1 + z j+N +2 -z ′ j+N +2 ≤ N ρ X 2 x j+N +1 -x ′ j+N +1 , z j+N +2 -z ′ j+N +2 + η
It follows from (4.6) that

z j+N +1 -z ′ j+N +1 + z j+N +2 -z ′ j+N +2 ≤ N ρ X 2 z j+N +1 -z ′ j+N +1 + η, z j+N +2 -z ′ j+N +2 + 2η ≤ N ρ X 2 2 b K j+N +1 + η + η, 2 b K j+N +2 + η + 2η.
Similarly, we can inductively find

m j+N +2 = u j+N +2 < • • • < m k = u k such that, k i=j+N +1 (z i -z ′ i ) ≤ 2 b N ρ X k-j-N K j+N +1 , . . . , K k ) + ω(1)
provided η is chosen small enough. Since Lemma 3.6 ensures the existence of C > 0 such that N ρ X n ≤ C ℓ n p for all n ∈ N the above inequality yields

k i=j+N +1 (z i -z ′ i ) ≤ 2C b k i=j+N +1 K p i 1/p + ω(1) ≤ 2CA b + 1 ω(1).
Finally, combining the above estimate with (4.7) and (4.8), we get that

f (m) -f (u) ≤ 3A + 2CA + b b ω (1)
.

As announced at the beginning of the proof, this yields a contradiction if N was initially chosen, as it was possible, so that ρ(N ) > 3A+2CA+b b ω(1).

Unlike reflexivity, quasi-reflexivity itself is not enough to prevent the Kalton graphs from embedding into a Banach space. We thank P. Motakis for showing us the next example.

Proposition 4.3 (Motakis).

There exists a quasi-reflexive Banach space X such that the family of graphs ([N] k , d k K ) k∈N equi-Lipschitz embeds into X. Proof. The proof relies on the existence of a quasi-reflexive Banach space X of order one which admits a spreading model, generated by a basis of X that is equivalent to the summing basis (s n ) ∞ n=1 of c 0 . This is shown in [START_REF] Freeman | On spreading sequences and asymptotic structures[END_REF] (Proposition 3.2) and based on a construction given in [START_REF] Bellenot | Quasi-reflexive and tree spaces constructed in the spirit of R.C. James[END_REF]. We refer the reader to [START_REF] Beauzamy | Modèles étalés des espaces de Banach[END_REF] for the necessary definitions. Consequently, there exists a sequence (x n ) ∞ n=1 in S X and constants A, B > 0 such that for all k ≤ n 1 < • • • < n k and all ε 1 , . . . , ε k in {-1, 0, 1} one has (4.9)

A k i=1 ε i s i c 0 ≤ k i=1 ε i x n i X ≤ B k i=1 ε i s i c 0 .
For k ∈ N and n = (n 1 , . . . , n k ) ∈ [N] k we define

g k (n) = k i=1 x 2k+n i .
It follows easily from Proposition 2.5, the inequality (4.9) and the fact that

(s n ) ∞ n=1 is a spreading sequence that A 2 d k K (n, m) ≤ g k (n) -g k (m) X ≤ Bd k K (n, m) for all n, m ∈ [N] k .
Remark 4.4. Let us mention that, more generally, it is proved in [START_REF] Argyros | A study of conditional spreading sequences[END_REF] that for any conditional normalized spreading sequence (e n ) ∞ n=1 , there exists a quasi-reflexive Banach space X of order 1 with a normalized basis (x i ) ∞ i=1 which generates (e n ) ∞ n=1 as a spreading model.

The James sequence spaces

Let p ∈ (1, ∞). We now recall the definition and some basic properties of the James space J p . We refer the reader to [START_REF] Albiac | Topics in Banach Space Theory[END_REF](Section 3.4) and references therein for more details on the classical case p = 2. The James space J p is the real Banach space of all sequences x = (x(n)) n∈N of real numbers with finite p-variation and verifying lim n→∞ x(n) = 0. The space J p is endowed with the following norm

