Coarse and Lipschitz universality
Florent P. Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht
aa r X i v : . [ m a t h . M G ] A p r COARSE AND LIPSCHITZ UNIVERSALITY
F. BAUDIER, G. LANCIEN, P. MOTAKIS, AND TH. SCHLUMPRECHTA
BSTRACT . In this paper we provide several metric universality results. We exhibit for certainclasses C of metric spaces, families of metric spaces ( M i , d i ) i ∈ I which have the property that a metricspace ( X , d X ) in C is coarsely, resp. Lipschitzly, universal for all spaces in C if the collection ofspaces ( M i , d i ) i ∈ I equi-coarsely, respectively equi-Lipschitzly, embeds into ( X , d X ) . Such familiesare built as certain Schreier-type metric subsets of c . We deduce a metric analog to Bourgain’stheorem, which generalized Szlenk’s theorem, and prove that a space which is coarsely universalfor all separable reflexive asymptotic- c Banach spaces is coarsely universal for all separable metricspaces. One of our coarse universality results is valid under Martin’s Axiom and the negation of theContinuum Hypothesis. We discuss the strength of the universality statements that can be obtainedwithout these additional set theoretic assumptions. In the second part of the paper, we study univer-sality properties of Kalton’s interlacing graphs. In particular, we prove that every finite metric spaceembeds almost isometrically in some interlacing graph of large enough diameter. C ONTENTS
1. Introduction 22. Preliminaries 32.1. Coarse and Lipschitz geometry 32.2. Trees, derivations, and Bourgain’s index theory 42.3. Schreier sets and higher order Tsirelson spaces 53. Metric universality via descriptive set theory 63.1. Lipschitz universality via a Lipschitz c -index 63.2. Coarse universality via a coarse c -index in MA+ ¬ CH 93.3. Coarse universality via strong boundedness 114. Coarse universality and barycentric gluing 125. Universality properties of interlacing graphs 165.1. Almost isometric universality of the interlacing graphs 165.2. Metric universality and metric elasticity 205.3. Separating interlacing graphs in Banach spaces with nonseparable biduals 22References 24
Mathematics Subject Classification.
1. I
NTRODUCTION
A metric space Y cu is said to be coarsely universal for a class M of metric spaces if everymetric space in M coarsely embeds into Y cu . By modifying the definition accordingly we canobviously consider universality in various categories: [Banach spaces ∼ isomorphic embeddings],[metric spaces ∼ bi-Lipschitz embeddings], etc. A natural question is thus the following: Given aclass of metric spaces can we find a metric space that is universal for this class with respect to a giventype of metric embedding? There are numerous embedding results that provide satisfactory answersto this broad question. That ℓ ∞ is isometrically universal for the class of separable metric spaces is areformulation of the (elementary but fundamental) Fr´echet-Kuratowski embedding theorem [Fr´e10,Kur35]. Note that ℓ ∞ is not separable and thus does not belong to the class it is a universal spacefor. This leads us to refine the question to, say: is there a member of the class that is universal forthe class itself? Urysohn’s space [Ury25] answers positively this question for the class of separablemetric spaces and isometric embeddings. However, it is not always possible to find a universal spacewithin the considered class. A (relatively) simple example is the class of separable super-reflexiveBanach spaces when universality refers to isomorphic embeddings. A much more difficult resultof Szlenk [Szl68] states that there is no separable reflexive Banach space that is isomorphicallyuniversal for the class of separable reflexive Banach spaces. Szlenk’s theorem was improved byBourgain [Bou80] who showed that a separable Banach space that is isomorphically universal for theclass of separable reflexive spaces is also isomorphically universal for all separable Banach spaces.So if we want to show that a separable Banach space contains an isomorphic copy of every separableBanach space we only need to show that it contains an isomorphic copy of every separable reflexiveBanach space. To prove this remarkable rigidity result in the context of isomorphic universality,Bourgain ingeniously incorporated techniques from descriptive set theory. Bourgain’s descriptiveset theoretic approach for universality problems, was further extended by Bossard [Bos02] to showthat a class of Banach spaces which is analytic, in the Effros-Borel structure of subspaces of C [ , ] ,and contains all separable reflexive Banach spaces, must contain a universal space.We will not discuss the numerous variants of the universality problem but instead we will focuson the following rigidity phenomenon in the context of universality. We voluntarily do not specifya specific type of embeddings. Problem 1.1.
For what classes C and D of metric spaces such that C ⊂ D , a universal space for C is also a universal space for D ? The first part of the article revolves around Problem 1.1 in the Lipschitz and coarse categories.Our first theorem says that a metric space is Lipschitzly universal for the class of all separable metricspaces, if it is universal for the uncountable collection C : = { ( S α ( Q ) , d ∞ ) : α < ω } , which we willrefer to as the collection of rational-valued smooth Schreier metric spaces. None of the metricspaces in C is coarsely universal, but since they are built as certain Schreier-type metric subsetsof c , their entire hierarchy captures enough structure of c , and thus confers its good universalityproperties. Theorem A.
If a complete separable metric space contains bi-Lipschitz copies of ( S α ( Q ) , d ∞ ) forevery countable ordinal α , then it is Lipschiztly universal for the class of all separable metric spaces.Theorem A should be thought of as a purely Lipschitz analogue of the linear universality resultthat states that if a Banach space X is isomorphically universal for the class of separable reflexiveasymptotic-c Banach spaces then X contains an isomorphic copy of c . This linear universality canbe found in [OSZ07], as it is explained at the end of section 1. Similarly to the linear setting we usean ordinal index `a la Bourgain. OARSE AND LIPSCHITZ UNIVERSALITY 3
In the context of coarse universality, technical difficulties arise and we need some additional set-theoretic axioms (Martin’s Axiom and the negation of the Continuum Hypothesis) to prove a coarseanalogue of Theorem A. Note that here we only consider integer-valued Schreier metric spaces.
Theorem B. (MA+ ¬ CH) If a separable metric space contains coarse copies of ( S α ( Z ) , d ∞ ) forevery countable ordinal α , then it is coarsely universal for the class of all separable metric spaces.We end the first part with several results which have statements which are somewhat weaker thanTheorem B, but can be shown without any further axioms. In particular, we show the following. Theorem C.
If a separable metric space ( M , d ) contains coarse copies of ( S α ( Z ) , d ∞ ) for everycountable ordinal α , then the class of all separable bounded metric spaces embeds equi-coarselyinto ( M , d ) .With the help of a deep result of Dodos [Dod09], we prove Theorem D below. Note that theassumption is formally stronger than that of Theorem B or Theorem C. Theorem D.
If a separable metric space is coarsely universal for the class of all reflexive asymptotic-c Banach spaces then it is coarsely universal for the class of all separable metric spaces.The second part of the article discusses some universality properties of the sequence of interlacinggraphs ([ N ] k , d I ) k and their applications to universality problems. The geometry of these graphs isintimately connected to the geometry of c via the summing norm, and we prove the followinguniversality property. Theorem E.
For every finite metric space X and every ε >
0, there exists k : = k ( X , ε ) ∈ N suchthat X admits a bi-Lipschitz embedding into ([ N ] k , d I ) with distortion at most 1 + ε .Note that it follows from this almost isometric universality property of the interlacing graphsand the work of Eskenazis, Mendel and Naor [EMN19] that the sequence of interlacing graphs ([ N ] k , d I ) k does not equi-coarsely embed into any Alexandrov space of nonpositive curvature.Then, we discuss the connection between metric universality, the geometry of the interlacinggraphs, and a nonlinear version of Johnson-Odell elasticity.In [Kal07], Kalton showed that a separable Banach X that is coarsely universal for all separablemetric spaces cannot have all its iterated duals separable. The argument is based on the existence ofuncountably many well separated copies of the interlacing graphs in c . We conclude the paper byshowing that it can be generalized to prove the following. Theorem F.
Let X be a separable Banach space with non separable bidual X ∗∗ and such that nospreading model generated by a normalized weakly null sequence in X is equivalent to the ℓ -unitvector basis. Assume that X coarsely embeds into a Banach space Y . Then there exists k ∈ N suchthat Y ( k ) is non separable.In connection with this last result, it is important to note that ℓ is known to coarsely embed into ℓ . 2. P RELIMINARIES
Coarse and Lipschitz geometry. If X and Y are two metric spaces, the Y -distortion of X ,denoted c Y ( X ) , is defined as the infimum of those D ∈ [ , ∞ ) such that there exist s ∈ ( , ∞ ) and amap f : X → Y so that for all x , y ∈ X (1) s · d X ( x , y ) ≤ d Y (cid:0) f ( x ) , f ( y ) (cid:1) ≤ s · D · d X ( x , y ) . F. BAUDIER, G. LANCIEN, P. MOTAKIS, AND TH. SCHLUMPRECHT
When (1) holds we say that X bi-Lipschitzly embeds into Y with distortion D . We introducesome convenient terminology and notation that will allow us to treat all at once various embeddingnotions. Definition 2.1.
Let X and Y be metric spaces. Let ρ , ω : [ , ∞ ) → [ , ∞ ) . We say that X ( ρ , ω ) -embeds into Y if there exists f : X → Y such that for all x , y ∈ X we have(2) ρ ( d X ( x , y )) ≤ d Y ( f ( x ) , f ( y )) ≤ ω ( d X ( x , y )) . If { X i } i ∈ I is a collection of metric spaces. We say that { X i } i ∈ I ( ρ , ω ) -embeds into Y if for every i ∈ I , X i ( ρ , ω ) -embeds into Y .We will say that { X i } i ∈ I equi-coarsely embeds into Y if there exist non-decreasing functions ρ , ω : [ , ∞ ) → [ , ∞ ) such that lim t → ∞ ρ ( t ) = ∞ and { X i } i ∈ I ( ρ , ω ) -embeds into Y . We say that { X i } i ∈ I equi-bi-Lipschiztly embeds into Y if { X i } i ∈ I ( ρ , ω ) -embeds into Y , where ρ and ω areincreasing and linear on [ , ∞ ) .Note that equi-bi-Lipschitz embeddability is a stronger condition than merely assuming thatsup i ∈ I c Y ( X i ) < ∞ since it does not allow for arbitrarily large or arbitrarily small scaling factorsin (1). However if Y is a Banach space rescaling is possible, and the two notions coincide.Aharoni’s embedding theorem [Aha74] states that there exists a universal constant K ∈ [ , ∞ ) such that every separable metric space bi-Lipschitzly embeds into c with distortion at most K . Theoptimal distortion in Aharoni’s embedding theorem is K = .2.2. Trees, derivations, and Bourgain’s index theory.
A tree T over a set X is a collection offinite sequences ( x , . . . , x n ) of elements of a set X with the property that whenever ( x , . . . , x n ) is in T then ( x , . . . , x n − ) is in T as well. A tree is well-founded if it has no infinite branch, i.e., thereis no sequence ( x k ) ∞ k = in X such that for all n ∈ N ( x , x , . . . , x n ) ∈ T . There is a classical ordinalderivation on trees which is defined transfinitely as follows: T = TT α + = { ( x , x , . . . , x n ) : ( x , x , . . . , x n , x n + ) ∈ T α } , for any ordinal α T β = ∩ α < β T α for any limit ordinal β .We definite o ( T ) , the order of a tree T , to be the least ordinal number such that T o ( T ) = /0, and byconvention we set o ( T ) = ∞ if such an ordinal does not exist. Note that if T is well-founded thenthe derivation produces a strictly decreasing sequence of trees and thus o ( T ) < ∞ . For every ordinal α it is easy to construct a tree T α such that o ( T α ) = α . In Section 2 we will need to strengthen acrucial result about trees on Polish spaces, which are complete, separable and metrizable spaces. Atree T on a topological space X is closed if for every n ∈ N , T ∩ X n is closed in X n equipped withthe product topology. The following proposition, which follows from [Kec95, Theorem 31.1], wasobserved by Bourgain [Bou80, Proposition 3]. Proposition 2.2.
