Chapter 1

RENORMINGS OF ℓ^{1} AND C_{0} AND FIXED POINT PROPERTIES

P. N. Dowling

Department of Mathematics and Statistics Miami University Oxford OH 45056

dowlinpn@muohio.edu

C. J. Lennard

Department of Mathematics and Statistics University of Pittsburgh Pittsburgh, PA 15260

lennard+@pitt.edu

B. Turett

Department of Mathematics and Statistics Oakland University Rochester, MI 48309

turett@oakland.edu

1. Preliminaries

As has been noted in previous chapters, there are many geometric conditions on a Banach space strong enough to imply that the Banach space has the fixed point property. Geometric conditions such as uniform rotundity, uniform smoothness, or normal structure together with reflexivity are sufficient to imply the fixed point property. Each of these conditions also implies (or assumes in the last case) that the Banach space is reflexive.

When one considers the classical Banach spaces, the reflexive ones (L^{p} or ℓ^{p} where 1 < p < ∞) have the fixed point property since each is uniformly rotund. The classical nonreflexive Banach spaces (c_{0}, ℓ^{1}, L^{1}, L^{∞}, and C[0, 1]) fail to have the fixed point property since each of these spaces contains an isometric copy of ℓ^{1} or c_{0} and we have already seen easy examples of isometries on nonempty closed bounded convex subsets of ℓ^{1} or c_{0} without fixed points. Thus it is natural to ask the following questions:

Question 1.1 If a Banach space X has the fixed point property, is X reflexive?

Question 1.2 If a Banach space X is reflexive, does X have the fixed point property?

In considering the first question, whether the fixed point property implies reflexivity, it is tempting to try to construct nonreflexive spaces with the fixed point property. Where however should one look for such spaces? Perhaps the most natural place to look is among the renormings of c_{0} or ℓ^{1} since these are, in a certain sense, the simplest of the nonreflexive Banach spaces. There is another advantage in considering these two classical spaces. It is well-known (Theorems 1.c.12 in [33] and 1.c.5 in [34]) that a Banach lattice or a Banach space with an unconditional basis is reflexive if and only if it contains no isomorphic copies of c_{0} or ℓ^{1}. Thus, if it can be shown that neither c_{0} nor ℓ^{1} can be renormed to have the fixed point property, it would follow that the fixed point property in either a Banach lattice or in a Banach space with an unconditional basis would imply reflexivity. Thus, success in considering the isomorphism classes of these two spaces would have extremely pleasant consequences for several large classes of Banach spaces.

If one attempts to renorm ℓ^{1} and c_{0} to have the fixed point property or if one attempts to show that ℓ^{1} and c_{0} can not be renormed to have the fixed point property, one is interested in knowing how distorted renormings of ℓ^{1} and c_{0} can be? Intuitively, if renormings of ℓ^{1} and c_{0} are not much different from ℓ^{1} and c_{0} with their usual norms, one may have the feeling that perturbations of the usual examples showing that ℓ^{1} and c_{0} fail the fixed point property may show all renormings of ℓ^{1} and c_{0} fail the fixed point property. If renormings of ℓ^{1} and c_{0} can look significantly different than ℓ^{1} and c_{0} with their usual norms, then one may have the feeling that one of these renormings may have the fixed point property. In fact, R. C. James has shown that neither ℓ^{1} nor c_{0} can be distorted (in the sense that any renormings of ℓ^{1} and c_{0} contain almost isometric copies of ℓ^{1} and c_{0} respectively). To be specific, James [29] proved the following theorems.

James’s Distortion Theorems. A Banach space X contains an isomorphic copy of ℓ^{1} if and only if, for every 0 < ε < 1, there exists a sequence (x_{n}) in X such that

(1 − ε) ∑_{n=1}^{∞} |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ ∑_{n=1}^{∞} |t_{n}|

for all (t_{n}) ∈ ℓ^{1}.

A Banach space X contains an isomorphic copy of c_{0} if and only if, for every 0 < ε < 1, there exists a sequence (x_{n}) in X such that

(1 − ε) sup_{n} |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ (1 + ε) sup_{n} |t_{n}|

holds for all (t_{n}) ∈ c_{0}.

In fact, the proofs that James provides shows a bit more.

James’s Distortion Theorems – Stronger versions. A Banach space X contains an isomorphic copy of ℓ^{1} if and only if, for every null sequence (ε_{n}) in (0, 1), there exists a sequence (x_{n}) in X such that

(1 − ε_{k}) ∑_{n=k}^{∞} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ ∑_{n=k}^{∞} |t_{n}|

for all (t_{n}) ∈ ℓ^{1} and for all k ∈ ℕ.

A Banach space X contains an isomorphic copy of c_{0} if and only if, for every null sequence (ε_{n}) in (0, 1), there exists a sequence (x_{n}) in X such that

(1 − ε_{k}) sup_{n≥k} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ (1 + ε_{k}) sup_{n≥k} |t_{n}|

holds for all (t_{n}) ∈ c_{0} and for all k ∈ ℕ.

Renormings of ℓ^{1} and c_{0}and fixed point properties 3

2. Renormings of ℓ^{1} and c_{0} and fixed point properties

To see how James’s theorems can be used to get results concerning fixed points, consider the class of uniformly lipschitzian maps on closed, bounded, convex subsets of X. Fixed point theorems for this class of mappings were considered in [24, 25].

Let X be a Banach space containing an isomorphic copy of ℓ^{1} and consider a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) guaranteed by the stronger version of James’s theorem for ℓ^{1}. Define C = { ∑_{n=1}^{∞} t_{n}x_{n} : 0 ≤ t_{n} and ∑_{n=1}^{∞} t_{n} = 1 } and T (∑_{n=1}^{∞} t_{n}x_{n}) = ∑_{n=1}^{∞} t_{n}x_{n+1}. Let z = ∑_{n=1}^{∞} t_{n}x_{n} and w = ∑_{n=1}^{∞} s_{n}x_{n} be elements of C. Then C is a closed, bounded, convex subset of X and, for any k ∈ ℕ,

||T^{k}z − T^{k}w|| = ∥∑_{n=1}^{∞}(t_{n} − s_{n})x_{n+k}∥

≤ ∑_{n=1}^{∞} |t_{n} − s_{n}|

≤ \frac{1}{1 − ε_{1}} ∥∑_{n=1}^{∞}(t_{n} − s_{n})x_{n}∥

= \frac{1}{1 − ε_{1}} ∥z − w∥ .

Thus T is uniformly lipschitzian and it is easily checked that T has no fixed point. This proves the following:

Proposition 2.1 If X is a Banach space containing an isomorphic copy of ℓ^{1} and ε > 0, then there exists a closed, bounded, convex subset C of X and an affine, uniformly lipschitzian map T : C → C such that the uniform Lipschitz constant of T is less than 1 + ε and T has no fixed point. Consequently, ℓ^{1} cannot be renormed to have the fixed point property for uniformly lipschitzian mappings.

A similar computation using the map defined in the proof of Theorem 2.27 below gives the analogous result for c_{0}: in particular, c_{0} cannot be renormed to have the fixed point property for uniformly lipschitzian mappings. This combines with Proposition 2.1 and the results mentioned in the introductory paragraphs to show:

Proposition 2.2 Let X be a Banach lattice or a Banach space with an unconditional basis. If, for some k > 1, X has the fixed point property for k-uniformly lipschitzian mappings, then X is reflexive.

