THE DIAMETER OF THE ISOMORPHISM CLASS OF A BANACH SPACE

W. B. JOHNSON AND E. ODELL August 20, 2004

Abstract. We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c_{0}. We call X elastic if for some K < ∞ for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K < ∞ every isomorph of X is K-elastic then X is finite dimensional.

1. Introduction.

Given a Banach space X, let D(X) be the diameter in the Banach-Mazur distance of the class of all Banach spaces which are isomorphic to X; that is,

D(X) = sup{d(X_{1}, X_{2}) : X_{1}, X_{2} are isomorphic to X}

where d(X_{1}, X_{2}) is the infimum over all isomorphisms T from X_{1} onto X_{2} of ∥T ∥·∥T^{−1}∥. It is well known that if X is finite (say, N) dimensional, then cN ≤ D(X) ≤ N for some positive constant c which is independent of N. The upper bound is an immediate consequence of the classical result (see e.g. [T-J, p. 54]) that d(Y, ℓ_{2}^{N}) ≤ \sqrt{N} for every N dimensional space Y. The lower bound is due to Gluskin [G], [T-J, p. 283].

It is natural to conjecture that D(X) must be infinite when X is infinite dimensional, but this problem remains open. As far as we know, this problem was first raised in print in the 1976 book of J. J. Schäffer [S, p. 99]. The problem was recently brought to the attention of the authors by V. I. Gurarii, who checked that every infinite dimensional super-reflexive space as well as each of the common classical Banach spaces has an isomorphism class whose diameter is infinite. To see these cases, note that if X is infinite dimensional and E is any finite dimensional space, then it is clear that X is isomorphic to E ⊕_{2} X_{0} for some space X_{0}. Therefore, if D(X) is finite, then X is finitely complementably universal; that is, there is

Johnson was supported in part by NSF DMS-0200690, Texas Advanced Research Program 010366-163, and the U.S.-Israel Binational Science Foundation. Odell was supported in part by NSF DMS-0099366 and was a participant in the NSF supported Workshop in Linear Analysis and Probability at Texas A&M University.

MR subject classification: 46G05, 46T20.

2 W. B. JOHNSON AND E. ODELL

a constant C so that every finite dimensional space is C-isomorphic to a C-complemented subspace of X. This implies that X cannot have non trivial type or non trivial cotype or local unconditional structure or numerous other structures. In particular, X cannot be any of the classical spaces or be super-reflexive.

In his unpublished 1968 thesis [Mc], McGuigan conjectured that D(X) must be larger than one when dim X > 1. Schäffer [S, p. 99] derived that D(X) ≥ 6/5 when dim X > 1 as a consequence of other geometrical results contained in [S], but one can prove directly that D(X) ≥ \sqrt{2}. Indeed, it is clearly enough to get an appropriate lower bound on the Banach-Mazur distance between X_{1} := Y ⊕_{1} ℝ and X_{2} := Y ⊕_{2} ℝ when Y is a non zero Banach space. Now X_{1} has a one dimensional subspace for which every two dimensional superspace is isometric to ℓ_{1}^{2}. On the other hand, every one dimensional subspace of X_{2} is contained in a two dimensional superspace which is isometric to ℓ_{2}^{2}. It follows that d(X_{1}, X_{2}) ≥ d(ℓ_{1}^{2}, ℓ_{2}^{2}) = \sqrt{2}.

The Main Theorem in this paper is a solution to Schäffer’s problem for separable Banach spaces:

Main Theorem. If X is a separable infinite dimensional Banach space, then D(X) = ∞.

Part of the work for proving the Main Theorem involves showing that if X is separable and D(X) < ∞, then X contains an isomorph of c_{0}. This proof is inherently not local in nature, and, strangely enough, local considerations, such as those mentioned earlier which yield partial results, play no role in our proof. We do not see how to prove that a nonseparable space X for which D(X) < ∞ must contain an isomorph of c_{0}. Our proof requires Bourgain’s index theory which in turn requires separability.

Our method of proof involves the concept of an elastic Banach space. Say that X is K-elastic provided that if a Banach space Y embeds into X then Y must K-embed into X (that is, there is an isomorphism T from Y into X with

∥y∥ ≤ ∥T y∥ ≤ K∥y∥

for all y ∈ Y). This is the same (by Lemma 2) as saying that every space isomorphic to X must K-embed into X. X is said to be elastic if it is K-elastic for some K < ∞.

Obviously, if D(X) < ∞ then X as well as every isomorph of X is D(X)-elastic. Thus the Main Theorem is an immediate consequence of

Theorem 1. If X is a separable Banach space and there is a K so that every isomorph of X is K-elastic, then X is finite dimensional.

THE DIAMETER OF THE ISOMORPHISM CLASS OF A BANACH SPACE 3

A key step in our argument involves showing that an elastic space X admits a normalized weakly null sequence having a spreading model not equivalent to either the unit vector basis of c_{0} or ℓ_{1}. To achieve this we first prove (Theorem 7) that if X is elastic then c_{0} embeds into X. It is reasonable to conjecture that an elastic infinite dimensional separable Banach space must contain an isomorph of C[0, 1]. Theorem 1 would be an immediate consequence of this conjecture and the “arbitrary distortability” of C[0, 1] proved in [LP]. Our derivation of Theorem 1 from Theorem 7 uses ideas from [LP] as well as [MR].

With the letters X, Y, Z, … we will denote separable infinite dimensional real Banach spaces unless otherwise indicated. Y ⊆ X will mean that Y is a closed (infinite dimensional) subspace of X. The closed linear span of the set A is denoted [A]. We use standard Banach space theory terminology, as can be found in [LT]. The material we use on spreading models can be found in [BL]. For simplicity we assume real scalars, but all proofs can easily be adapted for complex Banach spaces.

