Separation ratios of maps between Banach spaces

Under the weak assumption on a Banach space $E$ that $E\oplus E$ embeds isomorphically into $E$, we provide a characterisation of when a Banach space $X$ coarsely embeds into $E$ via a single numerical invariant.


Introduction
The concept of coarse embeddability between metric spaces can be viewed as a large scale analogue of uniform embeddability and may most easily be understood in terms of the moduli associated with a map.However, as we are exclusively concerned with Banach spaces, these moduli can further be reduced to a couple of numerical invariants.To avoid certain trivialities, we shall tacitly assume that all Banach spaces have dimension at least 2 and hence, in particular, that the infima and suprema above are taken over non-empty sets.Let us first note the obvious fact that ω(φ) = 0 if and only if φ is uniformly continuous.On the other hand, ω(φ) < ∞ if and only if φ is Lipschitz for large distances, that is, for some constant K and all x, y ∈ X.Similarly, assumptions on κ(φ) correspond to known conditions on the map φ.We summarise these as follows.
Note that the three coefficients above are all positive homogenous, in the sense that for all λ > 0 and similarly for κ(φ) and ω(φ).In particular, this means that the following quantity is invariant under rescaling φ.
Definition 2. The separation ratio of a map X φ −→ E is the quantity where we set a ∞ = 0 a = 0 for all a ∈ [0, ∞] and a 0 = ∞ a = ∞ 0 = ∞ for all 0 < a < ∞.Whereas φ being a uniform embedding cannot be directly expressed via the coefficients above, we note that φ is a coarse embedding provided that ω(φ) < ∞ and κ(φ) = ∞, that is, if φ is Lipschitz for large distances and is expanding.We thus see that R(φ) = ∞ if and only if φ is either uniformly continuous and uncollapsed (e.g., a uniform embedding) or if φ is a coarse embedding.
Motivated in part by the still open problem of deciding whether a Banach space X coarsely embeds into a Banach space E if and only it uniformly embeds, the papers [2,3,7,8,9,10] contain various constructions for producing uniform and coarse embeddings or obstructions to the same.In particular, in [10] (see Theorem 1.16) we showed that, provided that E ⊕ E isomorphically embeds into E, then a uniformly continuous and uncollapsed map X φ −→ E gives rise to a simultaneously uniform and coarse embedding of X into E.However, as shown by A. Naor [7], there are Lipschitz for large distance maps that are not even close to any uniformly continuous map.For the exclusive purpose of coarse embeddability, our main result, Theorem 3, removes the problematic assumption of uniform continuity of φ.Theorem 3. Suppose X and E are Banach spaces so that E ⊕ E isomorphically embeds into E. Then X coarsely embeds into E if and only if where the supremum is taken over all maps X φ −→ E.
The proof of Theorem 3 also allows us to address another issue, namely, the preservation of cotype under different forms of embeddability.For this, consider the following conditions on a map Also, the map φ is said to be solvent provided that there are constants R 1 , R 2 , . . .so that Provided that φ is Lipschitz for large distances, φ is solvent if and only if it is almost expanding (see Lemma 8 [8]).In analogy with Definition 2, we then define the exact separation ratio of φ to be As κ(φ) κ(φ), we then have R(φ) R(φ).Also, R(φ) = ∞ exactly when φ is either uniformly continuous and almost uncollapsed or is Lipschitz for large distances and solvent.
In connection with this, B. Braga [3] strengthened work by M. Mendel and A. Naor [6] to show that, if X maps into a Banach space E with non-trivial type by a map that is either uniformly continuous and almost uncollapsed or is Lipschitz for large distances and solvent, then cotype(X) cotype(E).
The following statement therefore covers both cases of Braga's result and seemingly provides the ultimate extension in this direction.
Theorem 4. Suppose X and E are Banach spaces so that and that E has non-trivial type.Then cotype(X) cotype(E).