If T is a closed and well founded tree on a Polish space, then o ( T ) < ω , where ω denotes the first uncountable ordinal. In order to facilitate the reading of Section 2, we recall Bourgain’s ordinal index “measuring”the presence of a given basic sequence in a Banach space. This idea was introduced in [Bou80] fora basis of C [ , ] , but can be (and has been extensively) applied for other basic sequences (see forinstance Definitions 3.1 and 3.6 in [AJO05] or [Ode04]). In this article we will be mostly interestedin the canonical basis of c . OARSE AND LIPSCHITZ UNIVERSALITY 5
Let ( e i ) i be a normalized basic sequence, X be a Banach space, and K ≥
1. Denote by T ( X , ( e i ) i , K ) the set of finite sequences ( x , x , . . . , x n ) of elements in X such that(3) 1 K k n ∑ k = a k x k k ≤ k n ∑ k = a k e k k ≤ K k n ∑ k = a k x k k . It is clear that T ( X , ( e i ) i , K ) is a closed tree on X . It is also straightforward that X contains a K -isomorphic copy of Y = span ( e i ) if and only if T ( X , ( e i ) i , K ) is not well founded (or in other wordshas an infinite branch). Moreover, if X is separable (and thus Polish), it follows from Proposition2.2 that X contains an K -isomorphic copy of Y = span ( e i ) if and only if o ( T ( X , ( e i ) i , K )) = ω . Atthe technical level, Bourgain constructed for every ordinal α , a separable reflexive Banach space X α such that for some universal constant K > T ( X α , ( e i ) i , K ) ≥ α , where ( e i ) i is a basis of C [ , ] . Ifa separable Banach space Z is isomorphically universal for all separable reflexive Banach spaces, itis easy to see that it must be C -isomorphically universal for all separable reflexive Banach spaces forsome C ≥
1. Indeed, if there exists a sequence of reflexive separable Banach spaces ( X n ) so that theembedding constants of them escape to infinity, then the reflexive separable space ( ∑ n X n ) wouldnot embed into Z . Thus Z will contain a C -isomorphic copy of all the X α ’s and thus T ( Z , ( e i ) i , D ) = ω for some D ≥
1, and based on the above discussion it follows that Z contains an isomorphiccopy of C [ , ] (which is well-known to be linearly isometrically universal for all separable Banachspaces thanks to Banach’s embedding theorem [Ban32]).Bourgain’s ( e i ) -index of X is defined as follows:I ( X , ( e i )) = sup { o ( T ( X , ( e i ) i , K )) : K ≥ } . We collect the key properties of the Bourgain’s index of the canonical basis of c , simply denotedby I c , that we will need later on. Proposition 2.3.
Let X , Y be separable Banach spaces. (1)
If X is a subspace of Y then I c ( X ) ≤ I c ( Y ) . (2) If X is isomorphically equivalent to Y then I c ( X ) = I c ( Y ) . (3) c embeds isomorphically into X if and only if I c ( X ) ≥ ω . Schreier sets and higher order Tsirelson spaces.
Schreier sets proved to be very useful tomeasure indices as well as to construct Banach spaces having certain indices. We will also use themin the more general metric context. We denote by [ N ] < ω the set of finite subsets of N . An element¯ n = { n , n , . . . , n k } ∈ [ N ] < ω will always be written in strictly increasing order, i.e., n < n < . . . < n k . If A and B are finite subsets of N we write n ≤ A < B if n ≤ min ( A ) ≤ max ( A ) < min ( B ) . Fora countable ordinal α we denote by S α ⊂ [ N ] < ω the Schreier family of order α which is definedrecursively as follows:S = (cid:8) { n } : n ∈ N } S α + = n S nj = E j : E j ∈ S α , for j = , . . . n and n ≤ E < E < . . . < E n o S β = (cid:8) A ∈ [ N ] < ω : ∃ n ∈ N , so that n ≤ A , and A ∈ S α n (cid:9) , if β is a limit ordinal, and ( α n ) ⊂ [ , α ) is a (fixed) sequence which increases to β .The above definition of S β , for β limit ordinal, is dependent on the choice of the sequence ( α n ) ,but for our purposes the specific choice of ( α n ) will be irrelevant. The Schreier sets ( S α ) α < ω are collections of finite subsets of N with increasing complexity which naturally generate trees T ( S α ) : = { ( n , n , . . . , n k ) : { n i } ki = ∈ S α } on N . It is not difficult to prove by transfinite inductionthat o ( T ( S α )) = ω α + G be a family offinite subsets of N and let E be a non-empty (finite or infinite) countable subset of R . We define the F. BAUDIER, G. LANCIEN, P. MOTAKIS, AND TH. SCHLUMPRECHT subset of c ( N ) X G , E = n ∑ i ∈ G c i e i : G ∈ G , c i ∈ E for i ∈ G o where ( e i ) is the canonical basis of c . We will endow X G , E with the metric d ∞ induced by thestandard c -norm k · k ∞ . When G = S α we will simply denote by ( S α ( E ) , d ∞ ) the metric spaceobtained. These metric spaces naturally embed into the higher order Tsirelson spaces T ∗ α , whichare reflexive Banach spaces whose duals T α have norms which are implicitly defined based on anadmissibility condition that involves the Schreier sets. Although the original space constructed byTsirelson [Tsi74] was T ∗ α , for α =
1, nowadays their duals T α , are usually referred to as Tsirelsonspaces , and it is easier to define T ∗ α by first defining T α . We recall the crucial properties of theBanach space T ∗ α (c.f. [OSZ07]), that are needed in this article. The separable reflexive Banachspace T ∗ α is asymptotic- c and has a 1-unconditional basis ( u i ) i with the property that for any G ∈ S α the sequence ( u i ) i ∈ G is 2-equivalent to the unit vector basis of ℓ | G | ∞ . From the latter property it followsthat the natural embedding of ( S α ( E ) , d ∞ ) in T ∗ α (mapping ∑ i ∈ G c i e i to ∑ i ∈ G c i u i , for G ∈ S α ) is a4-Lipschitz isomorphism. Moreover, it follows from [OSZ07] that Bourgain’s c -index of T ∗ α tendsto ω as α tends to ω .3. M ETRIC UNIVERSALITY VIA DESCRIPTIVE SET THEORY
This section is deeply inspired by the profound ideas introduced by Bourgain and Bossard inconnection with isomorphic universality, and the unification of these approaches initiated by Ar-gyros and Dodos [AD07]. The most natural approach to prove Theorem A (resp. Theorem B),is to mimic Bourgain’s strategy and construct an ordinal index that will detect the presence of abi-Lipschitz (resp. coarse) copy of c , and which behaves similarly to Bourgain c -index. We canindeed (though non-trivially) adjust Bourgain’s approach to prove the Lipschitz universality resultin Section 3.1. Unfortunately some difficulties arise in the coarse setting. On one hand, in Section3.2, we use additional set theoretic axioms to prove Theorem B. On the other hand, we need to re-sort to the delicate theory of strongly bounded classes of Banach spaces to prove Theorem D. Thisis carried over in Section 3.3 where we will use a deep theorem of Dodos. With this organization,we hope it will be clear what is the scope of application of Bourgain’s strategy and why it partiallyfails to work in the coarse framework.3.1. Lipschitz universality via a Lipschitz c -index. To detect the presence of a linear isomorphiccopy of C [ , ] Bourgain used a tree ordinal index where the trees are defined by a fixed basis of C [ , ] . By completeness, we only need to find a dense subset of c , in order to detect a Lipschitzcopy of c while to detect a coarse copy of c we only need to find a 1-net of c . Note that X [ N ] < ω , Q is a dense subset of c and that X [ N ] < ω , Z is a 1-net in c . It will be very useful to understand X G , E as the collection of all f : N → E for which there is G ∈ G so that supp ( f ) ⊂ G . To handle thenonlinearity of our universality problem we will introduce combinatorial objects called vines whichwill be a substitute for trees. The elements of a vine V will also be collections of elements of X ,but they will be indexed over collections of finitely supported functions f : N → E , where E is afixed countable subset of R , with 0 ∈ E . Such elements will be called bunches. For a collection V of bunches to be called a vine it must also be closed under a certain restriction operation. Formally,for a (finite or infinite) countable subset E of R , with 0 ∈ E , and finite subset G of N we call the set [ E , G ] = { f : N → E with supp ( f ) ⊂ G } OARSE AND LIPSCHITZ UNIVERSALITY 7 an E -bunch . Note that if G = /0, then [ E , G ] = { } , where 0 : N → E is the constant zero map. Weput c E = [ G ∈ [ N ] < ω [ G , E ] = (cid:8) ( ξ j ) ⊂ E : { j ∈ N : ξ j = } is finite (cid:9) which is dense in c if E is dense in R . Given a set X and a countable subset E of R , every elementof the form χ = ( x f ) f ∈ [ E , G ] in X [ E , G ] will be called an E -bunch over X . We define a partial order onthe set of E -bunches over X as follows. If χ = ( x f ) f ∈ [ E , F ] , ψ = ( y f ) f ∈ [ E , G ] we shall write χ (cid:22) ψ if F is an initial segment of G and for every f ∈ [ E , F ] we have y f = x f . This makes sense because [ E , F ] ⊂ [ E , G ] . If G = /0 then X [ E , G ] will be in an obvious way identified with X , and we note that for G ∈ [ N ] < ω and ( x f ) f ∈ [ E , G ] , x ≡ ( x f ) f ∈ [ E , /0 ] (cid:22) ( x f ) f ∈ [ E , G ] , or more generally ( x f ) f ∈ [ E , F ] (cid:22) ( x f ) f ∈ [ E , G ] for all initial segments F of G .A set V of E -bunches over X is called an E -vine over X if for all χ ∈ V the set [ ψ (cid:22) χ ] is asubset of V . Note that [ ψ (cid:22) χ ] is finite and totally ordered and hence ( V , (cid:22) ) is a tree in the abstractclassical sense. We shall say that the E -vine V is well founded if the tree ( V , (cid:22) ) is well founded, i.e., it contains no infinite totally ordered subsets. We define the derivatives of vines:For a vine V we put V ( ) = V \ { χ ∈ V : χ is (cid:22) -maxinal } , and recursively for any ordinal V ( α + ) = ( V ( α ) ) ( ) , and for a limit ordinal α , V ( α ) = \ β < α V ( β ) . Then, the ordinal index of V is o ( V ) = min { α : V ( α ) = /0 } . This is well defined if V is wellfounded. As for trees, under appropriate assumptions, being well founded is equivalent to havingcountable ordinal index. This will be proved in Proposition 3.2.For n ∈ N ∪ { } we define V ( n ) = (cid:8) χ = ( x f ) f ∈ [ E , G ] : | G | = n (cid:9) = V ∩ [ G ∈ [ N ] n X [ E , G ] . If X is a topological space then for each G ∈ [ N ] n the set X [ E , G ] can be equipped with the producttopology. Then the disjoint union ∪ G ∈ [ N ] n X [ E , G ] can be endowed with the induced topology. Inparticular, V ( n ) is a topological space. We shall call V a closed E -vine if V ( n ) is a closed subset of ∪ G ∈ [ N ] n X [ E , G ] for all n ∈ N . This is equivalent to saying that for all G ∈ [ N ] < ω the set V ∩ X [ E , G ] isclosed. Note that V being closed does not imply that the set ∪ χ =( x f ) f ∈ [ E , G ] ∈ V { x f : f ∈ [ E , G ] } is aclosed subset of X .We can define π n : V ( n + ) → V ( n ) as follows. If G ∈ [ N ] n + set G ′ = G \ { max ( G ) } . Given χ = ( x f ) f ∈ [ E , G ] in V ( n + ) we define π n ( χ ) = ( x f ) f ∈ [ E , G ′ ] , which is in V ( n ) . Note that a collection V of E -bunches over X is an E -vine if and only if for all n ∈ N we have that π n [ V ( n + ) ] ⊂ V ( n ) . Also,if X is a topological space then π n is a continuous function.The following is an analogue for vines of [Bou80, Lemma 2] and the proof is nearly identical. F. BAUDIER, G. LANCIEN, P. MOTAKIS, AND TH. SCHLUMPRECHT
Lemma 3.1.
