The coarse geometry of Tsirelson’s space and applications

By F. Baudier, G. Lancien, and Th. Schlumprecht

Abstract

The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson’s original space . Every Banach space that is coarsely embeddable into must be reflexive, and all of its spreading models must be isomorphic to . Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: coarsely contains neither nor for . We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into , and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to . Also, a purely metric characterization of finite dimensionality is obtained.

1. Introduction

1.1. Motivation

A natural and powerful approach to classify mathematical objects in a given category is to discover properties that are invariant with respect to the isomorphisms of the category. The notion of topological dimension, which allows us to distinguish between and as topological spaces, is a prime example. These invariants can take various forms, and in metric geometry one usually seeks properties that are stable under embeddings of a certain type, e.g., Lipschitz, uniform, or coarse. In this article we are concerned with a coarse invariant in the form of a concentration inequality for Lipschitz maps defined on infinite graphs with values into certain infinite-dimensional Banach spaces.

Discovering useful and tractable new concentration inequalities is usually a challenge. In the Lipschitz category, several fundamental inequalities have been discovered, partially motivated by the investigation of the Ribe program and its applications in geometric group theory or theoretical computer science. We refer to the survey of A. Naor Reference 25 for an enlightening account of this theory and to the recent work of Naor and Schechtman (Reference 28, Reference 27, and Reference 26) where fundamental and powerful new inequalities were proved.

Strong interest in the coarse geometry of infinite-dimensional Banach spaces arose in connection with the Novikov conjecture in topology and with a coarse version of the Baum–Connes conjecture in noncommutative geometry (see Reference 30, Reference 38, Reference 13, and Reference 36). A very efficient and powerful technique for proving the coarse geometric Novikov conjecture for a bounded geometry metric space is to coarsely embed this space into a Banach space with some strong enough convexity and smoothness properties. Indeed, G. Yu Reference 39 showed that a discrete metric space with bounded geometry that is coarsely embeddable into a Hilbert space satisfies the coarse geometric Novikov conjecture. This result was later generalized by Kasparov and Yu Reference 23 who proved that the conclusion holds if the space is merely coarsely embeddable into a super-reflexive Banach space. It is worth mentioning that the coarse geometric Novikov conjecture implies Gromov’s conjecture, stating that a uniformly contractible Riemannian manifold with bounded geometry cannot have uniformly positive scalar curvature, and the zero-in-the-spectrum conjecture, stating that the Laplace operator acting on the space of all -forms of a uniformly contractible Riemannian manifold has zero in its spectrum. Consequently, a very natural and intriguing question became whether the separable Hilbert space is the Banach space into which it is the hardest to embed. More precisely, the following problem was raised (Reference 15, Problem 14 and Reference 34, Problem 11.17).

Main Problem.

Does coarsely embed into every infinite-dimensional Banach space?

There is rather strong evidence that is the space into which it is the most difficult to embed. It was shown in Reference 33 that every locally finite metric subset of admits a bi-Lipschitz embedding into every infinite-dimensional Banach space, and in Reference 5 that every proper subset of (i.e., whose closed balls are compact) is almost Lipschitz embeddable into every infinite-dimensional Banach space. Both proofs use Dvoretzky’s theorem Reference 11 and the barycentric gluing technique introduced in Reference 3. Moreover, using Gaussian random variables, one can embed linearly isometrically into for all . Nowak Reference 29 proved that coarsely embeds into for all using Mazur maps. This technique was extended by Ostrovskii Reference 33 to embed coarsely into every Banach space with an unconditional basis and nontrivial cotype using the Mazur maps constructed by Odell and Schlumprecht in their solution to the distortion problem Reference 31. There are also Banach spaces with trivial cotype such as , or (which is a universal space for separable spaces and bi-Lipschitz embeddings Reference 1) that coarsely contain .

On the other hand, useful coarse invariants, such as asymptotic dimension or finite decomposition complexity, are not well suited to study the coarse geometry of infinite-dimensional Banach spaces, and the number of properties at our disposal to prevent coarse embeddability between infinite-dimensional Banach spaces is rather limited. There are essentially three obstructions. The first obstruction was discovered by Johnson and Randrianarivony Reference 18 when they showed that does not coarsely embed into when . Their approach relies heavily upon the fact that the target space is . The second obstruction is the metric cotype of Mendel and Naor Reference 24. For instance, the metric cotype argument can be used to rule out the coarse embeddability of into when . The last one is Kalton’s property Reference 19 which serves as an obstruction to coarse embeddability into reflexive (or stable) Banach spaces.⁠Footnote1 Since has the smallest possible cotype allowed for a Banach space and is both reflexive and stable, none of the above obstructions can provide a negative answer to the Main Problem.

1

A variant of Kalton’s property can be useful in the nonseparable setting Reference 20.

In this article the Main Problem is answered negatively, using the following approach. We show that there exists an infinite-dimensional Banach space in which the sequence of -dimensional Hamming graphs (over a countable set) does not equi-coarsely embed. Since it is easy, and well known, that equi-coarsely embeds into , the claim follows. Our work is inspired by a concentration inequality of Kalton and Randrianarivony Reference 21. They proved the following: Assume that is a Lipschitz map from into a reflexive Banach space , for which there exists , such that for any in and any weakly null sequence in we have

Then for every two vertices in some subgraph isometric to , one has

Since , inequality Equation 1.2 says that a Lipschitz map on concentrates on a “full” subgraph. This concentration inequality was originally used by Kalton and Randrianarivony as an obstruction to coarse Lipschitz embeddability, but it also provides information on compression exponents (cf. Reference 4 and Reference 9). Reflexivity is crucial in obtaining inequality Equation 1.2. Indeed, since is Lipschitz universal, inequality Equation 1.2 cannot hold for Lipschitz maps with values in , even though satisfies inequality Equation 1.1 for every . In order to prevent coarse embeddability, one would like to have for some , which does not depend on , the stronger inequality

If we could find a reflexive Banach space satisfying inequality Equation 1.2 for , that is,

then the concentration inequality Equation 1.3 would hold for . Unfortunately, if a Banach space satisfies inequality Equation 1.4, then must contain an isomorphic copy of (see Reference 22). Thus it cannot be reflexive and moreover it contains a bi-Lipschitz copy of every separable metric space, which precludes any concentration inequality. As we shall see, this transition phase phenomenon is not necessarily an obstruction. There are actually Banach spaces that have a -like” asymptotic smoothness property which is strong enough to obtain the concentration inequality Equation 1.3 but weak enough to coexist with reflexivity. A prime example of such space is Tsirelson’s original space.