Problem 7.4 in Braga's paper [3] asks what can be deduced about a space X that admits a map X φ −→ E that is just Lipschitz for large distances and almost uncollapsed, i.e. so that R(φ) > 0. That is, will restrictions on the geometry of E also lead to information about the geometry of X?In Example 10, we show that this is not always so.Indeed, if X is separable and E is infinite-dimensional, one can always find a map X φ −→ E that is both Lipschitz for large distances and uncollapsed, i.e., so that R(φ) > 0, and after renorming E one can even obtain R(φ) 1. On the other hand, Theorem 4 provides a positive answer to Braga's question under the alternative assumption sup φ R(φ) = ∞.
Acknowledgements: I am very grateful for the extensive feedback and criticisms I got from B. Braga on a first version of this paper and for suggesting a link with cotype that led to Theorem 4.

Proofs
Before proving our main results, let us introduce four functional moduli that lie behind the definitions of the (exact) compression and expansion coefficients.
Definition 5 (Compression moduli).For a (generally discontinuous and nonlinear) map X φ −→ E between two Banach spaces we define the exact compression modulus x − y = t and the compression modulus by Thus, κ φ is the pointwise largest map so that κ φ x − x φ(x) − φ(y) for all x, y ∈ X, while κ φ (t) = inf r t κ φ (r) is the pointwise largest monotone map satisfying the same inequality.Definition 6 (Expansion moduli).For a map X φ −→ E between Banach spaces, the exact expansion modulus and the expansion modulus The following are evident.
We recall that, to avoid trivialities, all Banach spaces are assumed to have dimension at least 2. Thus, suppose X φ −→ E is a map and that t > 0 and x, y ∈ X.Let n 1 be minimal so that x − y nt, whereby (n − 1)t x − y and pick In turn, this shows that for all s, t > 0 and so lim sup s→0+ ω φ (s) inf t>0 ω φ (t).Because ω φ is nondecreasing, the limit lim s→0+ ω φ (s) = inf s>0 ω φ (s) exists, whereby All in all, we find that In particular, we would obtain nothing new by introducing an exact expansion coefficient by ω(φ) = inf t>0 ω φ (t), since this is just the expansion coefficient itself.Furthermore, if ω(φ) < ∞, then φ is Lipschitz for large distances, that is, for some constant K and all x, y ∈ X.
Next, the definition of the separation ratio may initially be difficult to parse, so let us briefly restate it more explicitly.
Proof of Theorem 3. As noted, if X φ −→ E is a coarse embedding between arbitrary Banach spaces, then R(φ) = ∞, which proves one direction of implication.Also, under the stated assumption on E, by Theorem 1.16 [10], we have that X coarsely embeds into E if and only if R(φ) = ∞ for some map X φ −→ E. So suppose instead only that sup φ R(φ) = ∞.We then construct a coarse embedding X ψ −→ E as follows.
Because E ⊕ E embeds isomorphically into E, we may inductively construct two sequences E n , Z n of closed linear subspaces of E all isomorphic to E so that Concretely, we simply begin with an isomorphic copy E ⊕ E inside of E and let E 1 and Z 1 be respectively the first and second summand.Again, pick an isomorphic copy of E ⊕ E inside of Z 1 with first and second summand denoted respectively E 2 and Z 2 , etc.It thus follows that is a decreasing sequence of closed linear subspaces of E. Let We note that V is a closed linear subspace of E in which each E n is a closed subspace complemented by a bounded projection V Pn −→ E n so that so that E m ⊆ ker P n whenever n = m.On the other hand, we have no uniform bound on the norms P n .

Fix now a sequence of isomorphisms
Setting φ n = T n • θ n , we find that lim n

R(φn)
2 n Pn = ∞.The conclusion of the theorem therefore follows directly from Lemma 8 below.Lemma 8. Suppose X and E are Banach spaces and E Pn −→ E is a sequence of bounded linear projections onto subspaces E n ⊆ E so that E m ⊆ ker P n for all m = n.Assume also that there is a sequence of maps Then X coarsely embeds into E.