In studying the fixed point property for the class of nonexpansive self-maps of closed, bounded, convex subsets of a Banach space, it has been found to be useful if the terms (1 ± ε_{k}) in the stronger version of James’s theorems are “inside” the sum or the supremums. Definitions. A Banach space X is said to contain an asymptotically isometric copy of ℓ^{1} if there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that

∑_{n=1}^{∞} (1 − ε_{n}) |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n} ∥ ≤ ∑_{n=1}^{∞} |t_{n}| ,

for all (t_{n}) ∈ ℓ^{1}.

A Banach space X is said to contain an asymptotically isometric copy of c_{0} if there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that

sup_{n} (1 − ε_{n})|t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ sup_{n} |t_{n}| ,

for all (t_{n}) ∈ c_{0}.

It is easy to see that there is no difference in the above definitions if the phrase “there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X” is replaced by the phrase “for every null sequence (ε_{n}) in (0, 1), there exists a sequence (x_{n}) in X.”

These “better” copies of ℓ^{1} or c_{0} can be used to investigate the fixed point property for nonexpansive mappings.

Theorem 2.3 ([14]) If a Banach space X contains an asymptotically isometric copy of ℓ^{1}, then X fails the fixed point property for nonexpansive mappings on closed bounded convex subsets of X.

Proof. By assumption, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X so that

∑_{n=1}^{∞}(1 − ε_{n})|a_{n}| ≤ ∥∑_{n=1}^{∞} a_{n}x_{n}∥ ≤ ∑_{n=1}^{∞} |a_{n}| ,

for all (a_{n}) ∈ ℓ^{1}. Let (λ_{n}) be a strictly decreasing sequence in (1, ∞) with lim_{n→∞} λ_{n} = 1. By passing to subsequences if necessary, we can assume that

λ_{n+1} < (1 − ε_{n})λ_{n} .

Define y_{n} = λ_{n}x_{n}, for all n ∈ ℕ, and let C = \overline{co}\{y_{n} : n ∈ ℕ\}, the closed convex hull of the sequence (y_{n}). Clearly, C is a closed bounded convex subset of X whose elements are of the form z = ∑_{n=1}^{∞} t_{n}y_{n}, where t_{n} ≥ 0 for all n ∈ ℕ and ∑_{n=1}^{∞} t_{n} = 1. Define a mapping T : C → C by T (∑_{n=1}^{∞} t_{n}y_{n}) = ∑_{n=1}^{∞} t_{n}y_{n+1}, where t_{n} ≥ 0 for all n ∈ ℕ and ∑_{n=1}^{∞} t_{n} = 1. It is easily seen that T has no fixed points in C. We now show that T is nonexpansive. (In fact, we show that T is contractive).

Let z = ∑_{n=1}^{∞} t_{n}y_{n} and w = ∑_{n=1}^{∞} s_{n}y_{n} be elements of C with z ≠ w. Then

∥T z − T w∥ = ∥∑_{n=1}^{∞}(t_{n} − s_{n})y_{n+1}∥

≤ ∑_{n=1}^{∞} |t_{n} − s_{n}| ∥y_{n+1}∥

≤ ∑_{n=1}^{∞} |t_{n} − s_{n}|λ_{n+1}

< ∑_{n=1}^{∞} |t_{n} − s_{n}|λ_{n}(1 − ε_{n})

≤ ∥∑_{n=1}^{∞}(t_{n} − s_{n})λ_{n}x_{n}∥

= ∥z − w∥ .

This completes the proof of Theorem 2.3. ■

Similarly we obtain:

Theorem 2.4 ([15]) If a Banach space X contains an asymptotically isometric copy of c_{0}, then X fails the fixed point property for nonexpansive mappings on closed bounded convex subsets of X.

Proof. By assumption there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X so that

sup_{n}(1 − ε_{n})|a_{n}| ≤ ∥∑_{n=1}^{∞} a_{n}x_{n}∥ ≤ sup_{n} |a_{n}| ,

for all (a_{n}) ∈ c_{0}. Let (λ_{n}) be a strictly decreasing sequence in (1, ∞) converging to 1. By passing to subsequences if necessary, we can assume that λ_{n+1} < (1 − ε_{n})λ_{n}.

Define y_{n} = λ_{n}x_{n} for all n ∈ ℕ and

C = {∑_{n=1}^{∞} t_{n}y_{n} : (t_{n}) ∈ c_{0} , 0 ≤ t_{n} ≤ 1 for all n ∈ ℕ} .

C is clearly a closed bounded and convex subset of X. Define T : C → C by

T (∑_{n=1}^{∞} t_{n}y_{n}) = y_{1} + ∑_{n=1}^{∞} t_{n}y_{n+1} for ∑_{n=1}^{∞} t_{n}y_{n} ∈ C .

It is easily seen that T has no fixed points in C. To see that T is nonexpansive (in fact, contractive), let z = ∑_{n=1}^{∞} t_{n}y_{n} and w = ∑_{n=1}^{∞} s_{n}y_{n} be elements of C with z ≠ w.

Then

∥T z − T w∥ = ∥∑_{n=1}^{∞}(t_{n} − s_{n})y_{n+1}∥

= ∥∑_{n=1}^{∞}(t_{n} − s_{n})λ_{n+1}x_{n+1}∥

≤ sup_{n} |t_{n} − s_{n}|λ_{n+1}

< sup_{n} |t_{n} − s_{n}|λ_{n}(1 − ε_{n})

≤ ∥∑_{n=1}^{∞}(t_{n} − s_{n})λ_{n}x_{n}∥

= ∥z − w∥ .

This completes the proof of Theorem 2.4. ■

Given the preceding theorems, it is of interest to determine several things. First, do all renormings of ℓ^{1} or c_{0} contain asymptotically isometric copies of ℓ^{1} or c_{0} respectively? If the answer were “yes”, the preceding results would show that all renormings of ℓ^{1} and c_{0} fail the fixed point property. Second, if the answer to the first question is “no”, we would like to determine which Banach spaces contain asymptotically isometric copies of ℓ^{1} or c_{0} since these Banach spaces would fail to have the fixed point property.

A natural place to expect to find asymptotically isometric copies of ℓ^{1} or c_{0} is among the infinite-dimensional subspaces of ℓ^{1} and c_{0}. There one isn’t disappointed. Part (b) of the following theorem is from [16] .

Theorem 2.5 (a) If Y is a closed infinite-dimensional subspace of (ℓ^{1}, ∥ · ∥_{1}), then Y contains an asymptotically isometric copy of ℓ^{1}.

(b) If Y is a closed infinite-dimensional subspace of (c_{0}, ∥ · ∥_{∞}), then Y contains an asymptotically isometric copy of c_{0}.

Thus, a closed subspace of (ℓ^{1}, ∥ · ∥_{1}) or (c_{0}, ∥ · ∥_{∞}) has the fixed point property if and only if it is finite-dimensional.

Note the last sentence in the statement of Theorem 2.5 could just as well read: a closed subspace of ℓ^{1} or c_{0} has the fixed point property if and only if it is reflexive. This phrasing will occur frequently in this chapter.