Let E be a countable subset of R and V be a closed E -vine over a complete metricspace ( X , d ) . Assume that for all n ∈ N we have V ( n ) = π n [ V ( n + ) ] . Then either V = /0 or V is notwell founded.Proof. We fix an enumeration { ε i : i ∈ N } of E and for all n ∈ N we set E n = { ε , . . . , ε n } . As-suming V = ∅ , we can find an x ≡ ( x f ) f ∈ [ E , /0 ] ∈ V ∩ X = V ( ) . Since V ( ) = π ( V ( ) ) we find χ = ( x ( ) f ) f ∈ [ E , { k } ] ∈ V ( ) so that k π ( χ ) − x k <
1. By assumption there exists k > k and χ = ( x ( ) f ) f ∈ [ E , { k , k } ] ∈ V ( ) so that for f ∈ [ E , { k } ] we have d ( x ( ) f , x ( ) f ) ≤ /
2. Proceed induc-tively to find an increasing sequence of integers ( k m ) ∞ m = and a sequence ( χ m ) ∞ m = so that χ m =( x ( m ) f ) f ∈ [ E , { k , k ,..., k m } ] ∈ V ( m ) and for all m ∈ N and f ∈ [ E m , { k , . . . , k m } ] we have d ( x ( m ) f , x ( m + ) f ) ≤ / m . We conclude that for any m ∈ N and f ∈ [ E m , { k , . . . , k m } ] the sequence ( x ( m ) f ) m ≥ m isCauchy and we denote its limit by y f . Because V is an E -vine it is closed under taking projections π n and because V is assumed to be closed we deduce that ψ m = ( y f ) f ∈ [ E , { k ,..., k m } ] is in V for all m ∈ N . Because ( ψ m ) m is an infinite chain the E -vine V must be ill founded. (cid:3) The following is the analogue of Proposition 2.2 for vines.
Proposition 3.2.
Let E be a countable subset of R and V be a closed E -vine on a Polish space. If V is well founded then o ( V ) < ω .Proof. We will show that there is η < ω so that V ( η ) = /0. It is easily observed that for any n ∈ N and ordinal α we have(4) ( V ( α + ) ) ( n ) = π n [( V ( α ) ) ( n + ) ] , i.e., a χ of length n is in V ( α + ) if an only if it is the direct predecessor of a ψ of length n + V ( α ) .For n ∈ N , consider the decreasing hierarchy of closed sets ( V ( α ) ) ( n ) , α < ω of ∪ G ∈ [ N ] n X [ E , G ] .Because X is Polish, so is ∪ G ∈ [ N ] n X [ E , G ] and therefore there must exist an α n < ω so that forall β > α n we have ( V ( α n ) ) ( n ) = ( V ( β ) ) ( n ) . This is because in a Polish space there can be nostrictly increasing transfinite hierarchy of open sets of length ω . Take η = sup n α n and define W = ∪ ∞ n = ( V ( η ) ) ( n ) . We observe that W is an E -vine over X . We show that W satisfies theassumption of Lemma 3.1. Indeed, for n ∈ N we have W ( n ) = ( V ( η ) ) ( n ) = ( V ( η + ) ) ( n ) (by the choice of η ) = π n [( V ( η ) ) ( n + ) ] (by (4)) = π n h ( V ( η ) ) ( n + ) i (by continuity of π n ) = π n [ W ( n + ) ] . This means that either W = /0 or W is ill founded. Because V is closed W ⊂ V and because V iswell founded, so is W and hence W = /0. It follows that V ( η ) = ∪ n ∈ N ( V ( η ) ) ( n ) ⊂ W = /0. Therefore, o ( V ) ≤ η . (cid:3) We can now introduce an ordinal index that will capture the presence of a bi-Lipschitz copy ofc in a metric space. For any C >
0, any metric space ( M , d ) , and any countable subset E of R , it iseasy to verify that the set (think of [ E , G ] being a subset of c ) V ( M , E , C ) = (cid:26) ( x f ) f ∈ [ E , G ] : G ∈ [ N ] < ω , x f ∈ M for f ∈ [ E , G ] , and ∀ f , g ∈ [ E , G ] C k f − g k ∞ ≤ d ( x f , x g ) ≤ C k f − g k ∞ (cid:27) , OARSE AND LIPSCHITZ UNIVERSALITY 9 is a closed E -vine on M . We define the Lipschitz c -index of M asI Lipc ( M ) = sup { o ( V ( M , Q , C ) : C > } . Proposition 3.3.
Let M be a Polish space. Then, c bi-Lipschitzly embeds into M if and only if I Lipc ( M ) ≥ ω . Proof.
The necessary implication is easy. Indeed, if ψ is a Lipschitz embedding from c into M , define for G ∈ [ N ] < ω and f ∈ [ Q , G ] , x f = ψ ( ∑ i ∈ G f ( i ) e i ) . Then, for some C ≥
1, the set { ( x f ) f ∈ [ Q , G ] : G ∈ [ N ] < ω } is included in V ( M , Q , C ) which is therefore ill founded.Assume now that I Lipc ( M ) = ω , then for every countable ordinal α there exist C α > o ( V ( M , Q , C α ) ≥ α . Using a simple pigeonhole argument we can find C ≥ U of [ , ω ) , such that for all α ∈ U we have C α ≤ C . Since obvi-ously o ( V ( M , Q , C )) ≥ o ( V ( M , Q , C α )) ≥ α for every α ∈ U , it follows from Proposition 3.2that V ( M , Q , C ) is not well founded, i.e., there exists a strictly increasing sequence of integers ( k m ) m and for m ∈ N ∪ { } an M -bunch χ m = (cid:0) x ( m ) f : f ∈ [ { k , k , . . . , k m } , E ] (cid:1) ∈ V ( M , Q , C ) so that χ (cid:22) χ (cid:22) χ . . . . But this means that for every finitely supported f : { k , k , k , . . . , } → Q there isan x f ∈ M , so that χ m = (cid:0) x f : f ∈ [ { k , k , . . . , k m } , E ] (cid:1) , for m ∈ N . We define ψ : c Q → M by ψ (( q j ) j ) = x f where f : { k , k , . . . } → Q , is defined by f ( k j ) = q j . It follows that ψ is a bi-Lipschitz embedding from c Q (with the c -norm) into M . Since c Q is densein c and M is complete, ψ can be extended to a bi-Lipschitz embedding from c into M . (cid:3) To complete the proof of Theorem A it remains to show that if a complete separable metric space M is Lipschitz-universal for the collection of rational valued Schreier metrics then I Lipc ( M ) ≥ ω . Proof of Theorem A . Assume that for every ordinal α , ( M , d ) admits bi-Lipschitz embeddings of ( S α ( Q ) , d ∞ ) . Thus, after an eventual extraction argument, there exist a constant C >
0, an uncount-able A ⊂ [ , ω ) , and maps F α : ( S α ( Q ) , d ∞ ) → ( M , d ) , α ∈ A , such that for all f , g ∈ S α ( Q ) and α ∈ A (5) 1 C k f − g k ∞ ≤ d ( F α ( f ) , F α ( g )) ≤ C k f − g k ∞ . It follows that V ( M , Q , C ) has ordinal index at least o ( S α ) = ω α +
1, for all α ∈ A . To see this,define for every f in S α ( Q ) the vector x f = F α ( f ) and let W = { ( x f ) f ∈ [ Q , G ] : G ∈ S α } , which isthanks to (5) a sub-vine of V ( M , Q , C ) that has the same tree index as S α . (cid:3) Coarse universality via a coarse c -index in MA+ ¬ CH.
The technique from Section 3.1 donot seem to be robust enough to prove the statement of Theorem B without any further set theoreticassumptions. The main roadblock is that the simple extraction argument that provides equi-bi-Lipschitz embeddings from an uncountable collection of bi-Lipschitz embeddings does not hold inthe coarse setting. Under some additional set-theoretic axioms, MA+ ¬ CH, we can prove TheoremB. The advantage of assuming that Martin’s Axiom holds, but the Continuum Hypothesis fails, liesin the fact that the following diagonalization property of infinite subsets of N (cf. [Fre84, page 3ff]will be valid. Lemma 3.4. (MA+ ¬ CH) Let ( N α ) α < ω ⊂ [ N ] ω have the property that N β \ N α is finite whenever α < β (in which case we say that N β is almost contained in N α and write N β ⊂ a N α ). Then thereexists N in [ N ] ω so that N ⊂ a N α , for all α < ω . This diagonalization property will now be used to prove an “equi-regularization” principle forexpansion and compression moduli. Let us detail the case of the compression modulus. We firstneed some preparation. Denote I the class of all non decreasing maps f : N → N ∪ { } satisfying f ( ) =
0, lim n → ∞ f ( n ) = ∞ and f ( n + ) ≤ f ( n ) + n ∈ N . It will be useful to note that themap j : I → [ N ] ω , defined by j ( f ) = { n ∈ N , f ( n + ) > f ( n ) } is a bijection, and that its inverseis given by j − ( A )( n ) = ∑ i < n A ( i ) , for A ∈ [ N ] ω and n ∈ N . We shall also use the following easyfact. If j ( f ) = { m < m < · · · } and j ( g ) = { n < n < · · · } with n i ≤ m i for all i ∈ N , then f ≤ g .In particular, if j ( f ) ⊂ j ( g ) , then f ≤ g . We start with an easy lemma. Lemma 3.5.