Proof.By composing with a translation, we may suppose that φ n (0) = 0 for each n.Because lim n R(φn) 2 n Pn = ∞, we may also find constants ∆ n , δ n , Λ n , λ n > 0 so that For every n, we let Similarly, Also, ψ n (0) = 0 for all n, which shows that, for all x ∈ X, and so the series ∞ n=1 ψ n (x) is absolutely convergent in E. We may therefore define a map X ψ −→ E by letting We now verify that ψ is a coarse embedding of X into E. First, let m 1 be any given natural number and suppose that x, y ∈ X satisfy x − y m.Then we may find z 0 = x, z 1 , z 2 , . . ., z m = y so that z i−1 − z i 1 for all i and so, in particular, ψ n (z i−1 ) − ψ n (z i ) 2 −n for all n.It thus follows that In other words, for all m and x, y ∈ X, we have Conversely, if m is any given number, find n large enough so that Taken together, these two conditions show that ψ is a coarse embedding.
Proof of Theorem 4. Suppose X and E are Banach spaces so that sup φ R(φ) = ∞, and E have non-trivial type, i.e., type(E) > 1.We then note that also type ℓ 2 (E) = type(E) > 1 and cotype ℓ 2 (E) = cotype(E).Thus, if we can show that X maps into ℓ 2 (E) by a map that is Lipschitz for large distances and solvent, then, by the previously mentioned result of Braga (Theorem 1.3 [3]), we will have that cotype(X) cotype ℓ 2 (E) = cotype(E).
Another way to prove Theorem 4 is first to establish an analogue to Theorem 3 for the quantity sup φ R(φ) in place of sup φ R(φ).This is done by observing that the proof of Theorem 3 above can be changed to prove the following statement.Theorem 9. Suppose X and E are Banach spaces so that E ⊕ E isomorphically embeds into E. Assume also that then there is a map X φ −→ E that is Lipschitz for large distances and solvent.
In order to obtain Theorem 4, one then notes that ℓ 2 (E) ⊕ ℓ 2 (E) ∼ = ℓ 2 (E) and so, if sup φ R(φ) = ∞, where the supremum is taken over all maps X φ −→ E, we have a map X ψ −→ ℓ 2 (E) that is both Lipschitz for large distances and solvent.

Examples
Theorem 3 indicates that, to every pair of Banach spaces X and E, we may associate the constant that under very mild assumptions on E measures the extent of coarse embeddability of X into E.The number CR(X, E) will be termed the coarse embeddability ratio of X in E.
As the next example shows, the main interest lies in the case when CR(X, E) > 1, whereas CR(X, E) = 1 is easily obtained.
Example 10.If X is separable and E is a Banach space that admits an infinite equilateral set, that is, an infinite subset A ⊆ E so that, for some δ > 0, x − y = δ for all distinct x, y ∈ A, then we have CR(X, E) 1.To see this, let (Y x ) x∈A be a partition of X indexed by the set A into subsets Y x ⊆ X of diameter at most 1 and let X φ −→ E be defined by Observe that, if y − y ′ > 1, then y and y ′ must belong to different pieces of the partition and so φ(y) − φ(y ′ ) = δ.On the other hand, φ(y) − φ(y ′ ) δ for all y, y ′ ∈ X, so we see that κ φ (t) = δ for all t > 1, whereas ω φ (t) δ for all t > 0. So R(φ) 1.
In particular, this reasoning applies when E is one of the classical Banach spaces ℓ p , c 0 , L p or even the Tsirelson space T .Indeed, in these spaces, the standard unit basis (e n ) ∞ n=1 is an infinite equilateral set (or, in the case of Tsirelson's space, (e n ) ∞ n=2 is equilateral).Here we remark that T * is the reflexive space originally constructed and described by B. S. Tsirelson [11], while T is its ℓ 1 -asymptotic dual whose explicit construction was given by T. Figiel and W. B. Johnson [5].