x Jp = sup k-1 i=1 |x(p i+1 ) -x(p i )| p 1/p : 1 ≤ p 1 < p 2 < . . . < p k .
This is the historical example, constructed for p = 2 by R.C. James in [START_REF] James | Bases and reflexivity of Banach spaces[END_REF], of a quasi-reflexive Banach space which is isomorphic to its bidual. In fact J * * p can be seen as the space of all sequences x = (x(n)) n∈N of real numbers with finite p-variation, which is J p ⊕ R1, where 1 denotes the constant sequence equal to 1. The standard unit vector basis (e n ) ∞ n=1 (e n (i) = 1 if i = n and e n (i) = 0 otherwise) is a monotone shrinking basis for J p . Hence, the sequence (e * n ) ∞ n=1 of the associated coordinate functionals is a basis of its dual J * p . Then the weak * topology σ(J * p , J p ) is easy to describe. A sequence (x * n ) ∞ n=1 in J * p converges to 0 in the σ(J * p , J p ) topology if and only if it is bounded and lim n→∞ x * n (i) = 0 for every i ∈ N. For x ∈ J p , we define supp x = {i ∈ N : x(i) = 0}. For x, y ∈ J p , we denote: x ≺ y whenever max supp x < min supp y. Similarly, an element x * of J * p will be written x * = ∞ n=1 x * (n)e * n and supp x * = {i ∈ N : x * (i) = 0} and we shall denote x * ≺ y * whenever max supp x * < min supp y * .

The detailed proof of the following proposition can be found in [20] (Proposition 2.3). This a consequence of the following fact: there exists C ≥ 1 such that

* + y * | q * ≥ |x * | q * + |y * | q * .
In particular, | | * is q-AUC * for the weak * topology induced by J p and therefore | | is p-AUS on J p .

There is also a natural weak * topology on J p . Indeed, the summing basis (s n ) ∞ n=1 (s n (i) = 1 if i ≤ n and s n (i) = 0 otherwise) is a monotone and boundedly complete basis for J p . Thus, J p is naturally isometric to a dual Banach space: J p = X * with X being the closed linear span of the biorthogonal functionals (e

* n -e * n+1 ) ∞ n=1 in J * p associated with (s n ) ∞ n=1 . Note that X = {x * ∈ J * p , ∞ n=1 x * (n) = 0}
. Thus, a sequence (x n ) ∞ n=1 in J p converges to 0 in the σ(J p , X) topology if and only if it is bounded and lim n→∞ x n (i)x n (j) = 0 for every i = j ∈ N. The next proposition is easy (see Proposition 2.3 in [START_REF] Lancien | Réflexivité et normes duales possèdant la propriété uniforme de Kadec-Klee[END_REF] for the case p = 2). Proposition 5.2. The usual norm on J p is p-AUC * for the weak * topology induced by X. In other words, the restriction to X of the usual norm on J * p is q-AUS.

Then, since X is one codimensional in J * p , we have that J * p is isomorphic to X ⊕ R and therefore also admits an equivalent q-AUS norm.

The above remarks combined with Corollary 4.2 immediately yield the following.

Corollary 5.3. Let p ∈ (1, ∞). Then, the family ([N] k , d k K ) k∈N does not equi-coarsely embed into J p , nor does it equi-coarsely embed into J * p .

A Remark on the James tree space

Let us recall the construction of the James tree space J T . We denote T = 2 <ω the tree of all finite sequences with coefficients in {0, 1} equipped with its natural order: for s, t ∈ T , we say that s ≤ t if the sequence t extends s. The set of all infinite sequences with coefficients in {0, 1} will be denoted 2 ω . For s ∈ T , the length of s is denoted |s|. We call segment of T any set of the form {s ∈ T, t ≤ s ≤ t ′ } with t ≤ t ′ in T . For a map x : T → R, we define

x J T = sup n i=1 s∈S i x(s) 2 1/2
, where the supremum is taken over all pairwise disjoint segments S 1 , . . . , S n of T . Then the James tree space is the space J T = {x : T → R, x J T < ∞} equipped with the norm J T . For s ∈ T , we denote e s : T → R defined by e s (t) = δ s,t , t ∈ T . If ψ : N → T is a bijection such that |ψ(n)| ≤ |ψ(m)| whenever n ≤ m, then (e ψ(n) ) ∞ n=1 is a normalized, monotone and boundedly complete basis of J T . For s ∈ T , the coordinate functional e * s is defined by e * s (x) = x(s), x ∈ J T . Then the closed linear span of {e * s , s ∈ T } in J T * is denoted B and B * is isometric to J T . The space J T was built by R.C. James in [START_REF] James | A separable somewhat reflexive Banach space with nonseparable dual[END_REF] to serve as the first example of a separable Banach space with non separable dual, which does not contain an isomorphic copy of ℓ 1 .