Tsirelson’s original space was the first example of a Banach space that does not contain any isomorphic copies of or Reference 37. The validity of the concentration inequality Equation 1.3 for is pivotal in proving an unexpected rigidity result, which in turn implies that coarse and uniform versions of Tsirelson’s theorem hold.

1.2. Main results

All unexplained notation can be found in section 2. The main result of this article is the following rigidity result pertaining to the spreading model structure of Banach spaces coarsely embeddable into Tsirelson’s original space .

Theorem A.

Let be an infinite-dimensional Banach space. If coarsely embeds into or if its unit ball uniformly embeds into , then is reflexive and all its spreading models are isomorphic to .

Our rigidity result shows that Tsirelson’s construction is actually extremely robust in the sense that nonlinear versions of Tsirelson’s theorem hold. Indeed, Corollary B below is an immediate consequence of Theorem A since for the canonical basis of generates a spreading model that is linearly isometric to , while and are not even reflexive.

Corollary B.

and for (resp. and do not coarsely embed (resp. uniformly embed) into .

We will say that a Banach space is coarsely minimal if it coarsely embeds into every infinite-dimensional Banach space. The Main Problem asks whether is coarsely minimal. In fact, Theorem A provides a much stronger negative solution to the Main Problem. Indeed, a coarsely minimal Banach space embeds into and it must have nontrivial cotype using Mendel and Naor metric cotype notion Reference 24, but it also embeds into , and by Theorem A it must have trivial cotype, a contradiction.

Corollary C.

There is no coarsely minimal infinite-dimensional Banach space.

The proof of Theorem A relies essentially on two ingredients. The main ingredient, which is presented in section 4, is a new concentration inequality for Lipschitz maps on the -dimensional infinite Hamming graphs that take values into Tsirelson’s original space .

Theorem D.

Let and be a Lipschitz map. Then there exists such that for all one has

The concentration inequality above is strong enough to rule out coarse and uniform embeddings, and it can already be invoked to provide a negative solution to the Main Problem without the full strength of Theorem A.

The other ingredient, presented in section 3, uses the structure of spreading models of a Banach space to prove positive embeddability results. Such results are obtained for three different metrics: the tree metric, the Hamming metric, and the symmetric difference metric. Our embeddings are based on the existence of spreading models that are isomorphic, or not isomorphic, to .

The proof of Theorem A, given in section 4, is obtained by analyzing the tension between the concentration inequality for the Hamming graphs and the embeddability of the Hamming graphs into spaces which have spreading models that are not isomorphic to . For readers interested mostly in the geometry of Hilbert space we included in Remark 4.6 a short proof (which avoids the machinery of spreading models and only requires Theorem D) of the -case of Corollary B.

The analysis of the spreading model structure of infinite-dimensional Banach spaces in section 3 also leads us to a metric characterization of finite dimensionality in terms of equi-coarse embeddability of the sequence of countably branching trees of finite but arbitrarily large height, denoted by .

Theorem E.

For a Banach space the following assertions are equivalent:

(1)

is finite dimensional.

(2)

does not equi-coarsely embed into .

We mention a few related open problems in the last section.

2. Preliminaries

2.1. Nonlinear embeddings

Let and be two metric spaces, and let . One defines

and

Note that for every ,

The moduli and will be called the compression modulus and the expansion modulus of the embedding, respectively. We adopt the convention and . The map is a coarse embedding if and for all . A map is said to be a uniform embedding if and for all ; i.e., is an injective uniformly continuous map whose inverse is uniformly continuous.

If one is given a family of metric spaces , one says that equi-coarsely (resp. equi-uniformly) embeds into if there exist nondecreasing functions and, for all , maps such that , , and and for all (resp. and for all ).

2.2. Tree, symmetric difference, and Hamming metrics

In this section we define the metric spaces that will be needed and studied in the rest of this paper. For any infinite subset of , let and . For , we put and where denotes the cardinality of the set . Elements of and will always be listed in increasing order, i.e., if we write or , we assume that or , respectively. We define three metrics, , , and on . The restrictions of these metrics to the sets , for some and some , will still be denoted , , and .

The tree metric

For , , let

where if and .

We denote by the partial order of extension on , i.e., if and for all one has . One can then define a graph structure on , which is our set of vertices, by declaring two vertices to be adjacent if is the immediate predecessor of or is the immediate predecessor of . The graph obtained is the countably branching tree of infinite height, denoted or simply . The countably branching tree of height , , will also be denoted . It is easy to see that the tree metric coincides with the graph metric.

The symmetric difference metric

For , , let

where is the symmetric difference. At some point it will be useful for us to notice that one can define a graph structure on by defining an adjacency relation as follows: two vertices are adjacent if and only if . This graph is the -dimensional Johnson graph over a countable set and will be denoted or simply , and its graph metric coincides with the metric .

The Hamming metric

For , let

When restricted to the set , the metric , can be seen as the graph metric on the Hamming graph over a countable alphabet, denoted or simply , where two vertices are adjacent if they differ in exactly one coordinate.

Remark 2.1.

Both graphs and are infinite versions of finite graphs that arose from the fundamental work from the mid-twentieth century in coding theory of R. W. Hamming and S. M. Johnson, respectively.

2.3. Spreading models

For the convenience of the reader we briefly recall a few useful results from the theory of spreading models initiated by Brunel and Sucheston Reference 10 in the 1970s. We shall follow the exposition in Reference 8. An application of Ramsey’s theorem yields that every bounded sequence in a separable Banach space admits a subsequence such that for all , ,

exists, where the limit means that for all , there is such that for all ,

It is easy to see that that if denotes the canonical basis of , the vector space of sequences in which eventually vanish, the formula

defines a seminorm, and that is a spreading sequence in the sense that for all , , and integers ,

If does not have a converging subsequence, it is simple to show that the seminorm is actually a norm. The completion of for the norm is a Banach space called a spreading model of generated by the sequence , and we will refer to the sequence as the fundamental sequence of the spreading model. The fundamental sequence is not necessarily a basis of but if the generating sequence is normalized and weakly null, then is a normalized -suppression unconditional basis Reference 8, Proposition 1, p. 24. We call a basic sequence in a Banach space -suppression unconditional, for some , if for all and all ,

and in this case we denote by the smallest number satisfying Equation 2.7. The basic sequence is called -unconditional if for all and all , we have

and we denote in this case the smallest number satisfying Equation 2.8 by . It is easy to see that basic sequences are suppression unconditional if and only if they are unconditional, and that in this case

The function , where is the fundamental sequence of a spreading model of a Banach space, is usually called the fundamental function of . We deduce the following.

Proposition 2.2.

Let be a normalized weakly null sequence in a Banach space . Then for every there is a normalized weakly null basic subsequence of with basis constant , generating a spreading model with fundamental function , such that for all , for all , and for all one has

We will be particularly interested in -spreading models, where is the space of real valued sequences converging to equipped with the sup-norm. The following proposition follows from Reference 8, Lemma 1, p. 73 and Reference 8, Lemma 4, p. 75.