Let us also observe that, if E is infinite-dimensional, then E admits an equivalent renorming with respect to which it has an infinite equilateral set.Indeed, since E is infinite-dimensional, it contains a normalised basic sequence (e n ) ∞ Example 10 illustrates that the embeddability ratio CR(X, E) is sensitive to the specific norm on E, but not to the choice of norm on X.On the other hand, the condition CR(X, E) = ∞ only depends on the isomorphism class of E. Note also that, if X, Y and Z are Banach spaces so that CR(X, Y ) = ∞, then CR(X, Z) CR(Y, Z).
An important non-embeddability result was recently established by F. Baudier, G. Lancien and T. Schlumprecht [1], who showed that the separable Hilbert space ℓ 2 does not coarsely embed into Tsirelson's space T * .It is known that T * is minimal, that is, T * embeds isomorphically into all of its infinite-dimensional subspaces (see Chapter VI [4]).Also, T * has an unconditional basis and can therefore be written as a direct sum of two infinite-dimensional subspaces.It therefore follows that T * ⊕ T * embeds isomorphically into T * and thus E = T * satisfies the assumption of Theorem 3. It follows that the coarse embeddability ratio CR(ℓ 2 , T * ) is finite and we now proceed to give an upper bound.Proof.We rely on the analysis of [1], which also contains additional details about the construction below.For the proof, assume towards a contradiction that ℓ 2 φ −→ E satisfies R(φ) > 4. Then by pre and post-composing φ with dilations we can suppose that, for some constants ∆ > 0 and δ > 4, we have Let (e n ) ∞ n=1 be the standard unit vector basis for ℓ 2 and set ǫ = δ−4 2 .Let also k be large enough so that √ 2k ∆ and let [N] k be the collection of all k-element subsets of N equipped with the Johnson metric, Observe that d J is simply the shortest-path metric on the graph whose vertices is and so f (A) − f (B) 1.Thus, f is Lipschitz with constant 1.By Proposition 4.1 [1] there is an infinite subset M ⊆ N and some y ∈ T * so that, for any A ∈ [N] k with A ⊆ M, there are vectors y A 1 , . . ., y A k ∈ T * with y A i 1 so that y, y A 1 , . . ., y A k form a finite block basis of the standard unit vector basis for T * , k min supp(y A 1 ) and f (A) − (y + y A 1 + • • • + y A k ) < ǫ.In particular, for all A, B ∈ [N] k , A, B ⊆ M, we have that where the second bound follows from (2.13) in [1].On the other hand, for any two disjoint A, B ∈ [N] k , we have The following still unsolved problem provides the main theoretical motivation for our investigations here.
Problem 12. Suppose X and E are Banach spaces.Is it true that X coarsely embeds into E if and only if it uniformly embeds?Problem 13.Suppose X and E are Banach spaces so that CR(X, E) > 1.Does it follow that CR(X, E) = ∞?
n=1 .We define a new equivalent norm ||| • ||| on the closed linear space [e n ] ∞ n=1 by letting ∞ n=1 a n e n = sup n∈I a n e n + n∈J a n e n I, J are intervals and i < j for all i ∈ I and j ∈ J .As e n = 1 for all n, we find that |||e i − e j ||| = 2 for all i < j and so (e n ) ∞ n=1 is an equilateral set of the norm ||| • |||.It now suffices to notice that ||| • ||| extends to an equivalent norm on all of E.
[N]  k and where two vertices A and B are connected by an edge provided that|A △ B| = 2. Let then f : [N] k → T * be defined by f (A) = φ n∈A e n .Observe that, if d J (A, B) = 1, then n∈A e n − n∈B e n = |A △ B| = √ 2 f (A) − f (B)δ and thus contradicts the preceding upper bound.