2. The Main Result

The following lemma [Pe] shows that the two definitions of elastic mentioned in Section 1 are equivalent.

Lemma 2. Let Y ⊆ (X, ∥ · ∥) and let | · | be an equivalent norm on (Y, ∥ · ∥). Then | · | can be extended to an equivalent norm on X.

Proof. There exist positive reals C and d with d∥y∥ ≤ |y| ≤ C∥y∥ for y ∈ Y. Let F ⊆ CB_{X^{∗}} be a set of Hahn-Banach extensions of all elements of S_{(Y^{∗},|·|)} to all of X. For x ∈ X define

|x| = sup {|f(x)| : f ∈ F } ∨ d∥x∥ .

Let n ∈ ℕ and K < ∞. We shall call a basic sequence (x_{i}) block n-unconditional with constant K if every block basis (y_{i})_{i=1}^{n} of (x_{i}) is K-unconditional; that is,

∥ ∑_{i=1}^{n} ±a_{i}y_{i}∥ ≤ K∥ ∑_{i=1}^{n} a_{i}y_{i}∥

for all scalars (a_{i})_{i=1}^{n} and all choices of ±.

The next lemma is essentially contained in [LP]. In fact, by using the slightly more involved argument in [LP], the conclusion “with constant 2” can be changed to “with constant 1 + ε”, which implies that the constant in the conclusion of Lemma 4 can be changed from 2 + ε to 1 + ε.

Lemma 3. Let X be a Banach space with a basis (x_{i}). For every n there is an equivalent norm | · |_{n} on X so that in (X, | · |_{n}), (x_{i}) is block n-unconditional with constant 2.

4 W. B. JOHNSON AND E. ODELL

Proof. Let (P_{n}) be the sequence of basis projections associated with (x_{n}). We may assume, by passing to an equivalent norm on X, that (x_{n}) is bimonotone and hence ∥P_{j} −P_{i}∥ = 1 for all i < j. Let S_{n} be the class of operators S on X of the form S = ∑_{k=1}^{m}(−1)^{k}(P_{n_{k}} − P_{n_{k−1}}) where 0 ≤ n_{0} < ⋯ < n_{m} and m ≤ n. Define

|x|_{n} := sup{∥Sx∥ : S ∈ S_{n}} .

Thus ∥x∥ ≤ |x|_{n} ≤ n∥x∥ for x ∈ X. It suffices to show that for S ∈ S_{n}, |S|_{n} ≤ 2. Let x ∈ span (x_{n}) and |Sx|_{n} = ∥T Sx∥ for some T ∈ S_{n}. Then since T S ∈ S_{2n} ⊆ S_{n} + S_{n},

∥T Sx∥ ≤ 2|x|_{n} .

Lemma 4. For every separable Banach space X and n ∈ ℕ there exists an equivalent norm |·| on X so that for every ε > 0, every normalized weakly null sequence in X admits a block n-unconditional subsequence with constant 2 + ε.

Proof. Since C[0, 1] has a basis, the lemma follows from Lemma 3 and the the classical fact that every separable Banach space 1-embeds into C[0, 1].

Lemma 4 is false for some nonseparable spaces. Partington [P] and Talagrand [T] proved that every isomorph of ℓ_{∞} contains, for every ε > 0, a 1 + ε-isometric copy of ℓ_{∞} and hence of every separable Banach space.

Our next lemma is an extension of the Maurey-Rosenthal construction [MR], or rather the footnote to it given by one of the authors (Example 3 in [MR]). We first recall the construction of spreading models. If (y_{n}) is a normalized basic sequence then, given ε_{n} ↓ 0, one can use Ramsey’s theorem and a diagonal argument to find a subsequence (x_{n}) of (y_{n}) with the following property. For all m in ℕ and (a_{i})_{i=1}^{m} ⊂ [−1, 1], if m ≤ i_{1} < ⋯ < i_{m} and m ≤ j_{1} < ⋯ < j_{m}, then

|∥ ∑_{k=1}^{m} a_{k}x_{i_{k}} ∥ − ∥ ∑_{k=1}^{m} a_{k}x_{j_{k}} ∥| < ε_{m}.

It follows that for all m and (a_{i})_{i=1}^{m} ⊂ ℝ,

lim_{i_{1}→∞} … lim_{i_{m}→∞} ∥ ∑_{k=1}^{m} a_{k}x_{i_{k}} ∥ ≡ ∥ ∑_{k=1}^{m} a_{k}x̃_{k}∥

exists. The sequence (x̃_{i}) is then a basis for the completion of (span (x̃_{i}), ∥ · ∥) and (x̃_{i}) is called a spreading model of (x_{i}). If (x_{i}) is weakly null, then (x̃_{i}) is 2-unconditional. One shows this by checking that (x̃_{i}) is suppression 1-unconditional, which means that for all scalars (a_{i})_{i=1}^{m} and F ⊂ {1, …, m},

∥ ∑_{i∈F} a_{i}x̃_{i}∥ ≤ ∥ ∑_{i=1}^{m} a_{i}x̃_{i}∥.

Also, (x_{i}) is 1-spreading, which means that for all scalars (a_{i})_{i=1}^{m} and all n(1) < … n(m),

∥ ∑_{i=1}^{m} a_{i}x̃_{i}∥ = ∥ ∑_{i=1}^{m} a_{i}x̃_{n(i)}∥.