Proposition 2.3.

has a spreading model isomorphic to if and only if for all , has a spreading model whose fundamental sequence is -equivalent to the canonical basis of , i.e., that for all , ,

We will also need the following observation from Reference 8.

Proposition 2.4.

Reference 8, Proposition 3, p. 79 Let be a normalized weakly null sequence in a Banach space that generates a spreading model . Then is not isomorphic to if and only if there exists a subsequence of such that

2.4. Tsirelson spaces and

We recall a construction of a space originally designed by Tsirelson Reference 37, which is nowadays usually referred to as , while denotes its dual space and was described by Figiel and Johnson in Reference 14. Doing so, convenient notation and terminology that will be needed later on are introduced. Tsirelson’s original construction of has a geometric flavor while the Figiel–Johnson construction of is more analytic.

For and , we write if , and or if or , respectively. Here we set and . We call a sequence admissible if . For , we call the support of and for we write . We call a finite sequence of elements in a block sequence if for all . We define inductively for each a norm on . For we put , and assuming that has been defined for some , we let for

Then we put for

is then a norm on and is defined to be the completion of with respect to . As observed in Reference 14, satisfies the following implicit equation for all

and the unit vector basis is a -unconditional basis of . Then it was proven in Reference 14 that does not contain a subspace isomorphic to , which, together with the easy observation that certainly does not contain a subspace isomorphic to , yields by James’s theorem Reference 16, Theorem 2 that must be reflexive. Let us now denote the dual of by (the original Tsirelson’s space). We denote the unit basis of by (which is a -unconditional basis of ) and its unit ball by . Next to the property that is reflexive, the following property of will be essential for us:

Indeed, assume that is a block sequence in , with , and let , , be such that . By the -unconditionality of in we can assume that , where , for . Since is admissible, we obtain from 2.4 that

which proves claim Equation 2.13.

Tsirelson’s space is not the only reflexive space satisfying inequality Equation 2.13. For every countable ordinal , one can define (see Reference 32), a higher-order version of , where the admissibility condition refers to Schreier families of order . These Tsirelson-type spaces are all reflexive and satisfy inequality Equation 2.13, and they are incomparable in the sense that does not embed isomorphically into whenever .

Remark 2.5.

In this article all the results that are valid for also hold for every reflexive Banach space satisfying equation Equation 2.13, so in particular for for any countable ordinal .

3. A metric characterization of finite dimensionality

We start by proving some coarse embeddability results for the metric spaces introduced in section 2.2. Here, we do not need that the metrics can be seen as graph metrics. In Lemma 3.1 below we show that if is an infinite-dimensional Banach space admitting a spreading model , generated by a weakly null sequence, then there always exists a -Lipschitz map into , defined on any of the spaces , , or , whose compression modulus is essentially bounded from below by the fundamental function of the spreading model.

Lemma 3.1.

Let be any of the metric spaces , , or for . Let be an infinite-dimensional Banach space admitting a spreading model generated by a normalized weakly null sequence. Then for every there exists a map such that for all ,

Proof.

Let such that . By Proposition 2.2 there is a weakly null normalized basic sequence with basis constant not larger than , and thus the bimonotonicity constant is at most , generating a spreading model , such that for all , for all , and for all one has

We now consider the three different cases.

The symmetric difference metric case

Let be defined by , for . The map is clearly -Lipschitz since after cancellations it follows for that

for some and . If is even, then , and hence Equation 3.2 and the assumption on the bimonotonicity constant of yield

If is odd, the proof is similar, as will be seen while treating the next case.

The tree metric case

Let be a bijection. Let be defined by , for . The map is clearly -Lipschitz since after cancellations one has

for some and . Here we only detail the case when is odd, say , with . Note that . So again Equation 3.2 and the assumption on the bimonotonicity constant of imply that

The Hamming metric case

Let be a bijection, and let be defined by . The map is clearly -Lipschitz since after cancellations one has

for some and . Necessarily, , and hence, as in the other cases,

Remark 3.2.

In Lemma 3.1 we restricted the Hamming metric to , for , because is usually defined for strings of equal length. But it is not hard to show the existence of a map , satisfying condition Equation 3.1, if we replace the value by a larger number.

Lemma 3.1 yields a coarse embedding of , , and an equicoarse embedding of the sequence into every Banach space which admits at least one spreading model that is not isomorphic to and is generated by a weakly null sequence. Recall that Proposition 2.4 ensures that the associated fundamental function is unbounded. Then the following proposition follows simply from Lemma 3.1.

Corollary 3.3.

The metric spaces , , are all coarsely embeddable, and the sequence is equi-coarsely embeddable into a Banach space that has a spreading model generated by a normalized weakly null sequence, which is not isomorphic to .

More precisely, in all three cases, for any , there is a coarse embedding into such that is -Lipschitz and the modulus of compression of satisfies , for , where is the fundamental function of the spreading model .

For the tree metric we can deduce the following embeddability result, even if all spreading models are equivalent to .

Proposition 3.4.

If an infinite-dimensional Banach space has a spreading model isomorphic to then equi-bi-Lipschitzly embeds into .

More precisely, for every and every there exists a map , such that for all ,

Proof.

By Proposition 2.3 one may assume that there exists a normalized sequence such that for all , for all , and for all one has

Let be a bijection. For let be defined by for . After cancellations one has

for some and satisfying . So

and

which proves our claim.

Remark 3.5.

If an infinite-dimensional Banach space has a spreading model isomorphic to then, using a duality argument Reference 8, Proposition 1, p. 80, the relationship between the -weakly null index and the Szlenk index Reference 2, Theorem 4.2, and the embedding in Reference 6, Theorem 2.6, one can show that embeds bi-Lipschitzly into .

Moreover, since it is easy to see that every countable tree isometrically embeds into , we obtain that every countable tree coarsely embeds into any infinite-dimensional Banach space.

In the following theorem, which includes Theorem E, a purely metric characterization of finite dimensionality in terms of graph preclusion in the coarse category is given.

Theorem 3.6.

Let be a Banach space. The following assertions are equivalent:

(1)

is finite dimensional.

(2)

does not equi-coarsely embed into .

(3)

does not coarsely embed into .

Proof.

We start with a self-contained elementary proof of the main equivalence , which avoids the use of Remark 3.5.