Let ( g α ) α < ω ⊂ I . Then there exists ( f α ) α < ω ⊂ I such that (1) For all α < ω and all n ∈ N , f α ( n ) ≤ g α ( n ) . (2) For all α < β < ω , j ( f β ) ⊂ a j ( f α ) .Proof. We shall build ( f α ) α < ω by transfinite induction. So, set f = g and assume that β < ω issuch that we have found ( f α ) α < β satisfying (1) and (2). Since { α < β } is countable, a classicaldiagonal argument yields the existence of M ∈ [ N ] ω such that M ⊂ a j ( f α ) for all α < β . Let j ( g β ) = { n < n < · · · } . Then pick m < m < · · · ∈ M so that n i ≤ m i for all i ∈ N and set f β = j − ( { m , m , . . . } ) . We have that f β ≤ g β and j ( f β ) ⊂ M ⊂ a j ( f α ) for all α < β . Thisconcludes our induction. (cid:3) Armed with Lemma 3.4 we can now prove our “equi-regularization” principle below for com-pression moduli.
Proposition 3.6. (MA+ ¬ CH) Let ( ρ α ) α < ω be a family of non decreasing maps from [ , ∞ ) to [ , ∞ ) and so that lim t → ∞ ρ α ( t ) = ∞ for all α < ω . Then there exist an uncountable subset C of ω and ρ : [ , ∞ ) → [ , ∞ ) such that ρ ≤ ρ α for all α ∈ C and lim t → ∞ ρ ( t ) = ∞ .Proof. First, note that for all α < ω , we can find g α ∈ I such that g α ( n ) ≤ ρ α ( n ) , for all n ∈ N .Then consider the family ( f α ) α < ω associated to ( g α ) α < ω through Lemma 3.5. Next, we applyLemma 3.4 to get M ∈ [ N ] ω such that M ⊂ a j ( f α ) for all α < ω . For n ∈ N , denote C n = (cid:8) α < ω , M ∩ { n , n + , . . . } ⊂ j ( f α ) (cid:9) . Clearly, there exists n ∈ N such that C n is uncountable. We set C = C n and define f = j − ( M ∩{ n , n + , . . . } ) ∈ I . Then, for all α ∈ C , we have j ( f ) ⊂ j ( f α ) and therefore f ≤ f α . Finally, ρ defined by ρ = [ , ) and ρ = f ( n ) on [ n , n + ) , for n ∈ N , is the desired map. (cid:3) Similarly, for expansion moduli, we have.
Proposition 3.7. (MA+ ¬ CH) Let ( ω α ) α < ω be a family of non decreasing maps from [ , ∞ ) to [ , ∞ ) and so that ω α ( ) = . Then there exist an uncountable subset C of ω and ω : [ , ∞ ) → [ , ∞ ) suchthat ω ≥ ω α for all α ∈ C.Proof.
The argument is very similar. Let us just describe the few adjustments. We now considerthe class J of all functions f : N ∪ { } → N ∪ { } such that f ( ) = f ( n + ) ≥ f ( n ) + n ≥
0. The map k : J → [ N ] ω defined by k ( f ) = k ( N ) is a bijection. Then, for every α < ω ,there exists g α ∈ J such that g α ( n ) ≥ ω α ( n ) for all n ∈ N . Playing the same game as before, butwith the sets k ( g α ) instead of j ( g α ) , we obtain (under MA+ ¬ CH) the existence of an uncountablesubset C of ω and of g ∈ J such that g ≥ g α for all α ∈ C . The proof is then concluded by setting ω ( ) = ω = g ( n ) on ( n − , n ] , for n ∈ N . (cid:3) From Propositions 3.6 and 3.7, we deduce immediately.
OARSE AND LIPSCHITZ UNIVERSALITY 11
Proposition 3.8. (MA+ ¬ CH) If ( X α , d α ) α < ω is a collection of metric spaces such that for all α < ω , X α embeds coarsely into a metric space ( M , d ) , then there exists an uncountable subset Cof ω such that ( X α , d α ) α ∈ C embeds equi-coarsely into ( M , d ) .Proof of Theorem B . The argument goes along essentially the same lines as the proof of Theorem A,modulo the fact that we have to work with vines defined in terms of the compression and expansionmoduli. Let us outline the main steps and the place where (MA+ ¬ CH) is used.Let ρ , ω be two elements of the class F of all non decreasing functions from [ , ∞ ) → [ , ∞ ) that are vanishing at 0 and tending to ∞ at ∞ . Let also ( M , d ) be a complete separable metric space.Then, we define V ( M , Z , ρ , ω ) = n ( x f ) f ∈ [ Z , G ] : G ∈ [ N ] < ω , x f ∈ M for f ∈ [ E , G ] , and for f , g ∈ [ E , G ] we have ρ ( k f − g k ∞ ) ≤ d ( x f , x g ) ≤ ω ( k f − g k ∞ ) o , and the coarse c -index of M asI coarsec ( M ) = sup (cid:8) o ( V ( M , Z , ρ , ω ) : ρ , ω ∈ F (cid:9) . The next step is to prove the analogue of Proposition 3.3: c coarsely embeds into M if andonly if I coarsec ( M ) ≥ ω . For the non trivial implication, the pigeonhole argument yielding a uniformconstant C is replaced by Propositions 3.6 and 3.7 to prove the existence of ρ , ω ∈ F such that V ( M , Z , ρ , ω ) is not well founded (this is where (MA+ ¬ CH) is used). Then it implies the existenceof a coarse embedding of the integer grid of c (and therefore of c ) into M .Finally, assume that a separable metric space ( M , d ) , that we may assume to be complete, containsa coarse copy of all spaces ( S α ( Z ) , d ∞ ) , for α < ω . As in the proof of Theorem A, this implies thatI coarsec ( M ) ≥ ω , which finishes the proof. (cid:3) Remark 3.9.
We recall that a metric space
X coarse-Lipschitz embeds into a metric space Y if X ( ρ , ω ) -embeds into Y where, for all t ≥ ρ ( t ) = At − B and ω ( t ) = Ct + D for some constants A , B , C , D >
0. It follows clearly from the tools and arguments developed in the last two subsec-tions that we have, without assuming any further set theoretical axioms, the following statement: aseparable metric space containing coarse-Lipschitz all the spaces ( S α ( Z ) , d ∞ ) , for α < ω , must acontain a coarse-Lipschitz copy of c .It is natural to wonder if Theorem B holds without MA+ ¬ CH.
Problem 3.10.
If a separable metric space contains coarse copies of ( S α ( Z ) , d ∞ ) for every count-able ordinal α , is it coarsely universal for the class of all separable metric spaces? We discuss some positive partial results in Section 4.3.3.
Coarse universality via strong boundedness.
While we do not know how to prove TheoremB without further set axioms, we can prove Theorem D. Recall that the canonical embedding of ( S α ( Z ) , d ∞ ) in T ∗ α is a 4-Lipschitz isomorphism onto its image, and thus the stronger assumptionthat the metric space contains every separable reflexive asymptotic-c space, rather than merelythe collection of metric spaces ( S α ( Z )) α < ω , allows us to take advantage of the deep theory ofstrongly bounded classes of Banach spaces introduced by Argyros and Dodos [AD07]. A class C of separable Banach spaces is said to be strongly bounded if for every analytic subset A of C , thereexists Y ∈ C that contains isomorphic copies of every X ∈ A . Recall also that an infinite-dimensionalBanach space X is said to be minimal if X isomorphically embeds into every infinite-dimensionalsubspace of itself (e.g. the classical sequence space c is minimal). We will need the following deepresult of Dodos [Dod09, Theorem 7]. Theorem 3.11.
For any infinite-dimensional minimal Banach space Z not containing ℓ , the class NC Z : = { Y ∈ SB : Z does not linearly embed into Y } is strongly bounded.Proof of Theorem D . Denote R the set of all reflexive elements of SB and As c the set of all elementsof SB that are asymptotic- c . Let now M be a separable metric space such that every space in R ∩ As c coarsely embeds into M . If we denote CE M = { Y ∈ SB : Y coarsely embeds into M } , wehave that R ∩ As c ⊂ CE M . It is easily checked that CE M is analytic (see the proof of Theorem 1.7- section 7.1 in [dMB19]). Recall that we denoted NC c the set of all Y ∈ SB such that c does notlinearly embed into Y . If we assume, aiming for a contradiction that CE M ⊂ NC c , since CE M is ananalytic subset of NC c , which is strongly bounded by Theorem 3.11, there would exist X ∈ NC c such that any element of CE M , and therefore any element of R ∩ As c , linearly embeds into X . Thisis actually impossible since Bourgain’s c -index of the separable, reflexive and asymptotic- c space T ∗ α tends to ω as α tends to ω (see [OSZ07]). Therefore Bourgain’s c -index of X would beuncountable and X would contain an isomorphic copy of c ; a contradiction with X ∈ NC c . Sowe can now deduce the existence of Y ∈ CE M such that c linearly embeds into Y , and hence bycomposition c coarsely embeds into M . Since by a theorem of Aharoni [Aha74], every separablemetric space bi-Lipschitzly embeds into c , every separable metric space coarsely embeds into M .This concludes our proof. (cid:3) Remarks 3.12.
The same technique was used by B. de Mendonc¸a Braga in [dMB19] to prove thata Banach space which is coarsely universal for all reflexive separable Banach spaces is coarselyuniversal for all separable metric spaces.The reader will easily adapt the above proof to show that a Banach space that is Lipschitz uni-versal for R ∩ As c is Lipschitz universal for all separable metric spaces. But this was also a conse-quence of Theorem A.4. C OARSE UNIVERSALITY AND BARYCENTRIC GLUING
The motivation for this section is to provide a somewhat weaker statement than Theorem B,that does not require MA+ ¬ CH. We will show that containing coarse copies of the Schreier met-ric spaces ( S α ( Z ) , d ∞ ) for every α < ω , is a sufficient condition to equi-coarsely contain everyseparable bounded metric spaces. The reason we have the boundedness restriction is because with-out MA+ ¬ CH we only have the following equi-regularization principle (which is weaker than theequi-regularization principle obtained under MA+ ¬ CH).
Lemma 4.1.
Let C be an uncountable subset of ω . Assume that for each α ∈ C , we have increas-ing functions ρ α , ω α : [ , ∞ ) → [ , ∞ ) so that ρ α ( t ) ≤ ω α ( t ) for all t ∈ [ , ∞ ) and lim t → ∞ ρ α ( t ) = ∞ . Then there exist increasing functions ρ , ω : [ , ∞ ) → [ , ∞ ) and a decreasing nested sequence ( C k ) k ∈ N of uncountable subsets of ω so that (i) ρ ( t ) ≤ ω ( t ) for all t ∈ [ , ∞ ) , and lim t → ∞ ρ ( t ) = ∞ , (ii) for all k ∈ N , α ∈ C k , and ≤ t ≤ k we have ρ ( t ) ≤ ρ α ( t ) and ω α ( t ) ≤ ω ( t ) .Proof. For all k ∈ N and α ∈ C define s ( α , k ) = min { t ∈ N : ρ α ( t ) ≥ k } , t ( α , k ) = max { t ∈ N : ω α ( t ) ≤ k } and also define M ( α , k ) = min { n ∈ N : s ( α , k ) < s ( α , n ) } , N ( α , k ) = min { n ∈ N : t ( α , k ) < t ( α , n ) } . OARSE AND LIPSCHITZ UNIVERSALITY 13
Because, for each fixed k ∈ N , the sets { s ( α , k ) : α ∈ C } , { t ( α , k ) : α ∈ C } , { M ( α , k ) : α ∈ C } , { N ( α , k ) : α ∈ C } are all countable we may find uncountable sets C ⊃ C ⊃ C ⊃ · · · so thatfor each k ∈ N and α , β ∈ C k we have s ( α , k ) = s ( β , k ) = s k , t ( α , k ) = t ( β , k ) = t k , M ( α , k ) = M ( β , k ) = M k , and N ( α , k ) = N ( β k ) = N k . Clearly, we have s k ≤ s k + and t k ≤ t k + , for all k ∈ N .We also observe that lim k s k = lim k t k = ∞ . Indeed, it is easy to see that s k < s M k and t k < t N k . Pick k < k < · · · so that ( s k j ) j and ( t k j ) j are both strictly increasing.We now define ρ , ˜ ω : [ , ∞ ) → [ , ∞ ) as follows. ρ ( t ) = (cid:26) ≤ t < s k k j if s k j ≤ t < s k j + , j ∈ N , ˜ ω ( t ) = (cid:26) k if 0 ≤ t ≤ t k k j if t k j − < t ≤ t k j , j ≥ ω ( t ) = ρ ( t ) ∨ ˜ ω ( t ) . The conclusion follows straightforwardly after observing that s k j , t k j ≥ j for all j ∈ N . (cid:3) Using the concept of vines introduced in Section 3.2 in the coarse context we now deduce thefollowing.