It is not difficult to see that, when (x_{i}) is weakly null, (x̃_{i}) is not equivalent to the unit vector basis of c_{0} (respectively, ℓ_{1}) if and only if lim_{m} ∥ ∑_{i=1}^{m} x̃_{i}∥ = ∞ (respectively, lim_{m} ∥ ∑_{i=1}^{m} x̃_{i}∥/m = 0). All of these facts can be found in [BL].

Lemma 5. Let (x_{n}) be a normalized weakly null basic sequence with spreading model (x̃_{n}). Assume that (x̃_{n}) is not equivalent to either the unit vector basis of ℓ_{1} or the unit vector basis of c_{0}. Then for all C < ∞ there exist n ∈ ℕ, a subsequence (y_{i}) of (x_{i}), and an equivalent norm | · | on [(y_{i})] so that (y_{i}) is | · |-normalized and no subsequence of (y_{i}) is block n-unconditional with constant C for the norm | · |.

Proof. Recall that if (e_{i})_{1}^{n} is normalized and 1-spreading and bimonotone then ∥ ∑_{1}^{n} e_{i}∥ ∥ ∑_{1}^{n} e_{i}^{∗}∥ ≤ 2n where (e_{i}^{∗})_{1}^{n} is biorthogonal to (e_{i})_{1}^{n} [LT, p.118]. Thus e = \frac{∑_{1}^{n} e_{i}}{∥ ∑_{1}^{n} e_{i}∥} is normed by f = ^{\frac{∥ ∑_{1}^{n} e_{i}∥}{n}} ∑_{i=1}^{n} e_{i}^{∗}, precisely f(e) = 1 = ∥e∥, and ∥f∥ ≤ 2. These facts allow us to deduce that there is a subsequence (y_{i}) of (x_{i}) so that if F ⊆ ℕ is admissible (that is, |F | ≤ min F) then

f_{F} ≡ \frac{∥ ∑_{i∈F} y_{i}∥}{|F |}( ∑_{i∈F} y_{i}^{∗})

satisfies ∥f_{F} ∥ ≤ 5 and f_{F} (y_{F}) = 1, where

y_{F} ≡ \frac{∑_{i∈F} y_{i}}{∥ ∑_{i∈F} y_{i}∥} .

Indeed, (x̃_{i}) is 1-spreading and suppression 1-unconditional (since (x_{i}) is weakly null). Given 1/2 > ε > 0 we can find (y_{i}) ⊆ (x_{i}) so that if F ⊆ ℕ is admissible then (y_{i})_{i∈F} is 1 + ε-equivalent to (x̃_{i})_{i=1}^{|F |}. Furthermore we can choose (y_{i}) so that if F is admissible then for y = ∑ a_{i}y_{i}, ∥ ∑_{i∈F} a_{i}y_{i}∥ ≤ (2 + ε)∥y∥ (for a proof see [O] or [BL]). Hence ∥f_{F} ∥ ≤ (2 + ε)∥f_{F} |_{[y_{i}]_{i∈F}} ∥ < 5 for sufficiently small ε by our above remarks.

We are ready to produce a Maurey-Rosenthal type renorming. Choose n so that n > 7C and let ε > 0 satisfy n^{2}ε < 1. We choose a subsequence M = (m_{j})_{j=1}^{∞} of ℕ so that m_{1} = 1 and for i ≠ j and for all admissible sets F and G with |F | = m_{i} and |G| = m_{j},

a) \frac{∥ ∑_{k∈F} y_{k}∥}{∥ ∑_{k∈G} y_{k}∥} < ε , if m_{i} < m_{j} and

b) \frac{∥ ∑_{k∈F} y_{k}∥}{∥ ∑_{k∈G} y_{k}∥} \frac{m_{j}}{m_{i}} < ε , if m_{i} > m_{j} .

Indeed, we have chosen (y_{i}) so that

\frac{1}{2}∥ ∑_{i=1}^{|F |} x̃_{i}∥ ≤ ∥ ∑_{i∈F} y_{i}∥ ≤ 2∥ ∑_{i=1}^{|F |} x̃_{i}∥

and similarly for G. Since (x̃_{i}) is not equivalent to the unit vector basis of c_{0} (and is unconditional) lim_{m} ∥ ∑_{1}^{m} x̃_{i}∥ = ∞ so that a) will be satisfied if (m_{k}) increases sufficiently rapidly. Furthermore, since (x̃_{i}) is not equivalent to the unit vector basis of ℓ_{1}, lim_{m} ^{\frac{∥ ∑_{1}^{m} x̃_{i}∥}{m}} = 0 and so b) can also be achieved.

For i ∈ ℕ set A_{i} = {y_{F} : F is admissible and |F | = m_{i}} and A_{i}^{∗} = {f_{F} : F is admissible and |F | = m_{i}}. Let ϕ be an injection into M from the collection of all (F_{1}, …, F_{i}) where i < n and F_{1} < F_{2} < ⋯ < F_{i} are finite subsets of ℕ. Here F < G means max F < min G. Let

F ={ ∑_{i=1}^{n} f_{F_{i}} : F_{1} < ⋯ < F_{n}, |F_{1}| = m_{1} = 1, each F_{i} is admissible
and |F_{i+1}| = ϕ(F_{1}, …, F_{i}) for 1 ≤ i < n}

For y ∈ [(y_{i})] let

∥y∥_{F} = sup {|f(y)| : f ∈ F}

and set

|y| = ∥y∥_{F} ∨ ε∥y∥ .

This is an equivalent norm since for f ∈ F, ∥f∥ ≤ 5n. Also, |y_{i}| = 1 for all i.