By compactness, a finite-dimensional Banach space cannot contain an infinite sequence that is bounded and separated. On the other hand, if is an infinite-dimensional Banach space, then by Rosenthal’s -theorem, it either contains isomorphically or it has a weakly Cauchy sequence which is not norm converging. Since contains an isometric copy of , we may assume that contains a weakly Cauchy sequence which is not norm converging. After eventually passing to a subsequence of , still denoted , the sequence

is normalized and weakly null. Therefore, either has a subsequence that generates a spreading model isomorphic to and we apply Proposition 3.4 or has a subsequence that generates a spreading model not isomorphic to and we conclude with Corollary 3.3.

We now finish the circle of implications: is obvious, and follows from Corollary 3.3 and Remark 3.5.

4. The concentration inequalities

All the results of this section will be valid for the spaces , , equipped either with the Johnson metric or the Hamming metric . So throughout the section, will denote either or . We start with a general structural result on Lipschitz maps from into a reflexive space with a basis. For two vectors and in a Banach space with a basis, we shall use freely the convenient notation to mean that , and when , where the supports are with respect to the basis considered.

Proposition 4.1.

Let be a reflexive Banach space with a bimonotone basis . Then, for all , , , and Lipschitz maps there exist and satisfying the following.

For all , there exist , all with finite supports with respect to , such that

and

Proof.

We will show this result by induction on . The proof is based on a standard gliding hump procedure and a few diagonal arguments that we have chosen to detail. For , we shall denote the basis projection on the linear span of .

For assume that we are given , , a Lipschitz map , and . Since is bounded and is reflexive, by weak compactness, there exists and such that converges weakly to . From the weak lower semicontinuity of the norm of we deduce that for all . Since is weakly null, one can find such that

Then for all for all , , there exists such that

We now set and , for . For all , , has finite support and since the basis is bimonotone, . Finally, we deduce that from the triangle inequality.

Assume now that our statement is proved for , and let , a Lipschitz map , and be given. Using that is a bounded subset of the reflexive space and a diagonal argument, we infer the existence of such that for all , converges weakly to some . Using again that the norm of is weakly lower semicontinuous, we have that . This allows us to apply our induction hypothesis to the map and to find and , such that for all there exist , all with finite supports, such that for all and

We now fix . Note that the weak lower semicontinuity of the norm implies that for all , . Denote . Since is weakly null, there exists such that for all ,

Then, for all , , there exists so that

We now set . We have that, for all , : has finite support, , and

We conclude the proof with one last extraction argument.

If , we define recursively as follows. For all , where , and if has been chosen for then is picked such that and for all in the finite set . It is now easy to check that , , and for all and , , the vectors with finite support , satisfy the induction hypothesis for .

Remark 4.2.

We have assumed that the basis of is bimonotone only for convenience. In the general case, one gets a similar result, with the only difference being that , where is the bimonotonicity constant.

Remark 4.3.

Proposition 4.1 is actually true if we only assume that is a Banach space with a boundedly complete Schauder finite-dimensional decomposition. In that case, we just have to use weak-compactness instead of weak-compactness.

Our concentration result for Lipschitz maps with values in follows easily from Proposition 4.1. The following theorem is Theorem D when .

Theorem 4.4 (Concentration inequality).

Let , and let be a Lipschitz map. Then there exists such that for all one has

Proof.

We assume, as we may, that . Then it follows from Proposition 4.1 with , , and that there exists , such that for all there exist , all with finite supports, satisfying Equation 4.1 and Equation 4.2. And hence, for all ,

where for the last inequality we use property Equation 2.13 of .

The concentration inequalities stated above immediately provide an obstruction to equi-coarse embeddability of the Johnson graphs and the Hamming graphs . A rescaling argument can be used to provide an obstruction to equi-uniform embeddability of the rescaled metrics.

Corollary 4.5.
(1)

The sequence of Hamming graphs (resp. does not equi-coarsely (resp. equi-uniformly) embed into .

(2)

The sequence of Johnson graphs (resp. does not equicoarsely (resp. equi-uniformly) embed into .

Proof.

We only treat the case of the Hamming graphs, the case of the Johnson graphs being similar. Assume by contradiction that equi-coarsely embeds into , i.e., that there exist nondecreasing functions and for each a function , such that , , and for all . Since is the graph distance on , is actually -Lipschitz. By Theorem 4.4, for every , there exists such that for all one has

If and are chosen in such that then it implies that for every , , which contradicts the properties of and for large enough.

Assume now by contradiction that equi-uniformly embeds into , i.e., there exist nondecreasing functions and for each a function , such that , , and for all . Denote the identity map from to by . Then . By Theorem 4.4, for all , there exist such that for all . Therefore, considering again and in such that , we get that for all one has , which is a contradiction for large enough .

Remark 4.6.

As previously mentioned, Corollary B follows from the more involved rigidity phenomenon depicted in Theorem A. Nevertheless, for the reader only interested in the geometry of Hilbert space, we include an abridged proof of the -case. The proof follows simply from Corollary 4.5 and, in the coarse setting, the fact that the Johnson graphs equi-coarsely embeds into . Indeed, for consider the map

where denotes the canonical basis of . Then

For our claim in the uniform category, define for

then

and we again invoke Corollary 4.5 to conclude the proof of our claim.

Reflexivity is usually not preserved under nonlinear embeddings, as witnessed by Ribe’s example in Reference 35, where he showed that the two separable Banach spaces and , where is a sequence in such that , are uniformly homeomorphic. Using the Kalton–Randrianarivony concentration inequality, it was shown in Reference 6 that if coarse Lipschitz embeds into a reflexive Banach space that is asymptotically uniformly smooth, then must be reflexive. Actually, must have the Banach–Saks property as recently observed in Reference 9. Another consequence of the concentration inequality gives that reflexivity is stable under coarse embeddability into Banach spaces possessing the -like” asymptotic smoothness property of Tsirelson’s space . In order to preserve reflexivity, this trade-off between faithfulness of the embedding and smoothness properties of the target space is a rather interesting phenomenon.

Proposition 4.7.

Let be a Banach space. If coarsely embeds, or uniformly embeds, into then is reflexive.

Proof.

Assume that is not reflexive. Then, by James’s characterization of reflexive spaces Reference 17, there exists a sequence such that for all and ,

For every , the map defined as for , is clearly -Lipschitz. Assume by contradiction that there exists a coarse embedding . It follows again from the properties of graph distances that is -Lipschitz. Since , it follows from Theorem 4.4 that for all , there exists , such that for all . In particular for all , if are such that , we obtain by Equation 4.8 that , which is impossible for large enough .

For the proof in the uniform setting, we use the map defined as , which is clearly -Lipschitz. Assume by contradiction that there exists a uniform embedding . Then . By Theorem 4.4, for all , there exist such that for all . In particular for all , again if are such that , we get by Equation 4.8 that , which is again impossible if is sufficiently large.

Remark 4.8.