In [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF] it is shown that if a Banach space X coarsely contains c 0 then there exists k ∈ N such that X (k) , the dual of order k of X, is non separable. A close look at the proof of Theorem 3.5 in [START_REF] Kalton | Coarse and uniform embeddings into reflexive spaces[END_REF] allows to state the following. Theorem 6.1 (Kalton). Let X and Y be two Banach spaces such that X coarsely embeds into Y . Assume moreover that there exist an uncountable set I and for every i ∈ I and k ∈ N, a 1-Lipschitz map

f k i : ([N] k , d k K )) → X such that lim k→∞ inf i =j∈I inf M∈[N] ω sup n∈[M] k f k i (n) -f k j (n) = ∞.
Then there exists r ∈ N such that Y (r) is not separable.

As an application, we can show the following.

f k σ (m) -f k σ (n) = 1 √ k k i=1 s∈S i e * s ,
where S 1 , . . . , S k are pairwise disjoint segments in T . Note that for any segment S i the sum s∈S i e * s belongs to the unit ball of J T * . It then follows from the Cauchy-Schwarz inequality that f k σ is 1-Lipschitz on ([N] k , d k K ). Assume now that σ = τ ∈ 2 ω . Pick r ∈ N such that σ r = τ r . Then for any M ∈ [N] ω and any n = (n 1 , . . . , n k ) ∈ [M] k with n 1 ≥ r, we have

f k σ (n) -f k τ (n) B ≥ f k σ (n) -f k τ (n), e σ |n 1 ≥ √ k.
By Theorem 6.1 and the uncountability of 2 ω , this finishes our proof for B. s is in the unit ball of J T * . Therefore

g k σ (n) -g k τ (n) J T ≥ g k σ (n) -g k τ (n), x * ≥ √ k √ 2 .
This concludes our proof for J T .

  which finishes the proof of the fact. It is clear that d k K (n, m) = 1 if and only if max Fmin F = 1. Thus it only remains to prove that d is a graph metric. Now given n, m such that d(n, m) ≥ 2 we are looking for ℓ ∈ [M] k \ {m, n} such that d(m, n) = d(n, ℓ) + d(ℓ, m).

Lemma 3 . 4 .

 34 Let ϕ : [0, ∞) → [0, ∞) be an Orlicz function and p ∈ [1, ∞).

Theorem 6 . 2 .

 62 Let Y be a Banach space such that B or J T coarsely embeds into Y . Then there exists r ∈ N such that Y (r) is not separable.Proof. For σ ∈ 2 ω , we denote σ |n = (σ 1 , . . . , σ n ). Then, for k ∈ N, we definef k σ : [N] k → B as follows. For n = (n 1 , . . . , n k ) ∈ [N] k let Assume for instance that n 1 ≤ m 1 ≤ • • • n k ≤ m k .Then we can write

For σ ∈ 2 ωe σ |n i . It is easily checked that g k σ is 1 -

 21 and k ∈ N define now g k σ : [N] k → J T by ∀n = (n 1 , . . . , n k ) ∈ [N] k , g k σ (Lipschitz on ([N] k , d k K ). Assume now that σ = τ ∈ 2 ω . Pick r ∈ N such that σ r = τ r . Then for any M ∈ [N] ω and any n = (n 1 , . . . , n k ) ∈ [M] k with n 1 ≥ r, denote S = {s ∈ T, σ |n 1 ≤ s ≤ σ |n k }.The set S is a segment in T and x * = s∈S e *

Acknowledgements. We are grateful to P. Motakis for allowing us to include Proposition 4.3 and for very useful discussions. We also wish to mention that the formula describing the interlaced distance given in Proposition 2.2 has been independently obtained by F. Baudier, P. Motakis and Th. Schlumprecht.

The three authors are supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03).