Note that if f_{F} ∈ A_{i}^{∗} and y_{G} ∈ A_{j} with i ≠ j then

|f_{F} (y_{G})| = \frac{∥ ∑_{k∈F} y_{k}∥}{m_{i}} ∑_{k∈F} y_{k}^{∗} (\frac{∑_{k∈G} y_{k}}{∥ ∑_{k∈G} y_{k}∥}) ≤ \frac{∥ ∑_{k∈F} y_{k}∥}{∥ ∑_{k∈G} y_{k}∥} \frac{m_{i} ∧ m_{j}}{m_{i}} .

If m_{i} < m_{j} then |f_{F} (y_{G})| < ε by a). If m_{i} > m_{j} then |f_{F} (y_{G})| < ε by b).

It follows that if y = ∑_{i=1}^{n} y_{F_{i}} and f = ∑_{i=1}^{n} f_{F_{i}} ∈ F then |y| ≥ f(y) = n and if z = ∑_{i=1}^{n}(−1)^{i}y_{F_{i}}, then for all g = ∑_{j=1}^{n} f_{G_{j}} ∈ F, |g(z)| ≤ 6 + n^{2}ε < 7. Indeed, we may assume that g ≠ f and if F_{1} ≠ G_{1} then |G_{j}| ≠ |F_{i}| for all 1 < i ≤ n and 1 ≤ j ≤ n and so by a), b),

|g(z)| = | ∑_{j=1}^{n} f_{G_{j}}( ∑_{i=1}^{n}(−1)^{i}y_{F_{i}})| ≤ ∑_{j=1}^{n} ∑_{i=1}^{n} |f_{G_{j}}(y_{F_{i}})| < n^{2}ε .

Otherwise there exists 1 ≤ j_{0} < n so that F_{j} = G_{j} for j ≤ j_{0}, |F_{j_{0}+1}| = |G_{j_{0}+1}| and |F_{i}| ≠ |G_{j}| for j_{0} + 1 < i, j ≤ n. Using f_{G_{j_{0}+1}}(z) ≤ 5 + nε we obtain

\begin{aligned}|g(z)| & ≤ | ∑_{j=1}^{j_{0}} f_{G_{j}}(z)| + |f_{G_{j_{0}+1}}(z)| + | ∑_{j>j_{0}+1}^{n} f_{G_{j}}(z)| \\ & < 1 + 5 + nε + (n − (j_{0} + 1))nε < 6 + n^{2}ε . \end{aligned}

Hence |z| ≤ 7 follows and the lemma is proved since n/7 > C and such vectors y and z can be produced in any subsequence of (y_{i}).

Our next lemma follows from Proposition 3.2 in [AOST].

Lemma 6. Let X be a Banach space. Assume that for all n, (x_{i}^{n})_{i=1}^{∞} is a normalized weakly null sequence in X having spreading model (x̃_{i}^{n}) which is not equivalent to the unit vector basis of ℓ_{1}. Then there exists a normalized weakly null sequence (y_{i}) ⊆ X with spreading model (ỹ_{i}) such that (ỹ_{i}) is not equivalent to the unit vector basis of ℓ_{1}. Moreover, there exists λ > 0 so that for all n,

λ2^{−n}∥ ∑ a_{i}x̃_{i}^{n}∥ ≤ ∥ ∑ a_{i}ỹ_{i}∥

for all (a_{i}) ⊆ ℝ.

Theorem 7. Let X be elastic, separable and infinite dimensional. Then c_{0} embeds into X.

We postpone the proof to complete first the

Proof of Theorem 1.

Assume that X is infinite dimensional and every isomorph of X is K-elastic. Then by Theorem 7, c_{0} embeds into X. Choose k_{n} ↑ ∞ so that 2^{−n}k_{n} → ∞. Using the renormings of c_{0} by

|(a_{i})|_{n} = sup {| ∑_{F} a_{i}| : |F | = k_{n}}

and that X is K-elastic we can find for all n a normalized weakly null sequence (x_{i}^{n})_{i=1}^{∞} ⊆ X with spreading model (x̃_{i}^{n})_{i=1}^{∞} satisfying

∥ ∑ a_{i}x̃_{i}^{n}∥ ≥ K^{−1}|(a_{i})|_{n}

and moreover each (x̃_{i}^{n}) is equivalent to the unit vector basis of c_{0}. Thus by Lemma 6 there exists a normalized weakly null sequence (y_{i}) in X having spreading model (ỹ_{i}) which is not equivalent to the unit vector basis of ℓ_{1} and which satisfies for all n,

∥ ∑_{1}^{k_{n}} ỹ_{i}∥ ≥ λK^{−1}2^{−n}k_{n} → ∞ .

8 W. B. JOHNSON AND E. ODELL

Thus (ỹ_{i}) is not equivalent to the unit vector basis of c_{0} as well.

By Lemmas 2 and 5, for all C < ∞ we can find n ∈ ℕ and a renorming Y of X so that Y contains a normalized weakly null sequence admitting no subsequence which is block n-unconditional with constant C. By the assumption on X, the space Y must K-embed into every isomorph of X. But if C is large enough this contradicts Lemma 4.