The conclusion of Proposition 4.7 can be slightly strengthened as already observed in Reference 9. Indeed, using Reference 7, Proposition 2, p. 273 one can show that has the Banach–Saks property.

Recall that Tsirelson’s space has the following important properties:

is reflexive.

does not contain isomorphic copies of , for any nor of .

All the spreading models of are isomorphic to .

Theorem A shows that Tsirelson’s construction is surprisingly rigid.

Proof of Theorem A.

Assume first that is a coarse embedding. By Corollary 4.5 the sequence does not equi-coarsely embed into . Since Proposition 4.7 ensures that is reflexive, it follows from Rosenthal’s theorem that every spreading model of can be generated by a weakly null sequence. Finally, it follows from Corollary 3.3 that every spreading model of is isomorphic to .

Assume now that is a uniform embedding. Assume also by contradiction that admits a spreading model which is not isomorphic to . By Proposition 4.7 is reflexive. Thus, it follows from Rosenthal’s theorem and Proposition 2.4 that there exists a normalized weakly null sequence in such that , where

Define now by

Note that . For every there exists by Theorem 4.4 such that

But, it follows from the definition of that whenever . So for all , we have . This contradicts the fact that and .

5. Final comments and open problems

Since clearly has trivial cotype, it follows from the Mendel–Naor metric cotype obstruction that is not coarsely embeddable into any Banach space that coarsely embeds into a Banach space with nontrivial type. Therefore, and , , are coarsely incomparable in the sense that is not coarsely embeddable into and is not coarsely embeddable into . To the best of our knowledge this provides the first pairs of coarsely incomparable Banach spaces.

Corollary 5.1.

Assume that is a Banach space such that

(1)

(or equi-coarsely embeds into ,

(2)

coarsely embeds into a Banach space with nontrivial type.

Then and are coarsely incomparable.

Let us conclude with a few open questions.

Problem 5.1.

Does coarsely embed into every super-reflexive Banach space?

In view of Ostrovskii’s result Reference 33, which we referenced in the introduction, a counterexample to Problem 5.1 would have to be a Banach space that is super-reflexive not containing any unconditional basic sequence. It seems that the only known such space is Ferenczi’s space Reference 12.

Problem 5.2.

Does coarsely embed into any Banach space which has a spreading model that is not equivalent to ?

Acknowledgments

This work was completed while the second author was visiting Texas A&M University in College Station, Texas. He wishes to thank the mathematics department of Texas A&M University for its warm hospitality and the excellent working environment.

Mathematical Fragments

Equation (1.1)
Equation (1.2)
Equation (1.3)
Equation (1.4)
Theorem A.

Let be an infinite-dimensional Banach space. If coarsely embeds into or if its unit ball uniformly embeds into , then is reflexive and all its spreading models are isomorphic to .

Corollary B.

and for (resp. and do not coarsely embed (resp. uniformly embed) into .

Theorem D.

Let and be a Lipschitz map. Then there exists such that for all one has

Theorem E.

For a Banach space the following assertions are equivalent:

(1)

is finite dimensional.

(2)

does not equi-coarsely embed into .

Equation (2.7)
Equation (2.8)
Proposition 2.2.

Let be a normalized weakly null sequence in a Banach space . Then for every there is a normalized weakly null basic subsequence of with basis constant , generating a spreading model with fundamental function , such that for all , for all , and for all one has

Proposition 2.3.

has a spreading model isomorphic to if and only if for all , has a spreading model whose fundamental sequence is -equivalent to the canonical basis of , i.e., that for all , ,

Proposition 2.4.

Reference 8, Proposition 3, p. 79 Let be a normalized weakly null sequence in a Banach space that generates a spreading model . Then is not isomorphic to if and only if there exists a subsequence of such that

Equation (2.13)
Lemma 3.1.

Let be any of the metric spaces , , or for . Let be an infinite-dimensional Banach space admitting a spreading model generated by a normalized weakly null sequence. Then for every there exists a map such that for all ,

Equation (3.2)
Corollary 3.3.

The metric spaces , , are all coarsely embeddable, and the sequence is equi-coarsely embeddable into a Banach space that has a spreading model generated by a normalized weakly null sequence, which is not isomorphic to .

More precisely, in all three cases, for any , there is a coarse embedding into such that is -Lipschitz and the modulus of compression of satisfies , for , where is the fundamental function of the spreading model .

Proposition 3.4.

If an infinite-dimensional Banach space has a spreading model isomorphic to then equi-bi-Lipschitzly embeds into .

More precisely, for every and every there exists a map , such that for all ,

Remark 3.5.

If an infinite-dimensional Banach space has a spreading model isomorphic to then, using a duality argument Reference 8, Proposition 1, p. 80, the relationship between the -weakly null index and the Szlenk index Reference 2, Theorem 4.2, and the embedding in Reference 6, Theorem 2.6, one can show that embeds bi-Lipschitzly into .

Moreover, since it is easy to see that every countable tree isometrically embeds into , we obtain that every countable tree coarsely embeds into any infinite-dimensional Banach space.

Proposition 4.1.

Let be a reflexive Banach space with a bimonotone basis . Then, for all , , , and Lipschitz maps there exist and satisfying the following.

For all , there exist , all with finite supports with respect to , such that

and

Theorem 4.4 (Concentration inequality).

Let , and let be a Lipschitz map. Then there exists such that for all one has

Corollary 4.5.
(1)

The sequence of Hamming graphs (resp. does not equi-coarsely (resp. equi-uniformly) embed into .

(2)

The sequence of Johnson graphs (resp. does not equicoarsely (resp. equi-uniformly) embed into .

Remark 4.6.

As previously mentioned, Corollary B follows from the more involved rigidity phenomenon depicted in Theorem A. Nevertheless, for the reader only interested in the geometry of Hilbert space, we include an abridged proof of the -case. The proof follows simply from Corollary 4.5 and, in the coarse setting, the fact that the Johnson graphs equi-coarsely embeds into . Indeed, for consider the map

where denotes the canonical basis of . Then

For our claim in the uniform category, define for

then

and we again invoke Corollary 4.5 to conclude the proof of our claim.

Proposition 4.7.

Let be a Banach space. If coarsely embeds, or uniformly embeds, into then is reflexive.

Equation (4.8)
Problem 5.1.

Does coarsely embed into every super-reflexive Banach space?