Theorem 4.2.
Let ( M , d ) be a separable metric space and assume that for every α < ω the metricspace ( S α ( Z ) , d ∞ ) embeds coarsely into ( M , d ) . Then the class of all separable bounded metricspaces embeds equi-coarsely into ( M , d ) .More precisely, there exist m ∈ M and equi-coarse embeddings F n : B n = { x ∈ c : k x k ∞ ≤ n } → ( M , d ) so that for all n ∈ N we have F n ( ) = m .Proof. We will first find m ∈ M and ρ , ω : [ , + ∞ ) → [ , + ∞ ) tending to ∞ at ∞ , so that for any α < ω and n ∈ N there exists F α , n : B α , n = { x ∈ X S α , Z : k x k ∞ ≤ n } → M with F α , n ( ) = m andfor all x , y ∈ B α , n we have ρ ( k x − y k ∞ ) ≤ d ( F α , n ( x ) , F α , n ( y )) ≤ ω ( k x − y k ∞ ) . Let ˜ M be a countabledense subset M , since ( S α ( Z ) , d ∞ ) is a uniformly discrete metric space, it follows from a straight-forward perturbation argument that every ( S α ( Z ) , d ∞ ) embeds coarsely into ( ˜ M , d ) via a map f α with compression and expansion moduli ρ α , ω α . By passing to an uncountable set C ⊂ ω we canassume that there exists m ∈ ˜ M so that for all α ∈ C we have f α ( ) = m .Fix n ∈ N . Take the functions ρ , ω and the sets C ⊃ C ⊃ C ⊃ · · · given by Lemma 4.1. By theconclusion of that lemma, it follows that for every β ∈ C n the function f β : ( S β ( Z ) , d ∞ ) → ˜ M is a ( ρ , ω ) -coarse embedding on every subset of ( S β ( Z ) , d ∞ ) with diameter at most 2 n , and also because β ∈ C we have f β ( ) = m .Fix now α < ω . Because C n is uncountable we may pick β ∈ C n so that β > α . Then it is wellknown that there exists an infinite subset L = { ℓ i : ∈ N } of N so that for all G = { a , . . . , a d } ∈ S α the set { ℓ a , . . . , ℓ a d } is in S β . It follows that the map s L : ( S α ( Z ) , d ∞ ) → ( S β ( Z ) , d ∞ ) given by ∑ i ∈ G m i e i ∑ i ∈ G m i e ℓ i is an isometric embedding and maps 0 to 0. Therefore, F α , n = B α , n → M ,the restriction of f β ◦ s L to B α , n , has the desired properties.Next, we will use Proposition 3.2 to show that for all N ∈ N there exists a function F N : B N → M that is a ( ρ , ω ) -coarse embedding with the additional property that F N ( ) = m . More precisely, wewill define this F N on the subset B ( N , Z ) of B N consisting of all integer valued sequences in the set B N . Because this is a 1-net of B N and N is arbitrary we may then deduce the desired conclusion. Wedenote I N = [ − N , N ] ∩ Z and consider the closed I N -vine define by V : = n ( x f ) f ∈ [ I N , G ] : G ∈ [ N ] < ω , x f ∈ M for f ∈ [ I N , G ] , x = m , and for f , g ∈ [ I N , G ] we have ρ ( k f − g k ∞ ) ≤ d ( x f , x g ) ≤ ω ( k f − g k ∞ ) o Because for each α < ω the space ( S α ( Z ) , d ∞ ) ( ρ , ω ) -embeds into ( M , d ) via F α , N , which maps 0 to m , it follows that o ( V ) ≥ ω . Because we are only considering coarse embeddings we may assume that ( M , d ) is complete. By Proposition 3.2 the I N -vine V must be ill founded, i.e., there exists astrictly increasing sequence of integers ( k m ) m and for every finitely supported f : { k i : i ∈ N } → I N there exists x f ∈ M so that χ m = ( x f ) f ∈ [ I N , { k ,..., k m } ] is in V for all m ∈ N . By the definition of V itfollows that the map from B ( N , Z ) to ( M , d ) given by ∑ ∞ i = m i e i x f , where f : { k i : i ∈ N } → I n isthe function with f ( k i ) = m i , is a ( ρ , ω ) -embedding. (cid:3) The last result of this section is a variation of the barycentric gluing technique, which has aninterest on its own. With this gluing technique we can show that if we can equi-coarsely embedthe bounded subsets of c (or equivalently every separable bounded metric spaces) into a metricspace M then M is coarsely universal. In particular, an immediate consequence of Theorem 4.2and Theorem 4.4, below, is Corollary 4.3.
Let ( M , d ) be a separable metric space, If for every α < ω the metric space ( S α ( Z ) , d ∞ ) embeds coarsely into ( M , d ) , then c coarsely embeds into M . The original barycentric gluing technique (see [Bau07]), creates a coherent embedding of a metricspace into a Banach space, by pasting embeddings of balls of growing radii together. Here, theprocess is reversed in the sense that we will paste balls of Banach spaces into metric spaces, but ourproof has the caveat that it requires the gluing into M , rather than in M . Here is our general result. Theorem 4.4.
Let ( X , k · k ) be a Banach space and ( M , d ) be a metric space. Assume that thereexist increasing functions ρ , ω : [ , ∞ ) → [ , ∞ ) that are tending to ∞ at ∞ , m ∈ M, and for alln ∈ N , maps h n : nB X : → M, so that h n ( ) = m , and for all x , y ∈ nB X (6) ρ ( k x − y k ) ≤ d (cid:0) h n ( x ) , h n ( y ) (cid:1) ≤ ω ( k x − y k ) . We equip M with the ℓ ∞ -metric associated with d, that we still denote d. Then there is a ( ˜ ρ , ˜ ω ) -embedding of X into M where ˜ ρ ( t ) = ρ (cid:16) t (cid:17) and ˜ ω ( t ) = ω ( t ) , for all < t < ∞ .Proof. Choose inductively r = < r < r < . . . in N , so that(7) ρ ( r n + ) > ω ( r n ) and r n + ≥ r n For n ∈ N we define the following map α n : [ , ∞ ) → [ , ] (set r − = r − = r − = r − = r = α n ( t ) = t < r n − or t > r n , t − r n − r n − − r n − if r n − ≤ t < r n − ,1 if r n − ≤ t ≤ r n − , r n − tr n − r n − if r n − < t ≤ r n .The support of α n is ( r n − , r n ) , and { t : α n ( t ) = } = [ r n − , r n − ] .For i ∈ { , , , } we define F ( i ) : X → M as follows: For x ∈ X we choose l ∈ Z + , so that r ( l − )+ i ≤ k x k < r l + i and put F ( i ) ( x ) = h r l + i (cid:0) α r l + i ( k x k ) x (cid:1) . Then we define the map F : X → M , x (cid:0) F ( ) ( x ) , F ( ) ( x ) , F ( ) ( x ) , F ( ) ( x ) (cid:1) . We will show that the map F from X into M , satisfies:(8) 12 ρ (cid:16) k x − y k (cid:17) ≤ d (cid:0) F ( x ) , F ( y ) (cid:1) ≤ ω ( k x − y k ) . OARSE AND LIPSCHITZ UNIVERSALITY 15
Firstly, we estimate the compression function. Let x , y ∈ X , and assume without loss of generalitythat k x k ≤ k y k . Choose l ∈ N and i ∈ { , , , } , so that r l + i − ≤ k y k ≤ r l + i − . It is sufficientto show that d ( F ( i ) ( y ) , F ( i ) ( x )) ≥ ˜ ρ ( k x − y k ) . We first note that α l + i ( y ) = F ( i ) ( y ) = h r l + i ( y ) . We consider two cases.Case 1. r l + i − ≤ k x k , thus α l + i ( x ) = F ( i ) ( x ) = h r l + i ( x ) . It follows that d (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) = d (cid:0) ( h r l + i ( x ) , h r l + i ( y ) (cid:1) ≥ ρ ( k x − y k ) . Case 2. k x k < r l + i − . Thus, for some m ≤ ld (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) ≥ d (cid:0) F ( i ) ( y ) , m (cid:1) − d (cid:0) F ( i ) ( x ) , m (cid:1) = d (cid:0) h r l + i ( y ) , h r l + i ( ) (cid:1) − d (cid:0) h r m + i ( α m + i ( k x k ) x ) , h r m + i ( ) (cid:1) ≥ ρ ( k y k ) − ω ( k x k ) ≥ ρ ( k y k ) + ρ ( r l + i − ) − ω ( r l + i − ) ≥ ρ ( k y k ) ≥ ρ (cid:16) k x − y k (cid:17) . Secondly, we estimate the expansion function. We fix i ∈ { , , , } and we consider three cases.Case 1. For some n ∈ N we have r n − ≤ k x k ≤ k y k ≤ r n .If n = l + i − n = l + i −
2, then α l + i ( k y k ) = α l + i ( k x k ) =
1, and therefore d (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) = d (cid:0) h r l + i ( x ) , h r l + i ( y ) (cid:1) ≤ ω ( k x − y k ) . If n = l + i −
3, or n = l + i , then (cid:12)(cid:12) α l + i ( k x k ) − α l + i ( k y k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k x k − k y k r n − r n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r n k x − y k , and therefore (cid:13)(cid:13) α l + i ( k x k ) x − α l + i ( k y k ) y (cid:13)(cid:13) ≤ α l + i ( k x k ) k x − y k + k y k (cid:12)(cid:12) α l + i ( k x k ) − α l + i ( k y k ) (cid:12)(cid:12) ≤ k x − y k , which implies that d (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) = d (cid:0) h r l + i ( α l + i ( k x k ) x ) , h r l + i ( α l + i ( k y k ) y ) (cid:1) ≤ ω ( k x − y k ) . From now we assume that there are m , n ∈ N , m < n , so that r m ≤ k x k ≤ r m + ≤ r n ≤ k y k ≤ r n + . For j = , , . . . , n − m , let z j be the element on the the segment [ x , y ] ( i.e., points of the form x + t ( y − x ) with 0 ≤ t ≤
1) so that k z j k = r m + j , and put z = x and z n − m + = y .Case 2. n − m ≤ d (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) ≤ n − m + ∑ i = d (cid:0) F ( i ) ( z i − ) , F ( i ) ( z i ) (cid:1) ≤ n − m + ∑ i = ω ( k z i − − z i k ) ≤ ω ( k x − y k ) . Case 3. n − m ≥
4. It follows then from Case 2 that d (cid:0) F ( i ) ( x ) , F ( i ) ( y ) (cid:1) ≤ d (cid:0) F ( i ) ( x ) , m (cid:1) + d (cid:0) F ( i ) ( y ) , m (cid:1) = d (cid:0) F ( i ) ( x ) , F ( i ) ( z j ) (cid:1) + d (cid:0) F ( i ) ( y ) , F ( i ) ( z j ) (cid:1) ≤ ω ( k x − z k ) + ω ( k y − z k ) ≤ ω ( k x − y k ) . (cid:3)
5. U
NIVERSALITY PROPERTIES OF INTERLACING GRAPHS
In [Kal07], Kalton showed that a Banach space X that is coarsely universal for the class ofall separable metric spaces, or equivalently that coarsely contains c , cannot have separable iteratedduals, i.e., X ( r ) is nonseparable from some r ≥
2. Kalton’s argument is based on the metric propertiesof the interlacing graphs. As we will see in the next section, these graphs introduced by Kalton havesome remarkable universality properties.5.1.