It remains to prove Theorem 7. We shall employ an index argument involving ℓ_{∞}-trees defined on Banach spaces. If Y is a Banach space our trees T on Y will be countable. For some C the nodes of T will be elements (y_{i})_{1}^{n} ⊆ Y with (y_{i})_{1}^{n} bimonotone basic and satisfying 1 ≤ ∥y_{i}∥ and ∥ ∑_{1}^{n} ±y_{i}∥ ≤ C for all choices of sign. Thus (y_{i})_{1}^{n} is C-equivalent to the unit vector basis of ℓ_{∞}^{n}. T is partially ordered by (x_{i})_{1}^{n} ≤ (y_{i})_{1}^{m} if n ≤ m and x_{i} = y_{i} for i ≤ n. The order o(T) is given as follows. If T is not well founded (i.e., T has an infinite branch), then o(T) = ω_{1}. Otherwise we set for such a tree S, S′ = {(x_{i})_{1}^{n} ∈ S : (x_{i})_{1}^{n} is not a maximal node}. Set T_{0} = T, T_{1} = T′ and in general T_{α+1} = (T_{α})′ and T_{α} = ∩_{β<α}T_{β} if α is a limit ordinal. Then

o(T) = inf{α : T_{α} = ϕ} .

By Bourgain’s index theory [B1, B2] (see also [AGR]), if X is separable and contains for all β < ω_{1} such a tree of index at least β, then c_{0} embeds into X.

We now complete the

Proof of Theorem 7.

Without loss of generality we may assume that X ⊆ Z where Z has a bimonotone basis (z_{i}). Let X be K-elastic. We will often use semi-normalized sequences in X which are a tiny perturbation of a block basis of (z_{i}) and to simplify the estimates we will assume below that they are in fact a block basis of (z_{i}).

For example, if (y_{i}) is a normalized basic sequence in X then we call (d_{i}) a difference sequence of (y_{i}) if d_{i} = y_{k(2i)} − y_{k(2i+1)} for some k_{1} < k_{2} < · · ·. We can always choose such a (d_{i}) to be a semi-normalized perturbation of a block basis of (z_{i}) by first passing to a subsequence (y′_{i}) of (y_{i}) so that lim_{i→∞} z_{j}^{∗}(y′_{i}) exists for all j, where (z_{i}^{∗}) is biorthogonal to (z_{i}), and taking (d_{i}) to be a suitable difference sequence of (y′_{i}). We will assume then that (d_{i}) is in fact a block basis of (z_{i}).

We inductively construct for each limit ordinal β < ω_{1}, a Banach space Y_{β} that embeds into X. Y_{β} will have a normalized bimonotone basis (y_{i}^{β}) that can be enumerated as (y_{i}^{β})_{i=1}^{∞} = {y_{i}^{β,ρ,n} : ρ ∈ C_{β}, n ∈ ℕ, i ∈ ℕ} where C_{β} is some countable set. The order is such that (y_{i}^{β,ρ,n})_{i=1}^{∞} is a subsequence of (y_{i}^{β}) for fixed ρ and n.

THE DIAMETER OF THE ISOMORPHISM CLASS OF A BANACH SPACE 9

Before stating the remaining properties of (y_{i}^{β}) we need some terminology. We say that (w_{i}) is a compatible difference sequence of (y_{i}^{β}) of order 1 if (w_{i}) is a difference sequence of (y_{i}^{β}) that can be enumerated as follows,

(w_{i}) = {w_{i}^{β,ρ,n} : ρ ∈ C_{β}, n, i ∈ ℕ}

and such that for fixed ρ and n,

(w_{i}^{β,ρ,n})_{i} is a difference sequence of (y_{i}^{β,ρ,n+1})_{i} .

If (v_{i}) is a compatible difference sequence of (w_{i}) of order 1, in the above sense, (v_{i}) will be called a compatible difference sequence of (y_{i}^{β}) of order 2, and so on. (y_{i}^{β}) will be said to have order 0.

Let (v_{i}) be a compatible difference sequence of (y_{i}^{β}) of some finite order. We set

T ((v_{i})) = {(u_{i})_{1}^{s} : the u_{i}’s are distinct elements of {v_{i}}_{1}^{∞}, possibly in different order,
and ∥ ∑_{1}^{s} ±u_{i}∥ = 1 for all choices of sign} .

T ((v_{i})) is then an ℓ_{∞}-tree as described above with C = 1. The inductive condition on Y_{β}, or should we say on (y_{i}^{β,ρ,n}), is that for all compatible difference sequences (v_{i}) of (y_{i}^{β,ρ,n}) of finite order,

o(T ((v_{i}))) ≥ β .

Before proceeding we have an elementary but key

Sublemma. Let C < ∞ and let (w_{i}) be a block basis of a bimonotone basis (z_{i}) with 1 ≤ ∥w_{i}∥ ≤ C for all i and let

A = {F ⊆ ℕ : F is finite and ∥ ∑_{i∈F} ±w_{i}∥ ≤ C for all choices of sign}.

Then there exists an equivalent norm | · | on [(w_{i})] so that (w_{i}) is a bimonotone normalized basis such that for all F ∈ A,

| ∑_{F} ±w_{i}| = 1 .

Proof. Define | ∑ a_{i}w_{i}| = ∥(a_{i})∥_{∞} ∨ C^{−1}∥ ∑ a_{i}w_{i}∥.

We begin by constructing Y_{ω}. Let (x_{i}) ⊆ X be a normalized block basis of (z_{i}). For n ∈ ℕ, let | · |_{n} be an equivalent norm on [(x_{i})] given by the sublemma for C = 2^{n}. Thus | ∑_{F} ±x_{i}|_{n} = 1 if |F | ≤ 2^{n}.