References

[1]
I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of , Israel J. Math. 19 (1974), 284–291. MR0511661, Show rawAMSref\bib{Aharoni1974}{article}{ author={Aharoni, Israel}, title={Every separable metric space is Lipschitz equivalent to a subset of $c^{+}_{0}$}, journal={Israel J. Math.}, volume={19}, date={1974}, pages={284--291}, issn={0021-2172}, review={\MR {0511661}}, } Close amsref.
[2]
D. Alspach, R. Judd, and E. Odell, The Szlenk index and local -indices, Positivity 9 (2005), no. 1, 1–44. MR2139115, Show rawAMSref\bib{AJO2005}{article}{ author={Alspach, Dale}, author={Judd, Robert}, author={Odell, Edward}, title={The Szlenk index and local $l_1$-indices}, journal={Positivity}, volume={9}, date={2005}, number={1}, pages={1--44}, issn={1385-1292}, review={\MR {2139115}}, } Close amsref.
[3]
F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429. MR2363693, Show rawAMSref\bib{Baudier2007}{article}{ author={Baudier, Florent}, title={Metrical characterization of super-reflexivity and linear type of Banach spaces}, journal={Arch. Math. (Basel)}, volume={89}, date={2007}, number={5}, pages={419--429}, issn={0003-889X}, review={\MR {2363693}}, } Close amsref.
[4]
F. P. Baudier, Quantitative nonlinear embeddings into Lebesgue sequence spaces, J. Topol. Anal. 8 (2016), no. 1, 117–150. MR3463248, Show rawAMSref\bib{BaudierJTA2016}{article}{ author={Baudier, Florent P.}, title={Quantitative nonlinear embeddings into Lebesgue sequence spaces}, journal={J. Topol. Anal.}, volume={8}, date={2016}, number={1}, pages={117--150}, issn={1793-5253}, review={\MR {3463248}}, } Close amsref.
[5]
F. Baudier and G. Lancien, Tight embeddability of proper and stable metric spaces, Anal. Geom. Metr. Spaces 3 (2015), 140–156. MR3365754, Show rawAMSref\bib{BaudierLancien2015}{article}{ author={Baudier, F.}, author={Lancien, G.}, title={Tight embeddability of proper and stable metric spaces}, journal={Anal. Geom. Metr. Spaces}, volume={3}, date={2015}, pages={140--156}, issn={2299-3274}, review={\MR {3365754}}, } Close amsref.
[6]
F. Baudier, N. J. Kalton, and G. Lancien, A new metric invariant for Banach spaces, Studia Math. 199 (2010), no. 1, 73–94. MR2652598, Show rawAMSref\bib{BKL2010}{article}{ author={Baudier, F.}, author={Kalton, N. J.}, author={Lancien, G.}, title={A new metric invariant for Banach spaces}, journal={Studia Math.}, volume={199}, date={2010}, number={1}, pages={73--94}, issn={0039-3223}, review={\MR {2652598}}, } Close amsref.
[7]
B. Beauzamy, Banach-Saks properties and spreading models, Math. Scand. 44 (1979), no. 2, 357–384. MR555227, Show rawAMSref\bib{Beauzamy1979}{article}{ author={Beauzamy, Bernard}, title={Banach-Saks properties and spreading models}, journal={Math. Scand.}, volume={44}, date={1979}, number={2}, pages={357--384}, issn={0025-5521}, review={\MR {555227}}, } Close amsref.
[8]
B. Beauzamy and J.-T. Lapresté, Modèles étalés des espaces de Banach (French), Travaux en Cours. [Works in Progress], Hermann, Paris, 1984. MR770062, Show rawAMSref\bib{BeauzamyLapreste1984}{book}{ author={Beauzamy, B.}, author={Laprest\'e, J.-T.}, title={Mod\`eles \'etal\'es des espaces de Banach}, language={French}, series={Travaux en Cours. [Works in Progress]}, publisher={Hermann, Paris}, date={1984}, pages={iv+210}, isbn={2-7056-5965-X}, review={\MR {770062}}, } Close amsref.
[9]
B. M. Braga, Asymptotic structure and coarse Lipschitz geometry of Banach spaces, Studia Math. 237 (2017), no. 1, 71–97. MR3612891, Show rawAMSref\bib{BragaSM2017}{article}{ author={Braga, B. M.}, title={Asymptotic structure and coarse Lipschitz geometry of Banach spaces}, journal={Studia Math.}, volume={237}, date={2017}, number={1}, pages={71--97}, issn={0039-3223}, review={\MR {3612891}}, } Close amsref.
[10]
A. Brunel and L. Sucheston, On -convex Banach spaces, Math. Systems Theory 7 (1974), no. 4, 294–299. MR0438085, Show rawAMSref\bib{BrunelSucheston1974}{article}{ author={Brunel, Antoine}, author={Sucheston, Louis}, title={On $B$-convex Banach spaces}, journal={Math. Systems Theory}, volume={7}, date={1974}, number={4}, pages={294--299}, issn={0025-5661}, review={\MR {0438085}}, } Close amsref.
[11]
A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR0139079, Show rawAMSref\bib{Dvoretzky1961}{article}{ author={Dvoretzky, Aryeh}, title={Some results on convex bodies and Banach spaces}, conference={ title={Proc. Internat. Sympos. Linear Spaces}, address={Jerusalem}, date={1960}, }, book={ publisher={Jerusalem Academic Press, Jerusalem; Pergamon, Oxford}, }, date={1961}, pages={123--160}, review={\MR {0139079}}, } Close amsref.
[12]
V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225. MR1489106, Show rawAMSref\bib{Ferenczi1997}{article}{ author={Ferenczi, V.}, title={A uniformly convex hereditarily indecomposable Banach space}, journal={Israel J. Math.}, volume={102}, date={1997}, pages={199--225}, issn={0021-2172}, review={\MR {1489106}}, } Close amsref.
[13]
S. C. Ferry, A. Ranicki, and J. Rosenberg, A history and survey of the Novikov conjecture, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 7–66. MR1388295, Show rawAMSref\bib{FerryRanickiRosenberg1995}{article}{ author={Ferry, Steven C.}, author={Ranicki, Andrew}, author={Rosenberg, Jonathan}, title={A history and survey of the Novikov conjecture}, conference={ title={Novikov conjectures, index theorems and rigidity, Vol.\ 1}, address={Oberwolfach}, date={1993}, }, book={ series={London Math. Soc. Lecture Note Ser.}, volume={226}, publisher={Cambridge Univ. Press, Cambridge}, }, date={1995}, pages={7--66}, review={\MR {1388295}}, } Close amsref.
[14]
T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no , Compositio Math. 29 (1974), 179–190. MR0355537, Show rawAMSref\bib{FigielJohnson1974}{article}{ author={Figiel, T.}, author={Johnson, W. B.}, title={A uniformly convex Banach space which contains no $l_{p}$}, journal={Compositio Math.}, volume={29}, date={1974}, pages={179--190}, issn={0010-437X}, review={\MR {0355537}}, } Close amsref.