Proof of (b). Let (e_{n}) denote the standard unit vector basis in c_{0}. Let Z_{n} denote the closed linear span of (e_{j})_{j≥n} in (c_{0}, ∥ · ∥_{∞}). Since Y is an infinite-dimensional subspace of c_{0}, Y ∩ Z_{n} ≠ {0} for all n ∈ ℕ. Thus we can choose x_{1} ∈ Y ∩ Z_{1} with ∥x_{1}∥_{∞} = 1. We can write x_{1} = ∑_{n=1}^{∞} α_{n}^{1}e_{n}, where α_{n}^{1} ∈ ℝ for all n ∈ ℕ. Let n_{0} = 1. Choose n_{1} > 1 so that sup_{n≥n_{1}} |α_{n}^{1}| < 2^{−2}. Since Y ∩ Z_{n_{1}} ≠ {0}, choose x_{2} ∈ Y ∩ Z_{n_{1}} with ∥x_{2}∥_{∞} = 1. We can write x_{2} = ∑_{n=n_{1}}^{∞} α_{n}^{2}e_{n}, where α_{n}^{2} ∈ ℝ for all n ∈ ℕ. Choose n_{2} > n_{1} so that sup_{n≥n_{2}} |α_{n}^{i}| < 2^{−1} · 2^{−3} for i = 1, 2. Since Y ∩ Z_{n_{2}} ≠ {0}, choose x_{3} ∈ Y ∩ Z_{n_{2}} with ∥x_{3}∥_{∞} = 1. We can write x_{3} = ∑_{n=n_{2}}^{∞} α_{n}^{3}e_{n}, where α_{n}^{3} ∈ ℝ for all n ∈ ℕ. Continuing in this manner, we obtain a strictly increasing sequence (n_{k}) in ℕ and a sequence (x_{k}) in Y with x_{k} = ∑_{n=n_{k−1}}^{∞} α_{n}^{k}e_{n} and sup_{n≥n_{k−1}} |α_{n}^{i}| < (k − 1)^{−1} · 2^{−k−1} for all 1 ≤ i ≤ k − 1 with k ≥ 2. Fix (t_{k})_{k=1}^{∞} ∈ c_{0} and consider the element z = ∑_{k=1}^{∞} t_{k}x_{k} ∈ Y. Then

z = ∑_{k=1}^{∞} t_{k} ∑_{n=n_{k−1}}^{∞} α_{n}^{k}e_{n} = ∑_{k=1}^{∞} ∑_{n=n_{k−1}}^{n_{k}−1} (∑_{i=1}^{k} t_{i}α_{n}^{i}) e_{n} .

Since z ∈ c_{0}, we can also write z = ∑_{n=1}^{∞} z_{n}e_{n}. Hence, for n_{k−1} ≤ n ≤ n_{k} − 1, z_{n} = ∑_{i=1}^{k} t_{i}α_{n}^{i}. Let b_{k} = ∑_{n=n_{k−1}}^{n_{k}−1} α_{n}^{k}e_{n} and note that ∥b_{k}∥_{∞} = 1 for all k ∈ ℕ. Thus, for each k ∈ ℕ, we obtain

∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n} − t_{k}b_{k} ∥_{∞} = ∥ ∑_{n=n_{k−1}}^{n_{k}−1} (∑_{i=1}^{k} t_{i}α_{n}^{i}) e_{n} − t_{k} ∑_{n=n_{k−1}}^{n_{k}−1} α_{n}^{k}e_{n}∥_{∞}

= ∥ ∑_{n=n_{k−1}}^{n_{k}−1} (∑_{i=1}^{k−1} t_{i}α_{n}^{i}) e_{n}∥_{∞}

= max_{n_{k−1}≤n≤n_{k}−1} |∑_{i=1}^{k−1} t_{i}α_{n}^{i}|

≤ (k − 1) ∥t∥_{∞}( max_{1≤i≤k−1^{n_{k−1}≤n≤n_{k}−1}} |α_{n}^{i}|)

≤ (k − 1) ∥t∥_{∞} (\frac{1}{2^{k+1}(k − 1)})

= 2^{−k−1} ∥t∥_{∞}.

Also, for n_{k−1} ≤ n ≤ n_{k} − 1, we have:

|z_{n}| = |∑_{i=1}^{k} t_{i}α_{n}^{i}| ≥ |t_{k}α_{n}^{k}| − |∑_{i=1}^{k−1} t_{i}α_{n}^{i}|

≥ |t_{k}| |α_{n}^{k}| − (k − 1) ∥t∥_{∞} max_{1≤i≤k−1} |α_{n}^{i}|

≥ |t_{k}| |α_{n}^{k}| − (k − 1) ∥t∥_{∞} (\frac{1}{(k − 1)2^{k+1}} )

= |t_{k}| |α_{n}^{k}| − 2^{−k−1}∥t∥_{∞}.

Therefore,

max_{n_{k−1}≤n≤n_{k}−1} |z_{n}| ≥ |t_{k}| − 2^{−k−1}∥t∥_{∞},

and so

∥z∥_{∞} = sup_{k≥1} ( max_{n_{k−1}≤n≤n_{k}−1} |z_{n}|) ≥ sup_{k≥1} ( |t_{k}| − 2^{−k−1}∥t∥_{∞}) ≥ 2^{−1}∥t∥_{∞} .

Thus

∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n} − t_{k}b_{k}∥_{∞} ≤ 2^{−k−1}∥t∥_{∞} ≤ 2^{−k}∥z∥_{∞}

and therefore

| ∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n} ∥_{∞} − |t_{k}|| ≤ 2^{−k} ∥z∥_{∞} .

In particular, for each k ∈ ℕ,

∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n}∥_{∞} − |t_{k}| ≤ 2^{−k} ∥z∥_{∞} .

Choose k_{0} ∈ ℕ so that ∥ ∑_{n=n_{k_{0}−1}}^{n_{k_{0}}−1} z_{n}e_{n}∥_{∞} = ∥z∥_{∞} . Then ∥z∥_{∞} − |t_{k_{0}}| ≤ 2^{−k_{0}}∥z∥_{∞} .

Consequently,

∥z∥_{∞} ≤ \frac{1}{1 − 2^{−k_{0}}} |t_{k_{0}}| ≤ max_{k≥1} (\frac{1}{1 − 2^{−k}} ) |t_{k}|.

We also have that, for all k ∈ ℕ,

|t_{k}| − ∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n}∥_{∞} ≤ 2^{−k} ∥z∥_{∞} .

Hence

|t_{k}| ≤ ∥ ∑_{n=n_{k−1}}^{n_{k}−1} z_{n}e_{n}∥_{∞} + 2^{−k} ∥z∥_{∞} ≤ ∥z∥_{∞} + 2^{−k} ∥z∥_{∞}.

Therefore

sup_{k≥1} (\frac{1}{1 + 2^{−k}} ) |t_{k}| ≤ ∥z∥_{∞} .

Putting the above inequalities together we get that, for all (t_{k})_{k=1}^{∞} ∈ c_{0},

sup_{k≥1} (\frac{1}{1 + 2^{−k}} ) |t_{k}| ≤ ∥∑_{k=1}^{∞} t_{k}x_{k}∥_{∞} ≤ sup_{k≥1} (\frac{1}{1 − 2^{−k}} ) |t_{k}|.

This means that (x_{k}) spans an asymptotically isometric copy of c_{0} in Y.