Since X is K-elastic, for all n, ([(x_{i})], | · |_{n}) K-embeds into X. We thus obtain for n ∈ ℕ, a sequence (x_{i}^{n})_{i} ⊆ X with 1 ≤ ∥x_{i}^{n}∥ ≤ K for all i and such that 1 ≤ ∥ ∑_{i∈F} ±x_{i}^{n}∥ ≤ K for

10 W. B. JOHNSON AND E. ODELL

all |F | ≤ 2^{n} and all choices of sign. Furthermore (x_{i}^{n})_{i} is K-basic. By standard perturbation and diagonal arguments we may for each n pass to a difference sequence (d_{i}^{n})_{i} of (x_{i}^{n+1})_{i} so that enumerating, (d_{i}) = {d_{i}^{n} : n, i ∈ ℕ} is a block basis of (z_{i}) with 1 ≤ ∥d_{i}∥ ≤ K and with each (d_{i}^{n})_{i} being a subsequence of (d_{i}). We have that for |F | ≤ 2^{n} and all signs,

1 ≤ ∥ ∑_{i∈F} ±d_{i}^{n}∥ ≤ K .

We renorm [(d_{i})] by the sublemma for C = K and let the ensuing space be Y_{ω}. We change the name of (d_{i}) to (y_{i}^{ω}) in this new norm and let (y_{i}^{ω,1,n})_{i} = (d_{i}^{n})_{i}. (y_{i}^{ω}) has the property that if (w_{i}) is a compatible difference sequence of (y_{i}^{ω}) of finite order, then o(T ((w_{i}))) ≥ ω. Indeed, if (w_{i}) = {w_{i}^{ω,1,n} : n ∈ ℕ, i ∈ ℕ} then for |F | ≤ 2^{n}, ∥ ∑_{i∈F} ±w_{i}^{ω,1,n}∥ = 1.

Assume that Y_{β} has been constructed for the limit ordinal β with basis (y_{i}^{β}) = {y_{i}^{β,ρ,n} : p ∈ C_{β}, n, i ∈ ℕ} with the requisite properties above.

Let U : Y_{ω} → X and V : Y_{β} → X be K-embeddings. Since in total we are dealing with a countable set of sequences, namely (y_{i}^{ω,1,n})_{i} for n ∈ ℕ and (y_{i}^{β,ρ,n})_{i} for p ∈ C_{β}, n ∈ ℕ, by diagonalization and perturbation we can find a compatible difference sequence (w_{i}^{ω})_{i} of (y_{i}^{ω})_{i} of order 1 and a compatible difference sequence (w_{i}^{β})_{i} of (y_{i}^{β})_{i} of order 1 so that under a suitable reordering, (d_{i}) = (Uw_{i}^{ω})_{i} ∪ (V w_{i}^{β})_{i} is a block basis of (z_{i}). Moreover each (Uw_{i}^{ω,1,n})_{i} and (V w_{i}^{β,ρ,n})_{i} is a subsequence of (d_{i})_{i}.

Adjoin a new point p_{0} to C_{β} and set C_{β+ω} = C_{β} ∪{ρ_{0}}. Let d_{i}^{β+ω,ρ,n} = V w_{i}^{β,ρ,n} for ρ ∈ C_{β}

and d_{i}^{β+ω,ρ_{0},n} = Uw_{i}^{ω,1,n}. For |F | ≤ 2^{n} and G ⊆ C_{β}×ℕ×ℕ for which ∥ ∑_{(ρ,n,i)∈G} ±w_{i}^{β,ρ,n}∥ = 1,

we have by the triangle inequality that

(∗)  ∥ ∑_{i∈F} ±d_{i}^{β+ω,ρ_{0},n} + ∑_{(ρ,n,i)∈G} ±d_{i}^{β+ω,ρ,n}∥ ≤ 2K .

It follows that if we let (y_{i}^{β+ω}) be the basis (d_{i}^{β+ω}), renormed by the sublemma for C = 2K, that Y_{β+ω} = [(y_{i}^{β+ω})] has the required properties. Indeed (∗) yields that o(T ((y_{i}^{β+ω}))) ≥ β + 2^{n} for n ∈ ℕ and so o(T ((y_{i}^{β+ω}))) ≥ β + ω. Moreover by our induction hypothesis and choice of (d_{i}^{β+ω,p_{0},n}) the analogue of (∗) holds for a compatible difference sequence of order 1 or of any finite order. Thus o(T (z_{i}^{β+ω})) ≥ β + ω for any compatible difference sequence of (y_{i}^{β+ω}) of finite order.

If β is a limit ordinal not of the form α + ω we let U_{α} : Y_{α} → X be a K-embedding for each limit ordinal α < β. We again diagonalize to form compatible difference sequences (w_{i}^{α})_{i} of (y_{i}^{α}) of order 1 for each such α so that (U_{α}w_{i}^{α})_{α,i} is a block basis of (z_{i}) in some order. We let C_{β} be a disjoint union of the C_{α}’s and in the manner above obtain Y_{β}. Again by our induction hypothesis we will have that o(T ((y_{i}^{β}))) ≥ α for all limit ordinals α < β and the same holds for all compatible difference sequences of (y_{i}^{β}) of finite order.

3. Concluding remarks and Problems.

C[0, 1] is, of course, 1-elastic. By virtue of Lemma 4 (or Theorem 1), for all K, C[0, 1] can be renormed to be elastic but not K-elastic. Are there other examples of separable elastic spaces?

Problem 8. Let X be elastic (and separable, say). Does C[0, 1] embed into X?

Using index arguments, we have the following partial result.

Proposition 9. Let X be a separable Banach space, and suppose that Y = ∑ X_{n} is a symmetric decomposition of a space Y into spaces uniformly isomorphic to X. If Y is elastic, then C[0, 1] embeds into X.

In particular, if 1 ≤ p < ∞ and ℓ_{p}(X) is elastic, then C[0, 1] embeds into X.