Almost isometric universality of the interlacing graphs.
We define a slightly larger class ofinterlacing graphs than the ones introduced by Kalton. The set of vertices is [ N ] < ω , the set of finitesubsets of N , and we declare that two vertices A = { a , . . . , a n } and B = { b , . . . , b m } in [ N ] < ω areadjacent if and only if a = b and one of the following interlacing relations holds(i) n = m + a i b i a i + for 1 i m ,(ii) m = n + b i a i b i + for 1 i n ,(iii) n = m , a i b i a i + for 1 i < n , and a n b n , or(iv) n = m , b i a i b i + for 1 i < n , and b n a n .We also connect the empty set with all singletons. We refer to this graph as the universal interlac-ing graph, and we denote ([ N ] < ω , d I ) the universal interlacing graph equipped with its canonicalgraph metric. Kalton’s interlacing graph are defined in the same way besides only vertices withthe same length were considered. More precisely, Kalton’s interlacing graph of diameter k is thespace ([ N ] k , d ( k ) I ) , where the graph metric only refers to the interlacing relations ( iii ) or ( iv ) in thiscase. For A , B ∈ [ N ] k , although it is obvious that d ( k ) I ( A , B ) = d I ( A , B ) =
1, it is notimmediately clear that on [ N ] k the metrics d ( k ) I and d I coincide. As we shall see later, this is indeedthe case.The universality properties of the interlacing graphs stem from the fact that the interlacing metricadmits an interpretation in terms of the summing norm on c . For A , B in [ N ] < ω define the summingdistance d sum ( A , B ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ i ∈ A s i − ∑ i ∈ B s i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sum where ( s i ) i denotes the summing basis of c , endowed with the usual bimonotone version of thesumming norm, i.e. (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ i a i s i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sum = sup ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ∑ i = k a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : k , m ∈ N , k ≤ m ) . (9)In (9) one only needs to consider intervals at whose boundaries are sign-changes of the a i ’s. Moreprecisely for a sequence ( a i ) in c let 0 = m < m < . . . m s be chosen in N so that for all i ≤ s thesigns of a j on j ∈ [ m i − + , m i ] are the same ( i.e., all non negative or all non positive) then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∑ i a i s i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sum = sup ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ∑ i = k m i ∑ m i − + a j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ≤ l ≤ s } ) . (10)Thus for A , B ⊂ [ N ] < ω we write A △ B in increasing order as A △ B = { x , x , . . . , x n } and note that d sum ( A , B ) = max {| ( A ∩ E ) − ( B ∩ E ) | : E is an interval of N } (11) = max (cid:8)(cid:12)(cid:12) ( A ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) (cid:12)(cid:12) : 1 ≤ i ≤ j ≤ n (cid:9) . The above forms of the metric d sum will be used more often. We first show that the interlac-ing metric and the summing distance coincides. For fixed k , the coincidence of d ( k ) I ( A , B ) with OARSE AND LIPSCHITZ UNIVERSALITY 17 max {| ( A ∩ E ) − ( B ∩ E ) | : E is an interval of N } was already shown in [LPP20], where it was af-terwards used in connection with the canonical norm of c instead of k · k sum .For n ∈ N , A = { a , a , . . . , a n } ∈ N < ω , with a < a < . . . a n , we call A ′ = { a ′ , a ′ , . . . a ′ n } , with a ′ < a ′ < . . . < a ′ n a shift to the left of A if A ′ = A and a ≤ a ′ ≤ a ≤ a ′ ≤ . . . ≤ a n ≤ a ′ n . We notethat in this case(12) d I ( A , A ′ ) = d I ( A , A ′ \ { a ′ n } ) = d sum ( A , A ′ ) = d sum ( A , A ′ \ { a ′ n } ) . For another set B ∈ [ N ] < ω we say that a left shift A ′ of A is a shift towards B if A ′ \ A ⊂ B \ A . Theorem 5.1.
For A , B ∈ [ N ] < ω we have that d sum ( A , B ) = d I ( A , B ) .Moreover if k = A = B then there is a path of length d I ( A , B ) from A to B in the interlacinggraph, which stays in [ N ] k . Thus the restriction of d I to [ N ] k is d ( k ) I .Proof. We prove our statement by induction for all m ∈ N ∪ { } , and all A , B ∈ [ N ] < ω with m = d sum ( A , B ) .If m = d sum ( A , B ) = A = B , our claim is trivial. If m = d sum ( A , B ) = d I ( A , B ) =
1. Write A = { a , . . . , a n } , B = { b , . . . , b m } and assume, without lossof generality, that min ( A △ B ) = a i ∈ A . Note that | n − m | = | A − B | ≤ d sum ( A , B ) = ( A △ B ) = a i ∈ A we have 1 ≤ A ∩ [ a i , max { A ∪ B } ] − B ∩ [ a i , max { A ∪ B } ] =( n − i + ) − ( m − i ) , i.e., m ≤ n ≤ m +
1. Next, observe that for 1 ≤ i ≤ min { m , n } we have a i ≤ b i .Otherwise, set j = min { ≤ i ≤ min { m , n } : a i > b i } and note that a i < b i and thus i < j . Ifwe set E = [ b i , b j ] then d sum ( A , B ) = ≥ B ∩ E − A ∩ E = ( j − i + ) − ( j − i − ) = ≤ i ≤ min { m , n − } we have b i ≤ a i + . If this is not the case, set s = min { ≤ i ≤ min { m , n − } : b i > a i + } and observe that if E = [ a , a s + ] then A ∩ E = s + B ∩ E = s −
1, which is absurd. Finally we distinguish the cases n = m and n = m +
1. If n = m then we have demonstrated that (iii) of the definition of adjacency holds. If n = m + m ≥ ∈ N , and all A , B ∈ [ N ] < ω with d sum ( A , B ) < m it follows that d sum ( A , B ) = d I ( A , B ) , and that, d I ( A , B ) = d ( k ) I ( A , B ) if k = A = B .Let A , B ∈ [ N ] < ω with d sum ( A , B ) = m . If A ⊂ B , or B ⊂ A , and we assume without loss ofgenerality that B ( A , we put A ′ = A \ { a } , where a ∈ A \ B . Then d sum ( A , A ′ ) = d I ( A , A ′ ) = d sum ( A ′ , B ) = m −
1, and we deduce our claim from the induction hypothesisAssuming that A B and B A we write A △ B in increasing order as A △ B = { x , x , . . . , x l } . Itfollows that m = d sum ( A , B ) = max i ≤ j (cid:12)(cid:12) ( A ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) (cid:12)(cid:12) . Without loss of generality we can assume that x ∈ A \ B . There is a t ∈ N and numbers 1 ≤ i < i < . . . < i t < l so that { i s : s = , . . . t } = (cid:8) i ∈ { , , . . . , l − } : x i ∈ A and x i + ∈ B (cid:9) . We will now define A ′ ∈ [ N ] ω for which d I ( A , A ′ ) = d sum ( A , A ′ ) = d sum ( A ′ , B ) ≤ m −
1, andconsider the following two cases:Case 1. For all 1 ≤ j ≤ l we have ( A ∩ [ x j , x l ]) − ( B ∩ [ x j , x l ]) < m . Then we put A ′ = ( A \ { x i s : s ≤ t } ) ∪ { x i s + : s ≤ t } , which is a left shift of A towards B . Case 2. There is a j ≤ l so that ( A ∩ [ x j , x l ]) − ( B ∩ [ x j , x l ]) = m . It follows that x l ∈ A and we put A ′ = (cid:16) ( A \ { x i s : s ≤ t } ) ∪ { x i s + : s ≤ t } (cid:17) \ { x l } . We observe that if A = B the second case cannot happen. Indeed, assume that there is a j ≤ l so that ( A ∩ [ x j , x l ]) − ( B ∩ [ x j , x l ]) = m , then j > ( B ∩ [ x , x j − ]) − ( A ∩ [ x , x j − ]) = ( B \ A ) − ( B ∩ ( { x } ∪ [ x j , x l ]) − ( ( A \ B ) − ( A ∩ ( { x } ∪ [ x j , x l ]))= ( A ∩ ( { x } ∪ [ x j , x l ])) − ( B ∩ ( { x } ∪ [ x j , x l ])) = m + A ′ = A if A = B .From (12) it follows that d I ( A , A ′ ) = d sum ( A , A ′ ) =
1. We need to show that d sum ( A ′ , B ) ≤ m − d sum ( A ′ , B ) = m − i ∈ { , . . . , l } and define for i ≤ j ≤ lf ( j ) = ( A ′ ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) . Observe that f ( i ) ≤ ≤ m −
1. We claim that for all i < j ≤ l (13) f ( j ) ≤ min (cid:0) ( A ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) , m − (cid:1) . Since A ′ \ B ⊂ A \ B we have f ( j ) ≤ ( A ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) , for i ≤ j .Assume that our claim is not true, and let k be the minimum of all j > i so that f ( j ) = + min (cid:0) ( A ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) , m − (cid:1) . Since f ( k ) ≤ f ( k − ) + ( A ′ ∩ [ x i , x k − ]) − ( B ∩ [ x i , x k − ]) = m − A ′ △ B ⊂ A △ B it follows that x k ∈ A and thus ( A ∩ [ x i , x k − ]) − ( B ∩ [ x i , x k − ]) = m − ( A ∩ [ x i , x k − ]) − ( B ∩ [ x i , x k − ]) = m and thus ( A ∩ [ x i , x k ]) − ( B ∩ [ x i , x k ]) = m + ( A ∩ [ x i , x k ]) − ( B ∩ [ x i , x k ]) = m .Either k < l then x k + ∈ B , and since x k ∈ A it follows from the definition of A ′ that x k A ′ andthus f ( k ) = f ( k − ) = m −
1, which contradicts our assumption. Or k = l and thus ( A ∩ [ x i , x l ]) − ( B ∩ [ x i , x l ]) = m , which implies that x k = x l A ′ , since the second case in the definition of A ′ occurs, it would again follow that f ( k ) = m −
1, which is also a contradiction.Next we let j = , . . . , l , and put for i = , , . . . jg ( i ) = ( B ∩ [ x i , x j ]) − ( A ′ ∩ [ x i , x j ]) , and claim that g ( i ) ≤ min ( ( B ∩ [ x i , x j ]) − ( A ∩ [ x i , x j ]) , m − ) for all i ∈ { , , . . . , l } .Again since B △ A ′ ⊂ B △ A it follows that g ( i ) ≤ ( B ∩ [ x i , x j ]) − ( A ∩ [ x i , x j ]) for all i ∈ { , , . . . , j } .Assume our claim is not true and let k be the maximal k < j so that g ( k ) = m . So it follows that ( B ∩ [ x k , x j ]) − ( A ∩ [ x k , x j ]) = m , and thus x k ∈ B , and x k − ∈ A (note that k = x ∈ A )But this means that x k ∈ A ′ and thus ( B ∩ [ x k , x j ]) − ( A ′ ∩ [ x k , x j ]) = m −
1, which is again acontradiction.We therefore showed that for all 1 ≤ i < j ≤ l , | ( A ′ ∩ [ x i , x j ]) − ( B ∩ [ x i , x j ]) | ≤ m −
1, whichfinishes our proof. (cid:3)
The following Corollary could of course be also proven directly very easily.