Since the proof of (a) is similar, this completes the proof of Theorem 2.5. ■

For the next “example”, consider a result of Kadec and Pełczyński [30]: all nonreflexive subspaces of L^{1}[0, 1] contain isomorphic copies of ℓ^{1}. The form of the Kadec-Pełczyński result stated below can be proved by analyzing the proof of the original result (see [10]).

Theorem 2.6 ([14]) If X is a nonreflexive subspace of (L^{1}[0, 1], ∥ · ∥_{1}), then X contains an asymptotically isometric copy of ℓ^{1}. In particular, every nonreflexive subspace of (L^{1}[0, 1], ∥ · ∥_{1}) fails the fixed point property.

Since B. Maurey [35] has shown using ultrapower techniques that each reflexive subspace of L^{1}[0, 1] has the fixed point property, Theorem 2.6 used in tandem with Maurey’s result yields:

Corollary 2.7 Let X be a subspace of (L^{1}[0, 1], ∥ · ∥_{1}). Then X has the fixed point property if and only if X is reflexive.

Before continuing with other examples of spaces that have or fail to have the fixed point property using these methods, let us note that, for better or worse, there do exist renormings of ℓ^{1} which fail to contain asymptotically isometric copies of ℓ^{1} and renormings of c_{0} which fail to contain asymptotically isometric copies of c_{0}. Thus the hope that this method will lead to a proof that all renormings of ℓ^{1} or c_{0} fail to have the fixed point property is forlorn. In spite of this depressing sounding statement, we shall see that the use of asymptotically isometric copies of ℓ^{1} and c_{0} yields rich and interesting dividends.

Example 2.8 Let (γ_{n}) be a sequence in (0, 1) strictly increasing to 1. Define a norm on ℓ^{1} by

|||x||| = sup_{n} γ_{n} ∑_{k=n}^{∞} |ξ_{k}| for all x = (ξ_{n}) ∈ ℓ^{1} .

It is easy to check that ||| · ||| is equivalent to the usual ℓ^{1}-norm. We will show that (ℓ^{1}, ||| · |||) does not contain an asymptotically isometric copy of ℓ^{1}.

Let us assume that (ℓ^{1}, ||| · |||) contains an asymptotically isometric copy of ℓ^{1}. Then there is a null sequence (ε_{n}) in (0, 1) and a ||| · |||-normalized sequence (x_{n}) in ℓ^{1} so that

∑_{j=1}^{∞} (1 − ε_{j}) |t_{j}| ≤ |||∑_{j=1}^{∞} t_{j}x_{j}||| ≤ ∑_{j=1}^{∞} |t_{j}| for all t = (t_{j}) ∈ ℓ^{1} .  (*)

Without loss of generality we can assume that the sequence (x_{n}) is disjointly supported; i.e., that the support of x_{m} is disjoint from the support of x_{n} if m ≠ n. This is possible because, since the closed unit ball of ℓ^{1} is weak-star sequentially compact with respect to the predual c_{0}, by passing to a subsequence, we may suppose that (x_{n}) converges weak-star (and hence pointwise with respect to the canonical basis (e_{n})of ℓ^{1}) to some y ∈ ℓ^{1}. By replacing (x_{n}) by the |||·|||-normalization of the sequence ((x_{2n} −x_{2n−1})/2), we may suppose that y = 0. By the proof of the Bessaga-Pełczyński Theorem [5, 10], we can pass to a subsequence of (x_{n}) which is essentially disjointly supported. Truncating this subsequence appropriately, we obtain a disjointly supported sequence which, when normalized, satisfies (∗). Consequently, we can and do assume that (x_{n}) is disjointly supported.

By passing to subsequences if necessary, we can also assume that ε_{n} < \frac{1}{2n} for all n ∈ ℕ.

Let (m(k))_{k=0}^{∞} be a strictly increasing sequence in ℕ∪{0} with m(0) = 0 and (ξ_{j})_{j=1}^{∞} a sequence of scalars such that, for each k ∈ ℕ,

x_{k} = ∑_{j=m(k−1)+1}^{m(k)} ξ_{j}e_{j} .

For each N ∈ ℕ, we have

N + 1 − ε_{1} − Nε_{N} ≤ |||x_{1} + Nx_{N}|||

= sup_{m(N−1)+1≤i≤m(N)^{1≤j≤m(1)}} {γ_{j} (∑_{k=j}^{m(1)} |ξ_{k}| + N ∑_{k=m(N−1)+1}^{m(N)} |ξ_{k}|) , γ_{i}N ∑_{k=i}^{m(N)} |ξ_{k}|}

≤ sup_{m(N−1)+1≤i≤m(N)^{1≤j≤m(1)}} {γ_{j} ∑_{k=j}^{m(1)} |ξ_{k}| + Nγ_{m(1)} ∑_{k=m(N−1)+1}^{m(N)} |ξ_{k}| , Nγ_{i} ∑_{k=i}^{m(N)} |ξ_{k}|}

= sup_{m(N−1)+1≤i≤m(N)^{1≤j≤m(1)}} \{γ_{j} ∑_{k=j}^{m(1)} |ξ_{k}| + \frac{Nγ_{m(1)}}{γ_{m(N−1)+1}} (γ_{m(N−1)+1} ∑_{k=m(N−1)+1}^{m(N)} |ξ_{k}|) ,

Nγ_{i} ∑_{k=i}^{m(N)} |ξ_{k}|}

≤ max \{1 + \frac{Nγ_{m(1)}}{γ_{m(N−1)+1}} , N\}

Thus N + 1 − ε_{1} − Nε_{N} ≤ max \{1 + \frac{Nγ_{m(1)}}{γ_{m(N−1)+1}} , N\} for all N ∈ ℕ. Since ε_{1} < \frac{1}{2} and

Nε_{N} < \frac{1}{2}, we have N + 1 − ε_{1} − Nε_{n} > N, and hence

N + 1 − ε_{1} − Nε_{N} ≤ 1 + \frac{Nγ_{m(1)}}{γ_{m(N−1)+1}} for all N ∈ ℕ .

Therefore

1 + \frac{1}{N} − \frac{ε_{1}}{N} − ε_{N} ≤ \frac{1}{N} + \frac{γ_{m(1)}}{γ_{m(N−1)+1}} .

Letting N → ∞ yields 1 ≤ γ_{m(1)}, a contradiction since γ_{n} < 1 for all n ∈ ℕ.

In order to see the precise difference between a Banach space containing isomorphic copies of ℓ^{1} and a Banach space containing asymptotically isometric copies of ℓ^{1}, consider the following two theorems. The first is a classic due to A. Pełczyński [36] in the separable case and J. Hagler [28] in the nonseparable case. The second result was recently proven by S. Dilworth, M. Girardi, and J. Hagler.

Theorem 2.9 Let X be a Banach space. The following conditions are equivalent:

(a) X contains an isomorphic copy of ℓ^{1}.

(b) X^{∗} contains an isomorphic copy of L^{1}[0, 1].

(c) X^{∗} contains an isomorphic copy of C[0, 1]^{∗}.

If, in addition, X is separable, (a), (b), and (c) are equivalent to:

(d) X^{∗} contains an isomorphic copy of ℓ^{1}(Γ) for some uncountable set Γ.