Proof. Let us first observe that if C[0, 1] embeds into Y, then C[0, 1] embeds into X. Since this is surely well known, we just sketch a proof (which, incidentally, uses only that the decomposition Y = ∑ X_{n} of Y is unconditional): Let P_{n} be the projection from Y onto X_{n} and let W be a subspace of Y which is isomorphic to C[0, 1]. By a theorem of Rosenthal’s [R], it is enough to show that for some n, the adjoint P_{n}_{|W}^{∗} of the restriction of P_{n} to W has nonseparable range. This will be true if there is an m so that S_{m}_{|W}^{∗} has nonseparable range, where S_{m} := ∑_{i=1}^{m} P_{i}. Let Z be a subspace of W which is isomorphic to ℓ_{1}. If no such m exists, then for every m, the restriction of S_{m} to Z is strictly singular (that is, not an isomorphism on any infinite dimensional subspace of Z), and it then follows that Z contains a sequence (x_{n}) of unit vectors which is an arbitrarily small perturbation of a sequence (y_{n}) which is disjointly supported. The sequence (y_{n}), a fortiori (x_{n}), is then easily seen to be equivalent to the unit vector basis of ℓ_{1} and its closed span is complemented in Y since the decomposition is unconditional. It follows that ℓ_{1} is isomorphic to a complemented subspace of C[0, 1], which of course is false.

To complete the proof of Proposition 9, we assume that Y is K-elastic and prove that C[0, 1] embeds into Y. The proof is similar to, but simpler than, the proof of Theorem 7. First we recall the definition of certain canonical trees T_{α} of order α for α < ω_{1} (see e.g. [JO]). These form the frames upon which we will hang our bases. The tree T_{1} is a single node. If T_{α} has been defined, we choose a new node z ∉ T_{α} and set T_{α+1} := T_{α} ∪ {z}, ordered by z < t for all t ∈ T_{α} and with T_{α} preserving its order. If β < ω_{1} is a limit ordinal,

12 W. B. JOHNSON AND E. ODELL

we let T_{β} be the disjoint union of {T_{α} : α < β}. Then if s, t are in T_{β}, we say that s ≤ t if and only if s, t are both in T_{α} for some α < β and s ≤ t in T_{α}.

We shall prove by transfinite induction that if (x_{i})_{i=1}^{∞} is any normalized bimonotone basic sequence and α < ω_{1}, there is a Banach space Y_{α} = Y_{α}(x_{i}) with a normalized bimonotone basis (y_{t}^{α})_{t∈T_{α}} so that if (γ_{i})_{i=1}^{n} is any branch in T_{α} then (y_{γ_{i}}^{α})_{i=1}^{n} is 1-equivalent to (x_{i})_{i=1}^{n} and for all scalars (a_{r})_{t∈T_{α}}, ∥ ∑_{i=1}^{n} a_{γ_{i}}y_{γ_{i}}^{α}∥ ≤ ∥ ∑_{t∈T_{α}} a_{t}y_{t}^{α}∥. Furthermore, each Y_{α} will K-embed into Y. Just as in the proof of Theorem 7, it then follows from index theory that the Banach space spanned by (x_{i})_{i=1}^{∞} embeds into Y.

Fix any normalized bimonotone basic sequence (x_{i})_{i=1}^{∞}. Suppose that β is a limit ordinal and Y_{α} = Y_{α}(x_{i})_{i=1}^{∞} has been defined for all α < β. In view of the hypotheses, the space Y has a symmetric decomposition into spaces uniformly isomorphic Y, which we can index as Y = ∑_{α<β} X_{α}. For each α, there is an isomorphism L_{α} from Y_{α} into X_{α} so that ∥L_{α}∥ = 1 and ∥L_{α}^{−1}∥ is bounded independently of α. We can put an equivalent norm on Y = ∑_{α<β} X_{α} to make each L_{α} an isometry and make the decomposition 1-unconditional (but not necessarily 1-symmetric). Define Y_{β} to be the closed linear span of {L_{α}Y_{α} : α < β} in Y with its new norm. The space Y_{β} has the desired basis indexed by T_{β} and Y_{α} must K-embed into Y with its original norm because Y is K-elastic.

If β = α + 1, we let Y_{β}(x_{i})_{i=1}^{∞} = ℝ ⊕ Y_{α}(x_{i})_{i=2}^{∞} with the norm given by

∥(a, y)∥ := ∥y∥ ∨ sup{∥ax_{1} + ∑_{i=2}^{n} y_{γ_{i}}^{α∗}(y)y_{γ_{i}}^{α}∥ : (γ_{i})_{i=2}^{n} is a branch in T_{α}}.

Again it is clear that Y_{β} must K-embed into Y and that Y_{β} has the desired basis.

Given metric spaces X and Y, the Lipschitz distance d_{L}(X, Y) between them is defined to be the infimum of Lip (f) · Lip (f^{−1}), where the infimum is taken over all biLipschitz mappings f from X onto Y (the infimum of the empty set is ∞). Call a Banach space X Lipschitz K-elastic provided that every isomorph of X has Lipschitz distance at most K to some subset of X, and say that X is Lipschitz elastic if X is Lipschitz K-elastic for some K < ∞. It is interesting to note that the analogue of Theorem 1 for Lipschitz elasticity is false. Indeed, Aharoni [A], [BeLi, Theorem 7.11] proved that every separable metric space has, for each ϵ > 0, Lipschitz distance less than 3 + ϵ to some subset of c_{0}, while James [J], [BeLi, Proposition 13.6] proved that for each ϵ > 0, the space c_{0} is 1 + ϵ-equivalent to a subspace of every isomorph of c_{0}. Consequently, any separable Banach space which contains an isomorphic copy of c_{0} is Lipschitz 3 + ϵ-elastic.