[15]
G. Godefroy, G. Lancien, and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529–1583. MR3295641, Show rawAMSref\bib{GLZ2014}{article}{ author={Godefroy, G.}, author={Lancien, G.}, author={Zizler, V.}, title={The non-linear geometry of Banach spaces after Nigel Kalton}, journal={Rocky Mountain J. Math.}, volume={44}, date={2014}, number={5}, pages={1529--1583}, issn={0035-7596}, review={\MR {3295641}}, } Close amsref.
[16]
R. C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR0039915, Show rawAMSref\bib{James1950}{article}{ author={James, Robert C.}, title={Bases and reflexivity of Banach spaces}, journal={Ann. of Math. (2)}, volume={52}, date={1950}, pages={518--527}, issn={0003-486X}, review={\MR {0039915}}, } Close amsref.
[17]
R. C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Princeton Univ. Press, Princeton, N.J., 1972, pp. 159–175. Ann. of Math. Studies, No. 69. MR0454600, Show rawAMSref\bib{James1972}{article}{ author={James, Robert C.}, title={Some self-dual properties of normed linear spaces}, conference={ title={Symposium on Infinite-Dimensional Topology}, address={Louisiana State Univ., Baton Rouge, La.}, date={1967}, }, book={ publisher={Princeton Univ. Press, Princeton, N.J.}, }, date={1972}, pages={159--175. Ann. of Math. Studies, No. 69}, review={\MR {0454600}}, } Close amsref.
[18]
W. B. Johnson and N. L. Randrianarivony, does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1045–1050. MR2196037, Show rawAMSref\bib{JohnsonRandrianarivony2006}{article}{ author={Johnson, William B.}, author={Randrianarivony, N. Lovasoa}, title={$l_p\ (p>2)$ does not coarsely embed into a Hilbert space}, journal={Proc. Amer. Math. Soc.}, volume={134}, date={2006}, number={4}, pages={1045--1050}, issn={0002-9939}, review={\MR {2196037}}, } Close amsref.
[19]
N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Quart. J. Math. (Oxford) 58 (2007), no. 3, 393–414. MR2354924, Show rawAMSref\bib{Kalton2007}{article}{ author={Kalton, N. J.}, title={Coarse and uniform embeddings into reflexive spaces}, journal={Quart. J. Math. (Oxford)}, volume={58}, date={2007}, number={3}, pages={393--414}, issn={0033-5606}, review={\MR {2354924}}, } Close amsref.
[20]
N. J. Kalton, Lipschitz and uniform embeddings into , Fund. Math. 212 (2011), no. 1, 53–69. MR2771588, Show rawAMSref\bib{KaltonFM11}{article}{ author={Kalton, N. J.}, title={Lipschitz and uniform embeddings into $\ell _\infty $}, journal={Fund. Math.}, volume={212}, date={2011}, number={1}, pages={53--69}, issn={0016-2736}, review={\MR {2771588}}, } Close amsref.
[21]
N. J. Kalton and N. L. Randrianarivony, The coarse Lipschitz geometry of , Math. Ann. 341 (2008), no. 1, 223–237. MR2377476, Show rawAMSref\bib{KaltonRandrianarivony2008}{article}{ author={Kalton, Nigel J.}, author={Randrianarivony, N. Lovasoa}, title={The coarse Lipschitz geometry of $l_p\oplus l_q$}, journal={Math. Ann.}, volume={341}, date={2008}, number={1}, pages={223--237}, issn={0025-5831}, review={\MR {2377476}}, } Close amsref.
[22]
N. J. Kalton and D. Werner, Property , -ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. MR1324212, Show rawAMSref\bib{KaltonWerner1995}{article}{ author={Kalton, Nigel J.}, author={Werner, Dirk}, title={Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces}, journal={J. Reine Angew. Math.}, volume={461}, date={1995}, pages={137--178}, issn={0075-4102}, review={\MR {1324212}}, } Close amsref.
[23]
G. Kasparov and G. Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), no. 1, 1–56. MR2261750, Show rawAMSref\bib{KasparovYu2006}{article}{ author={Kasparov, Gennadi}, author={Yu, Guoliang}, title={The coarse geometric Novikov conjecture and uniform convexity}, journal={Adv. Math.}, volume={206}, date={2006}, number={1}, pages={1--56}, issn={0001-8708}, review={\MR {2261750}}, } Close amsref.
[24]
M. Mendel and A. Naor, Metric cotype, Ann. of Math. (2) 168 (2008), no. 1, 247–298. MR2415403, Show rawAMSref\bib{MendelNaor2008}{article}{ author={Mendel, Manor}, author={Naor, Assaf}, title={Metric cotype}, journal={Ann. of Math. (2)}, volume={168}, date={2008}, number={1}, pages={247--298}, issn={0003-486X}, review={\MR {2415403}}, } Close amsref.
[25]
A. Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012), no. 2, 167–233., Show rawAMSref\bib{Naor2012}{article}{ author={Naor, A.}, title={An introduction to the {R}ibe program}, journal={Jpn. J. Math.}, fjournal={Japanese Journal of Mathematics}, volume={7}, year={2012}, number={2}, pages={167--233}, issn={0289-2316}, mrclass={46-02 (46B07 46B80 46B85)}, mrnumber={2995229}, url={http://dx.doi.org/10.1007/s11537-012-1222-7}, } Close amsref.
[26]
A. Naor, Discrete Riesz transforms and sharp metric inequalities, Ann. of Math. (2) 184 (2016), no. 3, 991–1016. MR3549628, Show rawAMSref\bib{NaorAnnals2016}{article}{ author={Naor, Assaf}, title={Discrete Riesz transforms and sharp metric $X_p$ inequalities}, journal={Ann. of Math. (2)}, volume={184}, date={2016}, number={3}, pages={991--1016}, issn={0003-486X}, review={\MR {3549628}}, } Close amsref.
[27]
A. Naor and G. Schechtman, Pythagorean powers of hypercubes (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 1093–1116. MR3494167, Show rawAMSref\bib{NSFourier2016}{article}{ author={Naor, Assaf}, author={Schechtman, Gideon}, title={Pythagorean powers of hypercubes}, language={English, with English and French summaries}, journal={Ann. Inst. Fourier (Grenoble)}, volume={66}, date={2016}, number={3}, pages={1093--1116}, issn={0373-0956}, review={\MR {3494167}}, } Close amsref.
[28]
A. Naor and G. Schechtman, Metric inequalities, Forum Math. Pi 4 (2016), e3, 81. MR3456183, Show rawAMSref\bib{NSPi2016}{article}{ author={Naor, Assaf}, author={Schechtman, Gideon}, title={Metric $X_p$ inequalities}, journal={Forum Math. Pi}, volume={4}, date={2016}, pages={e3, 81}, issn={2050-5086}, review={\MR {3456183}}, } Close amsref.
[29]