Condition (a) in Theorem 2.9 always implies condition (d), but the converse can sometimes fail. For example, if X = c_{0}(Γ) for an uncountable set Γ, condition (d) clearly holds and yet X contains no isomorphic copy of ℓ^{1}. Indeed, if c_{0}(Γ) contains a subspace Y isomorphic to ℓ^{1}, then there would exist a countable subset Γ_{0} of Γ such that Y is a subspace of X = c_{0}(Γ_{0}). Since c_{0}(Γ_{0}) is isometric to c_{0} and each infinite-dimensional subspace of c_{0} contains another subspace isomorphic to c_{0}, this would imply that ℓ^{1} contains a subspace isomorphic to c_{0}, an impossible situation for ℓ^{1}.

Banach spaces containing asymptotically isometric copies of ℓ^{1} are characterized in the next theorem. Other equivalent conditions can be found in [11].

Theorem 2.10 ([11]) Let X be a Banach space. The following conditions are equivalent:

(a) X contains an asymptotically isometric copy of ℓ^{1}.

(b) X^{∗} contains an isometric copy of L^{1}[0, 1].

(c) X^{∗} contains an isometric copy of C[0, 1]^{∗}.

It is possible to use Theorem 2.10 to construct renormings of ℓ^{1} other than the one in Example 10 which contain no asymptotically isometric copy of ℓ^{1}. For example, see Corollary 12 in [11] or take any renorming of ℓ^{1} such that its dual space is a rotund renorming of ℓ^{∞} (as in Example 2 in [44]). In a similar vein, Dowling [22] has shown that a dual space X^{∗} contains an asymptotically isometric copy of c_{0} if and only if X^{∗} contains an isometric copy of ℓ^{∞}. Thus any rotund dual renorming of ℓ^{∞} fails to contain an asymptotically isometric copy of c_{0}. Specific renormings of ℓ^{1} which contain no asymptotically isometric copy of ℓ^{1} or specific renormings of c_{0} which contain no asymptotically isometric copy of c_{0} can be found in [13, 16].

Theorem 2.10 and Theorem 2.3 combine to show:

Corollary 2.11 If a Banach space X contains an asymptotically isometric copy of ℓ^{1}, then neither X nor X^{∗} has the fixed point property.

In fact, as we shall see in Corollary 2.15 below, a much stronger statement can be made.

Moreover note that Theorem 2.10 combines with Alspach’s example [1] to show that if a Banach space X contains an asymptotically isometric copy of ℓ^{1}, then X^{∗} also fails to have the weak fixed point property.

In order to give a proof of Theorem 2.10, we need some equivalent reformulations of the definitions of a Banach space containing an asymptotically isometric copy of ℓ^{1} or an asymptotically isometric copy of c_{0}. These reformulations will give us alternative ways to recognize when a Banach space contains asymptotically isometric copies of ℓ^{1} or c_{0} will prove extremely useful in many of the results below.

Theorem 2.12 Let X be a Banach space. The following conditions are equivalent:

(a) X contains an asymptotically isometric copy of ℓ^{1}.

(b) There is a sequence (x_{n}) in X and constants 0 < m ≤ M < ∞ such that, for all (t_{n}) ∈ ℓ^{1} ,

m ∑_{n=1}^{∞} |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ M ∑_{n=1}^{∞} |t_{n}|

and

lim_{n→∞} ∥x_{n}∥ = m .

(c) There is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that, for all (t_{n}) ∈ ℓ^{1} and for all k ∈ ℕ,

∑_{n=k}^{∞} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ (1 + ε_{k}) ∑_{n=k}^{∞} |t_{n}| .

Proof. Suppose condition (a) holds. Then there is a null sequence (ε_{n}) in (0, 1) and a sequence (y_{n}) in X so that

∑_{n=1}^{∞} (1 − ε_{n})|t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}y_{n}∥ ≤ ∑_{n=1}^{∞} |t_{n}| ;

for all (t_{n}) ∈ ℓ^{1}. Let x_{n} = (1 − ε_{n})^{−1}y_{n} for each n ∈ ℕ. Then, for all (t_{n}) ∈ ℓ^{1},

∑_{n=1}^{∞} |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ ∑_{n=1}^{∞} (1 − ε_{n})^{−1}| t_{n}| ≤ (1 − ε_{1})^{−1} ∑_{n=1}^{∞} |t_{n}| .

Also, since 1 ≤ ∥x_{n}∥ ≤ (1 − ε_{n})^{−1} , lim_{n→∞} ∥x_{n}∥ = 1. Thus conditions (b) holds.

Conversely, suppose that condition (b) holds. Fix a null sequence (ε_{n}) in (0, 1). By scaling if necessary, we can assume that m = 1. Since lim_{n→∞} ∥x_{n}∥ = m = 1, and ∥x_{n}∥ ≥ m = 1 for all n ∈ ℕ, by passing to subsequences, if necessary, we can assume that 1 ≤ ∥x_{n}∥ ≤ 1 + ε_{n} for all n ∈ ℕ. Define y_{n} = (1 + ε_{n})^{−1}x_{n} for all n ∈ ℕ. Then, since ∥y_{n}∥ ≤ 1, we have

∥∑_{n=1}^{∞} t_{n}y_{n}∥ ≤ ∑_{n=1}^{∞} |t_{n}| for all (t_{n}) ∈ ℓ^{1} .

Also

∥∑_{n=1}^{∞} t_{n}y_{n}∥ = ∥∑_{n=1}^{∞} t_{n}(1 + ε_{n})^{−1}x_{n}∥

≥ ∑_{n=1}^{∞} (1 + ε_{n})^{−1}|t_{n}|

≥ ∑_{n=1}^{∞} (1 − ε_{n})|t_{n}| .

Thus X contains an asymptotically isometric copy of ℓ^{1} so that (b) implies (a).

Since the equivalence of (b) and (c) is immediate, this completes the proof of Theorem 2.12. ■

Similarly:

Theorem 2.13 ([16]) Let X be a Banach space. The following conditions are equivalent:

(a) X contains an asymptotically isometric copy of c_{0}.

(b) There is a sequence (x_{n}) in X and constants 0 < m ≤ M < ∞ such that, for all (t_{n}) ∈ c_{0} ,

m sup_{n} |t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ M sup_{n} |t_{n}|

and

lim_{n→∞} ∥x_{n}∥ = M .

(c) There is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that for all (t_{n}) ∈ c_{0} and for all k ∈ ℕ,

(1 − ε_{k}) sup_{n≥k} |t_{n}| ≤ ∥∑_{n≥k} t_{n}x_{n}∥ ≤ sup_{n≥k} |t_{n}| .

Proof. As in the preceding theorem, we shall show the equivalence of (a) and (b). Their equivalence to condition (c) is then immediate.

Suppose that condition (a) holds. Then there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that

sup_{n} (1 − ε_{n}) |t_{n}| ≤ ∥∑_{n} t_{n}x_{n}∥ ≤ sup_{n} |t_{n}|

for all (t_{n}) ∈ c_{0} . Let m = inf_{n}(1 − ε_{n}). Then 0 < m < 1 and, for all (t_{n}) ∈ c_{0},

m sup_{n} |t_{n}| ≤ ∥∑_{n} t_{n}x_{n}∥ ≤ sup_{n} |t_{n}|.

Also, since 1 − ε_{n} ≤ ∥x_{n}∥ ≤ 1, we have lim_{n→∞} ∥x_{n}∥ = 1.