Corollary 5.2.
For all k , m ∈ N with k < m, ([ N ] k , d ( k ) I ) embeds isometrically into ([ N ] m , d ( m ) I ) . The following quantitative embedding result immediately implies Theorem E.
OARSE AND LIPSCHITZ UNIVERSALITY 19
Theorem 5.3.
Let ( X , d ) be a n-point metric space, and α : = α ( X ) = diam ( X ) sep ( X ) be its aspect ratio,where diam ( X ) = sup { d X ( x , y ) : x , y ∈ X } and sep ( X ) = inf { d X ( x , y ) : x , y ∈ X } . Then for every < ε < and every integer k > (cid:0) n + (cid:1)(cid:0) αε + diam ( X ) + (cid:1) , ( X , d ) embeds with distortion ( − ε ) − into ([ N ] k , d ( k ) I ) .In particular, for all ε > ( X , d ) embeds with distortion at most + ε into ([ N ] < ω , d I ) .Proof. The proof of the general situation can be reduced to the special case where ( X , d ) is a finitemetric space with even distances. Assuming that we have proven Claim 5.4.
Assume that for all x , y ∈ X , d ( x , y ) is an even integer and that k is an integer numbersuch that k > ( n + )( diam ( X ) + ) . Then, the space ( X , d ) embeds isometrically into ([ N ] k , d ( k ) I ) . Then we can finish the proof of the general case. Indeed, let ε > q suchthat ( sep ( X ) ε ) − ≤ q ≤ ( sep ( X ) ε ) − + x , y ∈ X define k x , y = max { k ∈ N ∪ { } : k / q d ( x , y ) } . Define a metric ˜ d on X with˜ d ( x , y ) = min ( ℓ ∑ i = k x i , x i − q : x i ∈ X for 0 i ℓ and x = x , y = y ℓ ) . One can check that ˜ d is indeed a metric and for all x , y ∈ X we have ( − ε ) d ( x , y ) ˜ d ( x , y ) d ( x , y ) , hence, it suffices to embed the space ( X , ˜ d ) with distortion 1 into ([ N ] k , d ( k ) I ) for appropriate k .Note that if we denote ˜ X = ( X , ˜ d ) , diam ( ˜ X ) diam ( X ) . By Claim 5.4, the space ( X , q ˜ d ) embedsisometrically into ([ N ] k , d ( k ) I ) , for k > ( n + )( q diam ( X ) + ) . Recall that q ( sep ( X ) ε ) − + ( n + )( q diam ( X ) + ) ( n + )( αε + diam ( X ) + ) . (cid:3) Proof of Claim . We will find an embedding Φ from X into the linear span of ( s i ) i , endowedwith the norm (9), so that for each x ∈ X the vector Φ ( x ) is of the for form ∑ i ∈ A ( x ) s i , with A ( x ) ≤ ( / )( n + / )( diam ( X ) + ) .We enumerate X = { x , . . . , x n + } . We add one more point x to obtain the set ˜ X = { x , x , . . . , x n + } .We extend d on ˜ X by setting for x ∈ X , d ( x , x ) = d ( x , x ) = D , where D is the minimal even integerwith 2 D > diam ( X ) . As the diameter of X is an even integer, we deduce D diam ( X ) / +
1. It isstraightforward to verify that the triangle inequality is still satisfied on ˜ X . For notational reasons,we add a “ghost” point x n + with the property d ( x , x n + ) = x ∈ X . We first define a map Φ : X → h{ s i : i ∈ N }i , the linear span of the s i ’s, by Φ ( x ) = n + ∑ i = ( d ( x , x i ) − d ( x , x i + )) s i . If we denote by ( s ∗ i ) i the sequence of coordinate functionals associated to ( s i ) i we observe that forall i ∈ N and x ∈ X , the number s ∗ i ( Φ ( x )) is an integer. We start by showing that Φ ( x ) is anisometric embedding. Let 1 k m n + k Φ ( x ) − Φ ( y ) k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ∑ i = k ( d ( x , x i ) − d ( x , x i + − d ( y , x i ) + d ( y , x i + )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | d ( x , x m + ) − d ( y , x m + ) − d ( x , x k ) + d ( y , x k ) | | d ( x , x m + ) − d ( y , x m + ) | + | d ( x , x k ) − d ( y , x k ) | d ( x , y ) . For the inverse inequality, let x = x j , y = x j ′ and let us assume without loss of generality j < j ′ .Define k = j and m = j ′ −
1. Then, k Φ ( x ) − Φ ( y ) k > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ∑ i = k ( d ( x , x i ) − d ( x , x i + ) − d ( y , x i ) + d ( y , x i + )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | d ( x , x m + ) − d ( y , x m + ) − d ( x , x k ) + d ( y , x k ) | = | d ( x , y ) − d ( y , y ) − d ( x , x ) + d ( y , x ) | = d ( x , y ) . Define Φ ( x ) = Φ ( x ) + D ∑ n + i = s i . Then Φ is an isometric embedding of X into h{ s i : i ∈ N }i so that for all i ∈ N and x ∈ X the number s ∗ i ( Φ ( x )) is a non-negative integer. For k = , . . . , n + N k = max { s ∗ k ( Φ ( x )) : x ∈ X } and M k = k ∑ j = N j . Also define M =
0. We are ready to define the desired embedding. For x ∈ X set Φ ( x ) = n + ∑ k = ∑ (cid:26) M k − < i M k − + s ∗ k ( Φ ( x )) (cid:27) s i . We deduce that Φ ( x ) is of the form ∑ i ∈ A ( x ) s i with A ( x ) = n + ∑ k = s ∗ k ( Φ ( x )) = n + ∑ k = (cid:0) d ( x , x i ) − d ( x , x i + ) (cid:1) + D ( n + )= ( d ( x , x ) − d ( x , x n + )) + D ( n + ) = D + D ( n + )= (cid:0) n + (cid:1) D (cid:0) n + (cid:1)(cid:0) diam ( X ) + (cid:1) Applying (10) to m i = M i , i = , . . . , n we deduce for x , y ∈ X that k Φ ( x ) − Φ ( y ) k = max { (cid:12)(cid:12)(cid:12) q ∑ i = p M i ∑ j = M i − + s ∗ j ( Φ ( x ) − Φ ( y )) (cid:12)(cid:12)(cid:12) : 1 ≤ p ≤ q ≤ n } (14) = max n q ∑ i = p s ∗ i ( Φ ( x ) − Φ ( y )) o = d ( x , y ) . (15)Finally our conclusion follows therefore from Theorem 5.1 and Corollary 5.2. (cid:3) Metric universality and metric elasticity.
It is a well known and long standing open problemwhether c isomorphically embeds into a Banach space whenever it bi-Lipschitzly embeds into it.Due to Aharoni’s theorem this fundamental rigidity problem in nonlinear Banach space geometrycan be reformulated as the following universality question. Problem 5.5.
Let X be a Banach space. If X is Lipschitz universal for the class of separable metricspaces, does X contain an isomorphic copy of c ? It is possible to answer positively Problem 5.5 for Banach lattices using Kalton’s work on theinterlacing graphs. This fact seems to have been overlooked and we describe the argument in theensuing discussion. Recall that a Banach space Y has Kalton’s property Q if there exists C ∈ ( , ∞ ) OARSE AND LIPSCHITZ UNIVERSALITY 21 such that for all k ∈ N and every Lipschitz map f from ([ N ] k , d ( k ) I ) to Y , there exists M ∈ [ N ] ω suchthat for all ¯ m , ¯ n ∈ [ M ] k we have(16) (cid:13)(cid:13) f ( ¯ m ) − f ( ¯ n ) (cid:13)(cid:13) Y ≤ C Lip ( f ) . Kalton showed that reflexive Banach spaces [Kal07, Theorem 4.1] and more generally, Banachspaces whose unit ball uniformly embeds into a reflexive Banach space [Kal07, Corollary 4.3] haveProperty Q . It follows from (16) (and the fact that coarse embeddings f whose domains are graphsmust be ω ( ) -Lipschitz) that the sequence of interlacing graphs cannot equi-coarsely embed into aBanach space with property Q . Therefore if a Banach space X equi-coarsely contains the interlacinggraphs, it must fail property Q . By [Kal07, Corollary 4.3], the unit ball of X does not uniformlyembed into a reflexive Banach space. But Kalton also proved [Kal07, Theorem 3.8] that for aseparable Banach lattice X , B X uniformly embeds into a reflexive Banach space if and only if X does not contain any subspace isomorphic to c . Thus, it follows from the above discussion that: Theorem 5.6 ([Kal07]) . If X is a separable Banach lattice and if ([ N ] k , d I ) k equi-coarsely embedsinto X , then X contains an isomorphic copy of c . The following statement is an immediate consequence of Theorem 5.6.
Corollary 5.7.
If a separable Banach lattice X is coarsely universal for the class of separablemetric spaces, then X contains an isomorphic copy of c . Thus, Problem 5.5 (as well as its coarse analogue) has a positive solution for Banach lattices.It is worth pointing out that the coarse (resp. uniform) analogue of Problem 5.5 does not hold ingeneral since using the theory of H ¨older free spaces, it is proved in [Kal04] that c coarsely (resp.uniformly) embeds into a Schur space. Recall that a Banach space has the Schur property if everyweakly null sequence converges to 0 in the norm topology, and hence a Schur space cannot containany isomorphic copy of c . Remark 5.8.
The Lipschitz version of Corollary 5.7 can be proven for a Banach space with anunconditional basis using classical linear and nonlinear Banach space theory. Indeed, for Banachspaces with an unconditional basis the following dichotomy holds: either the unconditional basis isnot boundedly-complete, and by a result of James [Jam50] X will contain an isomorphic copy of c ,or the unconditional basis is boundedly-complete and hence X will be isomorphic to a dual spaceand thus X will have the Radon-Nikod´ym property. Note that the two possibilities are mutuallyexclusive. In the first situation the conclusion of Corollary 5.7 already holds, and in the second sit-uation we can use classical differentiability theory and obtain a contradiction. A similar dichotomyargument fails for Banach lattices since L is a Banach lattice that does not linearly contain c andyet L does not have the Radon-Nikod´ym property.Theorem 5.6 has also an application to metric analogues of the linear notion of elasticity. In 1976Sch¨affer raised the problem whether the isomorphism class of every infinite dimensional Banachspace X is unbounded in the sense that D ( X ) : = sup { d BM ( Y , Z ) : Y , Z are isomorphic to X } = ∞ where d BM denotes the Banach-Mazur distance . Johnson and Odell introduced the notion of elas-ticity for their solution of Sch¨affer’s problem for separable Banach spaces.