THE DIAMETER OF THE ISOMORPHISM CLASS OF A BANACH SPACE 13

Problem 10. If X is a separable, infinite dimensional Banach space and there is a K so that every isomorph of X is Lipschitz K-elastic, must X contain a subspace isomorphic to c_{0}?

We also do not know whether the Main Theorem has a Lipschitz analogue:

Problem 11. If X is a separable Banach space and there exists K < ∞ so that d_{L}(X, Y) < K for all spaces Y which are isomorphic (or biLipschitz equivalent) to X, then must X be finite dimensional?

The space c_{0} does not have the property mentioned in Problem 11 (use, for example, the fact [GKL] that for every K < ∞ there exists K′ < ∞ so that if a Banach space X is K biLipschitz equivalent to c_{0} then d(X, c_{0}) ≤ K′).

Since Schäffer’s problem remains open in the nonseparable setting, we mention it explicitly as:

Problem 12. Does there exist a nonseparable Banach space X for which D(X) < ∞?

Also open is:

Problem 13. Does there exist a nonseparable Banach space X and a K < ∞ so that all isomorphs of X are K-elastic?

Under the generalized continuum hypothesis, it follows from [E] that for every cardinal ℵ there is a Banach space X of density character ℵ so that every Banach space of density character ℵ is isometrically isomorphic to a subspace of X. These provide the only known examples of elastic Banach spaces.

Problem 14. If X is elastic and infinite dimensional, does X contain an isomorph of every Banach space whose density character is the same as the density character of X?

As was mentioned earlier, the space ℓ_{∞} is, for every ϵ > 0, 1 + ϵ-isomorphic to a subspace of every isomorph of itself (see [P] and [T]). However, ℓ_{∞} is not elastic. To see this, take a biorthogonal system (x_{α}, x_{α}^{∗})_{α<2^{ℵ_{0}}} in ℓ_{∞} with ∥x_{α}∥ = 1 and ∥x_{α}^{∗}∥ bounded. Given a natural number m, renorm ℓ_{∞} by

∥x∥_{m} := max \{\frac{1}{m}∥x∥_{∞}, sup_{|σ|=m} (∑_{α∈σ} |x_{α}^{∗}(x)|^{2})^{1/2}\} .

It is not hard to see that (ℓ_{∞}, ∥ · ∥_{m}) is not better than \sqrt{m}-normed by any countable set of norm one functionals, which implies that d((ℓ_{∞}, ∥ · ∥_{m}), Y) ≥ \sqrt{m} for any subspace Y of ℓ_{∞}. A similar argument shows that ℓ_{∞}(ℵ) is not elastic for any infinite cardinal number ℵ.

References

[A] I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c_{0}, Israel J. Math. 19 (1974), 284–291.

[AGR] S. Argyros, G. Godefroy and H. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the geometry of Banach spaces, vol. 2 (W.B. Johnson and J. Lindenstrauss, eds.) North Holland (2003), 1007–1069.

[AOST] G. Androulakis, E. Odell, Th. Schlumprecht, N. Tomczak-Jaegermann, On the structure of the spreading models of a Banach space, Canadian J. Math., (to appear).

[BL] B. Beauzamy and J.-T. Lapresté, Modeles etales des espaces de Banach, Travaux en Cours, Hermann, Paris, 1984.

[BeLi] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, vol. 1, American Math. Soc. Colloquium Publications 48 (2000).

[B1] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Bel. 32 (1980), 235–249.

[B2] J. Bourgain, On separable Banach spaces, universal for all separable reflexive Banach spaces, Proc. A.M.S. 79 (1980), 241–246.

[E] A. S. Esenin-Vol’pin, On the existence of a universal bicompactum of arbitrary weight, Doklady Akad. Nauk SSSR (N.S.) 68 (1949), 649–652.

[G] E. D. Gluskin, The diameter of the Minkowski compactum is roughly equal to n, Functional Anal. Appl. 15 (1981), 72–73.

[GKL] G. Godefroy, N. Kalton, and G. Lancien, Subspaces of c_{0}(N) and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000), 798–820.

[J] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.

[JO] R. Judd and E. Odell, Concerning the Bourgain ℓ_{1} index of a Banach space, Israel J. Math. 108 (1998), 145–171.

[LP] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249.

[LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Sequence spaces, Springer-Verlag, (1977).

[Mc] R. A. McGuigan, Jr., Near isometry of Banach spaces and the Banach-Mazur distance, Thesis, University of Maryland, College Park (1968).

[MR] B. Maurey and H. P. Rosenthal, Normalized weakly null sequences with no unconditional subsequence, Studia Math., 61 (1977), 77–98.

[O] E. Odell, On Schreier unconditional sequences, Contemp. Math. 144 (1993), 197–201.

[P] J. R. Partington, Subspaces of certain Banach sequence spaces, Bull. London Math. Soc. 13 (1981), 162–166.

[Pe] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228.

[R] H. P. Rosenthal, On factors of C([0, 1]) with non-separable dual, Israel J. Math. 13 (1972), 361–378.

[S] J. J. Schäffer, Geometry of spheres in normed spaces, Lecture notes in pure and applied mathematics 20, Marcel Dekker, Inc. (1976).

[T] M. Talagrand, Sur les espaces de Banach contenant ℓ^{1}(τ), Israel J. Math. 40 (1981), 324–330. [T-J] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman

Monographs and Surveys in Pure and Applied Mathematics 38, Longman (1989).

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

   E-mail address: johnson@math.tamu.edu

Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257

   E-mail address: odell@math.utexas.edu