Conversely, suppose that condition (b) holds and let (ε_{n}) be a null sequence in (0, 1). By considering x_{n}/M rather than x_{n}, we can assume that M = 1. In particular, m ≤ ∥x_{n}∥ ≤ 1 for all n ∈ ℕ and lim_{n→∞} ∥x_{n}∥ = 1. Hence, by passing to subsequences if necessary, we can assume that 1 − ε_{n} ≤ ∥x_{n}∥ ≤ 1, for all n ∈ ℕ.

By passing to subsequences again, if necessary, we can assume that ε_{n} < m/4 for all n ∈ ℕ. Define δ_{1} = m and δ_{n} = (4/m)ε_{n} for all n ≥ 2.

Consider the expression ∥ ∑_{k=1}^{n} t_{k}x_{k}∥ for scalars t_{1}, t_{2}, …, t_{n}. By assumption we have

m max_{1≤k≤n} |t_{k}| ≤ ∥∑_{k=1}^{n} t_{k}x_{k}∥ ≤ max_{1≤k≤n} |t_{k}|.

By scaling, assume that max_{1≤k≤n} |t_{k}| = 1. Then

m ≤ ∥∑_{k=1}^{n} t_{k}x_{k}∥ ≤ 1.

To show that X contains an asymptotically isometric copy of c_{0}, it suffices to show that

max_{1≤k≤n} (1 − δ_{k})|t_{k}| ≤ ∥∑_{k=1}^{n} t_{k}x_{k}∥ ≤ 1.

Since we already have the right-hand inequality, it remains only to show the left-hand inequality. First, note that if |t_{j}| < m, then

(1 − δ_{j})|t_{j}| ≤ (1 − ε_{j})|t_{j}| < m ≤ ∥∑_{k=1}^{n} t_{k}x_{k}∥.

Secondly, if |t_{j}| ≥ m, choose c_{j} with c_{j}t_{j} = |t_{j}|. By convexity we have

1 − ε_{j} ≤ ∥x_{j}∥

≤ \frac{1}{2} ∥x_{j} + ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥ + \frac{1}{2} ∥x_{j} − ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥

≤ \frac{1}{2} ∥x_{j} + ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥ + \frac{1}{2}.

Hence ∥x_{j} + ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥ ≥ 1 − 2ε_{j}. By convexity again we have

1 − 2ε_{j} ≤ ∥x_{j} + ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥

≤ \frac{1}{2} ∥∑_{k=1}^{n} c_{j}t_{k}x_{k}∥ + \frac{1}{2} ∥(2 − c_{j}t_{j})x_{j} + ∑_{1≤k≤n,k≠j} c_{j}t_{k}x_{k}∥

≤ \frac{1}{2} ∥∑_{k=1}^{n} t_{k}x_{k}∥ + \frac{1}{2} (2 − |t_{j}|) .

Thus ∥∑_{k=1}^{n} t_{k}x_{k}∥ ≥ |t_{j}| − 4ε_{j}. Therefore, since |t_{j}| ≥ m, we have

∥ ∑_{k=1}^{n} t_{k}x_{k} ∥ ≥ |t_{j}| − 4ε_{j} ≥ |t_{j}| − (4/m)|t_{j}|ε_{j} = (1 − δ_{j})|t_{j}|.

Putting the two pieces together we have

∥∑_{j=1}^{n} t_{j}x_{j}∥ ≥ max_{1≤j≤n}(1 − δ_{j})|t_{j}| .

Thus (b) implies (a) and the proof of Theorem 2.13 is complete. ■

We can now give a proof of Theorem 2.10. The proof given below is for real Banach spaces. The theorem also holds for complex Banach spaces and the proof of (b) implies (a) in the complex case may be found in [11].

Proof of Theorem 2.10. Suppose that (a) holds. Then, by Theorem 2.12, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that, for all (t_{n}) ∈ ℓ^{1} and for all k ∈ ℕ,

∑_{n=k}^{∞} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ (1 + ε_{k}) ∑_{n=k}^{∞} |t_{n}| .

Let Y = \overline{span}(x_{n}) and let (z_{n}) be a dense sequence in the unit sphere of C[0, 1]. Define a linear operator S from Y into C[0, 1] so that S(x_{n}) = z_{n} for each n ∈ ℕ. Using the left-hand inequality above, it is easy to check that ∥S∥ = 1. It is also easy to check that S maps Y onto C[0, 1]. (This is similar to the proof that every separable Banach space is a quotient of ℓ^{1}; see e.g., [10] p.73.) Identifying C[0, 1] with its canonical image in C[0, 1]^{∗∗} and using the fact that C[0, 1]^{∗∗} has the Hahn-Banach extension property (i.e., C[0, 1]^{∗∗} is a P_{∞}-space), there exists a linear mapping T : X → C[0, 1]^{∗∗} such that ∥T ∥ = ∥S∥ = 1. The adjoint T^{∗} maps C[0, 1]^{∗∗∗} into X^{∗} and ∥T^{∗}∥ = 1. Identifying C[0, 1]^{∗} with its canonical image in C[0, 1]^{∗∗∗}, if μ ∈ C[0, 1]^{∗}, then

∥T^{∗}(μ)∥ = sup{μ(T x) : x ∈ X, ∥x∥ = 1}
≥ sup{μ(T y) : y ∈ Y, ∥y∥ = 1}
= sup{μ(Sy) : y ∈ Y, ∥y∥ = 1}
= sup{μ(f) : f ∈ C[0, 1], ∥f∥ = 1}
= ∥μ∥ .

Combined with ∥T^{∗}∥ = 1, it follows that ∥T^{∗}(μ)∥ = ∥μ∥. Thus T^{∗} maps C[0, 1]^{∗} isometrically into X^{∗}.

Since (c) implies (b) is obvious, we need only show that (b) implies (a) in order to complete the proof. So assume that T : L^{1}[0, 1] → X^{∗} is a linear isometry. Then T^{∗} : X^{∗∗} → L^{∞}[0, 1] is a weak^{∗}-weak^{∗} continuous, norm-one map onto L^{∞}[0, 1]. By Goldstine’s Theorem, W = T^{∗}(B_{X}) is dense in the unit ball B_{L^{∞}[0,1]} of L^{∞}[0, 1].

The idea is now easy to explain. Since the span of the Rademacher functions (r_{n}) in L^{∞}[0, 1] is easily seen to be isometric to ℓ^{1}, we shall get a sequence (T^{∗}(x_{n})) in W which mimics the action of the Rademacher functions closely enough so that the closed span of the sequence (T^{∗}(x_{n})) is an asymptotically isometric copy of ℓ^{1} in L^{∞}[0, 1]. It will then be easy to check that the closed span of (x_{n}) is an asymptotically isometric copy of ℓ^{1} in X.