Definition 5.9 ([JO05]) . Let K ∈ [ , ∞ ) . A Banach space Y is K-elastic provided that if a Banachspace X isomorphically embeds into Y then X must be K -isomorphically embeddable into Y , and Y will be elastic if it is K -elastic for some K . This widely use terminology can be misleading since log ( d BM ) (and not d BM ) is a semi-metric. The connection with Sch¨affer’s problem comes from the observation that if D ( X ) < ∞ then X aswell as every isomorphic copies of X is D ( X ) -elastic. Elasticity is intimately connected to univer-sality. First of all, it is immediate that every Banach space is crudely finitely representable into anyelastic Banach space, in particular every elastic Banach space has trivial cotype. Second of all, aconsequence of Banach-Mazur embedding theorem is that C [ , ] is 1-elastic. Moreover, a key stepin [JO05] is the following theorem. Theorem 5.10. [JO05, Theorem 7]
Let X be a separable infinite-dimensional Banach space. If Xis elastic then c embeds isomorphically into X . The conjecture from [JO05] that a separable elastic Banach space must contain an isomorphiccopy of C [ , ] , was recently solved positively by Alspach and Sari [AS16].We now discuss a metric analogue of Theorem 5.10. According to Johnson and Odell a Banachspace Y is said to be Lipschitz K-elastic provided that if a Banach space is isomorphic to Y then X must bi-Lipschitzly embed into Y with distortion at most K . Johnson and Odell definition ofLipschitz elasticity is motivated by the fact that a Banach space X is K -elastic if and only if everyisomorphic copy of X is K -isomorphic to a subspace of X (the proof uses an Hahn-Banach extensionargument that goes back to Pelczy´nski [Peł60]). It was observed in [JO05] that it follows fromAharoni’s embedding theorem and James’ distortion theorem, that there exists K ≥ must be Lipschitz K -elastic. The constant K isrelated to the optimal distortion in Aharoni’s embedding and can be taken to be 2 + ε for every ε > Definition 5.11.
Let K ∈ [ , ∞ ) . A metric space Y is metric K-elastic provided that if a metric space X bi-Lipschitzly embeds into Y then X must be bi-Lipschitzly embeddable into Y with distortion atmost K , and Y will be metric elastic if it is metric K -elastic for some K .It is immediate that a Banach space that is metric K -elastic (as a metric space) is Lipschitz K -elastic. With this stronger nonlinear notion of elasticity we obtain the following theorem, whichcontains a strong nonlinear analogue of Theorem 5.10 in the context of Banach lattices. Theorem 5.12.
Let X be a separable infinite-dimensional Banach lattice. The following assertionsare equivalent. (1) c isomorphically embeds into X . (2) c bi-Lipschitzly embeds into X . (3) c coarsely embeds into X . (4) X is metric elastic. (5) ([ N ] k , d I ) k ∈ N embeds equi-bi-Lipschitzly into X . (6) ([ N ] k , d I ) k ∈ N embeds equi-coarsely into X .Proof. (1) implies (2) implies (3) is trivial. (3) implies (1) is Corollary 5.7. (2) implies (4) followsfrom Aharoni’s embedding theorem and the fact that separability is a Lipschitz invariant. For (4)implies (5), observe that an infinite-dimensional Banach space X has a 1-separated sequence of unitvectors, and thus for all k ∈ N , the k -dimensional interlacing graph ([ N ] k , d I ) (which is countable,1-separated, and has diameter k ) embeds bi-Lipschitzly into X with distortion at most k . Since X ismetric K -elastic for some K ≥
1, it follows that sup k ∈ N c X (([ N ] k , d I )) ≤ K . For Banach spaces, (5)implies (6) always holds. An appeal to Corollary 5.6 gives the remaining implication. (cid:3) Separating interlacing graphs in Banach spaces with nonseparable biduals.
The follow-ing concentration result for interlacing graphs was shown by Kalton [Kal07].
OARSE AND LIPSCHITZ UNIVERSALITY 23
Theorem 5.13. [Kal07, Theorem 3.5] . Let k ∈ N and Y be a Banach space such that Y ( k ) , theiterated dual of order k of Y , is separable. Assume that ( g i ) i ∈ I is an uncountable family of -Lipschitz maps from ([ N ] k , d ( k ) I ) to Y . Then there exist i = j ∈ I and M ∈ [ N ] ω such that for all ¯ n ∈ [ M ] k we have (cid:13)(cid:13) g i ( ¯ n ) − g j ( ¯ n ) (cid:13)(cid:13) ≤ . Vaguely speaking, it follows from Theorem 5.13 that if a Banach space X contains uncountablymany well-separated 1-Lipschitz images of the interlacing graphs and if X coarsely embeds into aBanach space Y , then Y cannot have all its iterated duals separable. This idea was devised by Kaltonin [Kal07] to show that if c coarsely embeds into a Banach space Y , then one of the iterated dualsof Y is non separable (in particular, Y cannot be reflexive). It was adapted in [LPP20] to show thatthe same conclusion holds if the James tree space JT or its predual coarsely embeds into Y . In theseproofs the non separability of the bidual of the embedded space plays an important role. However,since ℓ coarsely embeds into ℓ , this is not a sufficient condition. We will prove that a certainpresence of ℓ in the embedded space is essentially the only obstruction. Theorem 5.14. [Theorem F]
Let X be a separable Banach space with non separable bidual X ∗∗ , ℓ X , and such that no spreading model generated by a normalized weakly null sequence in X isequivalent to the ℓ -unit vector basis. Assume that X coarsely embeds into a Banach space Y . Thenthere exists k ∈ N such that Y ( k ) is non separable.Proof. We start with the construction of our well separated 1-Lipschitz maps from ([ N ] k , d ( k ) I ) to X .Since X is separable and X ∗∗ is not, using Riesz Lemma and an easy transfinite induction, we canbuild ( x ∗∗ α ) α < ω in S X ∗∗ such that ∀ α < ω , d (cid:0) x ∗∗ α , sp ( X ∪ { x ∗∗ β , β < α } (cid:1) > . Fix now α < ω . Since ℓ X it follows from a result by Odell and Rosenthal [OR75, Equiva-lences (1)-(5) on page 376] that for each α < ω there is a sequence ( x α , n ) ∞ n = in S X , which convergesweak ∗ in X ∗∗ to x ∗∗ α . In particular, the sequence ( x α , n ) ∞ n = is weakly Cauchy. Since d ( x ∗∗ α , X ) > ,we may as well assume, after extracting a subsequence, that k x α , n − x α , m k > , for all n = m .After passing to a further subsequence we can also assume that ( x α , n ) ∞ n = has a spreading model.But this means, that all the sequences of the form ( x α , n j − x α , n j − ) ∞ j = ⊂ B X , with ( n j ) ∞ j = increas-ing sequence in N , have the same spreading model ( e α j ) ∞ j = . Define now λ α k = k ∑ kj = e α j k . Sincespreading models generated by weakly null sequences are 1-suppression unconditional we have thatfor any α < ω , the sequence ( λ α k ) k is non decreasing. It also follows from our assumptions on X and from the fact that ( x α , m − x α , n ) n = m is semi-normalized that ∀ α < ω , lim k → ∞ k λ α k = ∞ . For fixed α < ω and k ∈ N , we can apply Ramsey’s Theorem and, after passing to a further sub-sequence, we can assume that for all ¯ m , ¯ n ∈ [ N ] k , with k ≤ m < n < m < n < . . . < m k < n k , wehave (cid:13)(cid:13)(cid:13) k ∑ j = x α , n j − x α , m j (cid:13)(cid:13)(cid:13) ≤ λ α k . Using then the usual diagonalization argument we can assume that for all k ∈ N and for all ¯ m , ¯ n ∈ [ N ] k , with m < n < m < n < . . . < m k < n k ,(17) (cid:13)(cid:13)(cid:13) k ∑ j = x α , n j − x α , m j (cid:13)(cid:13)(cid:13) ≤ λ α k . Then, we define for α < ω and k ∈ N , f ( k ) α : [ N ] k → X , ¯ n λ α k k ∑ i = x α , n i . It follows from the definition of the interlaced distance, equation (17) and the monotonicity of ( λ α k ) k that f ( k ) α is 1-Lipschitz.Consider now α < β ∈ [ , ω ) . Since dist ( x ∗∗ β , sp ( x ∗∗ α )) > /
4, by Hahn-Banach, there exists an x ∗∗∗ α , β ∈ S X ∗∗∗ with x ∗∗∗ α , β ( x ∗∗ α ) = x ∗∗∗ α , β ( x ∗∗ β ) = dist ( x ∗∗ β , sp ( x ∗∗ α )) > /
4. By the principle of localreflexivity (applied to the space X ∗ ) there exists x ∗ α , β ∈ S X ∗ with x ∗∗ α ( x ∗ α , β ) = x ∗∗ β ( x ∗ α , β ) > / M ∈ [ N ] ω :sup ¯ n ∈ [ M ] k (cid:13)(cid:13) f ( k ) α ( ¯ n ) − f ( k ) β ( ¯ n ) (cid:13)(cid:13) (18) ≥ lim sup n ∈ M , n → ∞ lim sup n ∈ M , n → ∞ . . . lim sup n k ∈ M , n k → ∞ x ∗ α , β (cid:16) λ β k k ∑ i = x β , n i − λ α k k ∑ i = x α , n i (cid:17) ≥ λ β k k = k λ β k . This finishes our construction of uncountably many well separated X -valued Lipschitz maps.Assume now that X coarsely embeds into a Banach space Y such that all the iterated duals of Y areseparable and let g : X → Y be such a coarse embedding. Of course, we may assume that ω g ( ) ≤ α < ω and k ∈ N , we define g ( k ) α = g ◦ f ( k ) α . We have that g ( k ) α is 1-Lipschitz from ([ N ] k , d ( k ) I ) to Y . For a fixed k ∈ N , we can therefore apply Theorem 5.13 to any uncountable sub-family of ( g ( k ) α ) α < ω . We then deduce from (18) that for any k ∈ N , { α < ω , ρ g ( k λ α k ) > } iscountable. This implies that the set { α < ω ∃ k ∈ N , ρ g ( k λ α k ) > } is also countable and since [ , ω ) is uncountable, there exists α < ω such that for all k ∈ N , ρ g ( k λ α k ) ≤
3. This is in contra-diction with the fact that for this given α < ω , k λ α k ր ∞ , if k ր ∞ and lim t → ∞ ρ g ( t ) = ∞ . (cid:3) Understanding quantitatively what is the order of the non-separable iterated dual in Theorem Fis a very interesting problem.
Problem 5.15.
Assume that X is c , or any separable Banach space with non separable bidualX ∗∗ and ℓ X such that no spreading model generated by a normalized weakly null sequence isequivalent to the ℓ -unit vector basis. If X coarsely embeds into a Banach space Y , does it implythat Y ∗∗ is non separable? R EFERENCES [Aha74] I. Aharoni,
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