P. W. Nowak, On coarse embeddability into -spaces and a conjecture of Dranishnikov, Fund. Math. 189 (2006), no. 2, 111–116. MR2214573, Show rawAMSref\bib{Nowak2006}{article}{ author={Nowak, Piotr W.}, title={On coarse embeddability into $l_p$-spaces and a conjecture of Dranishnikov}, journal={Fund. Math.}, volume={189}, date={2006}, number={2}, pages={111--116}, issn={0016-2736}, review={\MR {2214573}}, } Close amsref.
[30]
P. W. Nowak and G. Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR2986138, Show rawAMSref\bib{NowakYu2012}{book}{ author={Nowak, Piotr W.}, author={Yu, Guoliang}, title={Large scale geometry}, series={EMS Textbooks in Mathematics}, publisher={European Mathematical Society (EMS), Z\"urich}, date={2012}, pages={xiv+189}, isbn={978-3-03719-112-5}, review={\MR {2986138}}, } Close amsref.
[31]
E. Odell and T. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281., Show rawAMSref\bib{OdellSchlumprecht1994}{article}{ author={Odell, E.}, author={Schlumprecht, T.}, title={The distortion problem}, journal={Acta Math.}, volume={173}, date={1994}, pages={259\ndash 281}, } Close amsref.
[32]
E. Odell, Th. Schlumprecht, and A. Zsák, Banach spaces of bounded Szlenk index, Studia Math. 183 (2007), no. 1, 63–97. MR2360257, Show rawAMSref\bib{OSZ2007}{article}{ author={Odell, E.}, author={Schlumprecht, Th.}, author={Zs\'ak, A.}, title={Banach spaces of bounded Szlenk index}, journal={Studia Math.}, volume={183}, date={2007}, number={1}, pages={63--97}, issn={0039-3223}, review={\MR {2360257}}, } Close amsref.
[33]
M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009), 163–183. MR2471569, Show rawAMSref\bib{Ostrovskii2009}{article}{ author={Ostrovskii, M. I.}, title={Coarse embeddability into Banach spaces}, journal={Topology Proc.}, volume={33}, date={2009}, pages={163--183}, issn={0146-4124}, review={\MR {2471569}}, } Close amsref.
[34]
M. I. Ostrovskii, Metric embeddings: Bilipschitz and coarse embeddings into Banach spaces, De Gruyter Studies in Mathematics, vol. 49, De Gruyter, Berlin, 2013. MR3114782, Show rawAMSref\bib{Ostrovskiibook}{book}{ author={Ostrovskii, Mikhail I.}, title={Metric embeddings}, series={De Gruyter Studies in Mathematics}, volume={49}, subtitle={Bilipschitz and coarse embeddings into Banach spaces}, publisher={De Gruyter, Berlin}, date={2013}, pages={xii+372}, isbn={978-3-11-026340-4}, isbn={978-3-11-026401-2}, review={\MR {3114782}}, } Close amsref.
[35]
M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), no. 2-3, 139–147. MR770696, Show rawAMSref\bib{Ribe1984}{article}{ author={Ribe, M.}, title={Existence of separable uniformly homeomorphic nonisomorphic Banach spaces}, journal={Israel J. Math.}, volume={48}, date={1984}, number={2-3}, pages={139--147}, issn={0021-2172}, review={\MR {770696}}, } Close amsref.
[36]
J. Rosenberg, Novikov’s conjecture, Open Problems in Mathematics, Springer, Cham, 2016, pp. 377–402. MR3526942, Show rawAMSref\bib{Rosenberg2016}{article}{ author={Rosenberg, Jonathan}, title={Novikov's conjecture}, conference={ title={Open Problems in Mathematics}, }, book={ publisher={Springer, Cham}, }, date={2016}, pages={377--402}, review={\MR {3526942}}, } Close amsref.
[37]
B. S. Tsirelson, It is impossible to imbed of into an arbitrary Banach space (Russian), Funkcional. Anal. i Priložen. 8 (1974), 57–60., Show rawAMSref\bib{Tsirelson1974}{article}{ author={Tsirel{$^\prime $}son, B. S.}, title={It is impossible to imbed $l\sb {p}$ of $c\sb {0}$ into an arbitrary Banach space}, language={Russian}, journal={Funkcional. Anal. i Prilo\v zen.}, volume={8}, date={1974}, pages={57\ndash 60}, } Close amsref.
[38]
A. Valette, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. From notes taken by Indira Chatterji, With an appendix by Guido Mislin, DOI 10.1007/978-3-0348-8187-6., Show rawAMSref\bib{Valette2002}{book}{ author={Valette, A.}, title={Introduction to the {B}aum-{C}onnes conjecture}, series={Lectures in Mathematics ETH Z\"urich}, note={From notes taken by Indira Chatterji, With an appendix by Guido Mislin}, publisher={Birkh\"auser Verlag, Basel}, year={2002}, pages={x+104}, isbn={3-7643-6706-7}, mrclass={58J22 (19K35 22D25 46L80 46L87)}, mrnumber={1907596 (2003f:58047)}, mrreviewer={Paul D. Mitchener}, doi={10.1007/978-3-0348-8187-6}, url={http://dx.doi.org/10.1007/978-3-0348-8187-6}, } Close amsref.
[39]
G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR1728880, Show rawAMSref\bib{Yu2000}{article}{ author={Yu, Guoliang}, title={The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space}, journal={Invent. Math.}, volume={139}, date={2000}, number={1}, pages={201--240}, issn={0020-9910}, review={\MR {1728880}}, } Close amsref.

Article Information

MSC 2010
Primary: 46B20 (Geometry and structure of normed linear spaces), 46B85 (Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science), 46T99 (None of the above, but in this section), 05C63 (Infinite graphs), 20F65 (Geometric group theory)
Author Information
F. Baudier
Department of Mathematics, Texas A&M University, College Station, Texas 77843
florent@math.tamu.edu
MathSciNet
G. Lancien
Laboratoire de Mathématiques de Besançon, CNRS UMR-6623, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cédex, France
gilles.lancien@univ-fcomte.fr
MathSciNet
Th. Schlumprecht
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, and Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 16627, Prague, Czech Republic
schlump@math.tamu.edu
MathSciNet
Additional Notes

The first author was partially supported by the National Science Foundation under grant number DMS-1565826.

The second author was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03) and as a participant of the “NSF Workshop in Analysis and Probability” at Texas A&M University.

The third author was supported by the National Science Foundation under grant number DMS-1464713.

Journal Information
Journal of the American Mathematical Society, Volume 31, Issue 3, ISSN 1088-6834, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2018 American Mathematical Society
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