Let (ε_{n}) be a null sequence in (0, 1). Consider the weak^{∗} neighborhood N_{∞} in L^{∞}[0, 1] about the first Rademacher function

r_{1} = χ_{[0, \frac{1}{2}]} − χ_{(\frac{1}{2} ,1]}

determined by the functions χ_{[0, \frac{1}{2} ]} and χ_{( \frac{1}{2} ,1]} in L^{1}[0, 1] and the number ε_{1}/2. Since W is dense in B_{L^{∞}[0,1]}, there exists f_{1} ∈ W ∩ N_{∞}. It is easy to check that both A_{1}^{1} = \{t ∈ [0, \frac{1}{2}] : 1 − ε_{1} < f_{1}(t) ≤ 1\} and A_{2}^{1} = \{t ∈ (\frac{1}{2}, 1] : −1 ≤ f_{1}(t) < −1 + ε_{1}\} have positive measure. Thus there exist pairwise disjoint sets D_{j}^{1} where j = 1, 2, 3, 4 such that each D_{j}^{1} has positive measure and D_{1}^{1} ∪ D_{2}^{1} = A_{1}^{1} and D_{3}^{1} ∪ D_{4}^{1} = A_{2}^{1}.

Consider next the “Rademacher-like” function

r̃_{2} = χ_{D_{1}^{1}} − χ_{D_{2}^{1}} + χ_{D_{3}^{1}} − χ_{D_{4}^{1}}.

Let N_{∈} be the weak^{∗} neighborhood in L^{∞}[0, 1] about r̃_{2} determined by the L^{1}-functions χ_{D_{1}^{1}}, χ_{D_{2}^{1}}, χ_{D_{3}^{1}}, and χ_{D_{4}^{1}} and the number δ_{2} = min{ε_{2} · μ(D_{j}^{1})/2 : j = 1, 2, 3, 4}. Again the density of W in B_{L^{∞}[0,1]} implies that there exists a function f_{2} ∈ W ∩ N_{∈}. As before, it is then easy to check that the sets

A_{1}^{2} = {t ∈ A_{1}^{1} : 1 − ε_{2} < f_{2}(t) ≤ 1} ∩ D_{1}^{1}
A_{2}^{2} = {t ∈ A_{1}^{1} : −1 ≤ f_{2}(t) < −1 + ε_{2}} ∩ D_{2}^{1}
A_{3}^{2} = {t ∈ A_{2}^{1} : 1 − ε_{2} < f_{2}(t) ≤ 1} ∩ D_{3}^{1}
A_{4}^{2} = {t ∈ A_{2}^{1} : −1 ≤ f_{2}(t) < −1 + ε_{2}} ∩ D_{4}^{1}

are pairwise disjoint sets with positive measure.

Continuing as above, we obtain subsets

{A_{k}^{n}}_{k=1,··· ,2^{n}^{n∈ℕ}}

of [0, 1] with positiive measure and functions (f_{n}) in W such that

A_{2k−1}^{n+1} ∪ A_{2k}^{n+1} ⊂ A_{k}^{n}
A_{2k−1}^{n+1} ∩ A_{2k}^{n+1} = ∅
1 − ε_{n} < f_{n}(t) ≤ 1  if t ∈ A_{k}^{n}, k odd
−1 ≤ f_{n}(t) < −1 + ε_{n}  if t ∈ A_{k}^{n}, k even

Let x_{n} ∈ B_{X} such that T^{∗}(x_{n}) = f_{n}. For each finite sequence (α_{n})_{n=1,··· ,N}, choose t ∈ A_{j_{1}}^{1} ∩ A_{j_{2}}^{2} ∩ ⋯ ∩ A_{j_{N}}^{N} where j_{k} is odd if α_{k} ≥ 0 and j_{k} is even if α_{k} < 0. Then, since ∥T^{∗}∥ = 1,

∑_{n=1}^{N} |α_{n}| ≥ ∥∑_{n=1}^{N} α_{n}x_{n}∥ ≥ ∥T^{∗}(∑_{n=1}^{N} α_{n}x_{n})∥

= ∥∑_{n=1}^{N} α_{n}f_{n}∥

≥ |∑_{n=1}^{N} α_{n}f_{n}(t)|

= ∑_{n=1}^{N} α_{n}f_{n}(t)

= ∑_{n=1}^{N} |α_{n}||f_{n}(t)|

≥ ∑_{n=1}^{N} |α_{n}|(1 − ε_{n})

Thus (x_{n}) spans an asymptotically isometric copy of ℓ^{1} in X. This completes the proof of Theorem 2.10. ■

The reformulations of the definitions of a Banach space having an asymptotically isometric copy of ℓ^{1} or c_{0} are also useful in identifying other Banach spaces which fail to have the fixed point property. In fact, as we shall see momentarily, several “large” Banach spaces can not even be renormed to have the fixed point property.

Theorem 2.14 ([15, 16]) Let Γ be an uncountable set. Then the following hold:

(a) Every renorming of ℓ^{1}(Γ) contains an asymptotically isometric copy of ℓ^{1}.

(b) Every renorming of c_{0}(Γ) contains an asymptotically isometric copy of c_{0}.

Consequently, if Γ is uncountable, neither ℓ^{1}(Γ) nor c_{0}(Γ) can be renormed to have the fixed point property.

Proof. Let e_{γ} be the element in ℓ^{1}(Γ) with e_{γ}(γ) = 1 and e_{γ}(α) = 0 if α ≠ γ. Let ||| · ||| be an equivalent norm on ℓ^{1}(Γ). Then there exists constants 0 < m ≤ M < ∞ such that

m ∑_{γ∈F} |a_{γ}| ≤ |||∑_{γ∈F} a_{γ}e_{γ}||| ≤ M ∑_{γ∈F} |a_{γ}| ,

for all finite subsets F of Γ and for all scalars a_{γ}, γ ∈ F. Define

m_{A} = inf {|||∑_{γ∈F} a_{γ}e_{γ}||| : ∑_{γ∈F} |a_{γ}| = 1, F is a finite subset of A}

where A is an uncountable subset of Γ. Note that m ≤ m_{A} ≤ M for all uncountable subsets of A of Γ and m_{A} increases as A decreases. Let ω_{1} be the first uncountable ordinal and let (A_{α})_{α<ω_{1}} be a decreasing chain of uncountable subsets of Γ with ∩_{α<ω_{1}}A_{α} = ∅. Then (m_{A_{α}})_{α<ω_{1}} is a nondecreasing transfinite sequence of real numbers and hence eventually constant. Thus there exists α_{0} such that if α ≥ α_{0}, then m_{A_{α}} = m_{A_{α_{0}}} ≡ m_{0}.

Consider A_{α_{0}}. There exists n_{1} ∈ ℕ, and real numbers a_{j}^{1} and elements γ_{j}^{1} ∈ A_{α_{0}} for j = 1, ..., n_{1} such that

∑_{j=1}^{n_{1}} |a_{j}^{1}| = 1 and m_{0} ≤ |||∑_{j=1}^{n_{1}} a_{j}^{1}e_{γ_{j}^{1}} ||| ≤ m_{0} + 2^{−1}.

Since ∩_{α<ω_{1}}A_{α} = ∅, there exists α_{1} ≥ α_{0} such that γ_{j}^{1} ∉ A_{α_{1}}, for j = 1, ..., n_{1}. Since m_{A_{α_{1}}} = m_{0}, there exists n_{2} ∈ ℕ, and real numbers a_{j}^{2} and elements γ_{j}^{2} ∈ A_{α_{1}} for j = 1, ..., n_{2} such that

∑_{j=1}^{n_{2}} |a_{j}^{2}| = 1 and m_{0} ≤ |||∑_{j=1}^{n_{2}} a_{j}^{2}e_{γ_{j}^{2}} ||| ≤ m_{0} + 2^{−2}.

Continuing in this manner we obtain a block basic sequence (x_{k}) of (e_{γ}) where

x_{k} = ∑_{j=1}^{n_{k}} a_{j}^{k}e_{γ_{j}^{k}} and m_{0} ≤ |||x_{k}||| ≤ m_{0} + 2^{−k}.

Then, for all scalars a_{1}, ..., a_{n}, we have

m_{0} ∑_{k=1}^{n} |a_{k}| ≤ |||∑_{k=1}^{n} a_{k}x_{k} ||| ≤ M ∑_{k=1}^{n} |a_{k}|

and lim_{n→∞} |||x_{k}||| = m_{0}. Hence, by Theorem 2.11, (ℓ^{1}(Γ), ||| · |||) contain an asymptotically isometric copy of ℓ^{1}. This completes the proof of (a).

Since the proof of (b) is analogous using Theorem 2.13, we leave the rest of the proof to the reader. ■

One application of the above theorem is to improve Corollary 2.11 which stated that the dual of a Banach space containing an asymptotically isometric copy of ℓ^{1} fails the fixed point property. In fact, much more can be said. Note the weakened hypothesis and the stronger conclusion in the next corollary.

Corollary 2.15 If X is a Banach space containing an isomorphic copy of ℓ^{1}, then X^{∗} cannot be renormed to have the fixed point property. In particular, ℓ^{∞} cannot be renormed to have the fixed point property.

Proof. Following the statement of the result of Pełczyński and Hagler (Theorem 2.9), we noted that if a Banach space X contains an isomorphic copy of ℓ^{1}, then X^{∗} contains an isomorphic copy of ℓ^{1}(Γ) for some uncountable set Γ independent of the separability of X. This and an application of Theorem 2.14 completes the proof. ■

Before continuing with further applications of these theorems, we pause for a somewhat unexpected example. As just noted, every renorming of ℓ^{∞} contains an asymptotically isometric copy of ℓ^{1}. One might expect that every renorming of ℓ^{∞} would also contain an asymptotically isometric copy of c_{0}. This turns out to be false.

Example 2.16 (ℓ^{∞}, ||| · |||) does not contain an asymptotically isometric copy of c_{0}, where the ||| · ||| norm is defined on ℓ^{∞} as follows: if x = (ξ_{j})_{j=1}^{∞} ∈ ℓ^{∞} then

|||x||| = sup_{j} |ξ_{j}| + ∑_{j=1}^{∞} 2^{−j}|ξ_{j}|.

Note that |||·||| is an equivalent norm on ℓ^{∞} and ||x||_{∞} ≤ |||x||| ≤ 2||x||_{∞} for all x ∈ ℓ^{∞}.

Proof. In order to obtain a contradiction, assume that (ℓ^{∞}, ||| · |||) does contain an asymptotically isometric copy of c_{0}. That is, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in ℓ^{∞} such that

max_{1≤j≤n}(1 − ε_{j})|t_{j}| ≤ |||∑_{j=1}^{n} t_{j}x_{j}||| ≤ max_{1≤j≤n} |t_{j}| ,

for all scalars t_{1}, t_{2}, …, t_{n} and for all n ∈ ℕ.

Without loss of generality, we can assume that the sequence (x_{n}) converges pointwise to 0. For each n ∈ ℕ, let x_{n} = (ξ_{j}^{n})_{j=1}^{∞}. Since |||x_{1}||| ≥ 1 − ε_{1} > 0, there exists j ∈ ℕ such that ξ_{j}^{1} ≠ 0. Let k = min\{j : ξ_{j}^{1} ≠ 0\} and α = \frac{1}{3·2^{k}} |ξ_{k}^{1}|. Choose N_{1} ≥ k so that ∑_{j=N_{1}+1}^{∞} 2^{−j} < \frac{α}{4}. Choose N_{2} ∈ ℕ so that ε_{n} < α, for all n ≥ N_{2}. Since (x_{n}) converges pointwise to 0, choose N ≥ N_{2} such that |ξ_{j}^{n}| < \frac{α}{4} for j = 1, 2, …, N_{1} and for all n ≥ N. Hence, for each n ≥ N,

∥x_{n}∥_{∞} ≤ |||x_{n}|||

= ∥x_{n}∥_{∞} + ∑_{j=1}^{∞} 2^{−j}|ξ_{j}^{n}|

= ∥x_{n}∥_{∞} + ∑_{j=1}^{N_{1}} 2^{−j}|ξ_{j}^{n}| + ∑_{j=N_{1}+1}^{∞} 2^{−j}|ξ_{j}^{n}|

≤ ∥x_{n}∥_{∞} + ∑_{j=1}^{N_{1}} 2^{−j} \frac{α}{4} + ∑_{j=N_{1}+1}^{∞} 2^{−j}

≤ ∥x_{n}∥_{∞} + \frac{α}{2}

By the convexity of ∥ · ∥_{∞}, we have ∥x_{n}∥_{∞} ≤ \frac{1}{2} (∥x_{1} + x_{n}∥_{∞} + ∥x_{1} − x_{n}∥_{∞}). Thus, either ∥x_{1} + x_{n}∥_{∞} ≥ ∥x_{n}∥_{∞} or ∥x_{1} − x_{n}∥_{∞} ≥ ∥x_{n}∥_{∞}. If ∥x_{1} + x_{n}∥_{∞} ≥ ∥x_{n}∥_{∞}, it follows that:

1 ≥ |||x_{1} + x_{n}|||

= ∥x_{1} + x_{n}∥_{∞} + ∑_{j=1}^{∞} 2^{−j}|ξ_{j}^{1} + ξ_{j}^{n}|

≥ ∥x_{n}∥_{∞} + 2^{−k} | ξ_{k}^{1} + ξ_{k}^{n} |

≥ |||x_{n}||| − \frac{α}{2} + 2^{−k} ( |ξ_{k}^{1}| − |ξ_{k}^{n}|)

≥ |||x_{n}||| − \frac{α}{2} + 2^{−k} ( |ξ_{k}^{1}| − \frac{α}{4} )

≥ |||x_{n}||| − α + 2^{−k} |ξ_{k}^{1}|

≥ 1 − ε_{n} − α + 2^{−k} |ξ_{k}^{1}|

≥ 1 − α − α + 2^{−k} |ξ_{k}^{1}|

= 1 + α,

which is clearly impossible.

A similar contradiction occurs if we assume that ∥x_{1} − x_{n}∥_{∞} ≥ ∥x_{n}∥_{∞}. This completes the proof that (ℓ^{∞}, ||| · |||) contains no asymptotically isometric copy of c_{0}. ■

The usefulness of the above results in relating the fixed point property with reflexivity can further be illustrated in the context of Orlicz spaces. The basic definitions and facts about Orlicz spaces can be found in [31] or [41]. For now, let us recall a few specific results that we shall need. Hopefully these few comments will make it possible to follow the proof of Theorem 2.17 even for those readers unfamiliar with Orlicz spaces.

First, if (Ω, Σ, μ) is a finite nonatomic measure space, the Orlicz space L^{Φ}(μ) is reflexive if and only if the Young function Φ and its conjugate function Ψ both satisfy a Δ_{2}-condition for large values. (This is a result due to W. A. J. Luxemburg ([32],[41]).) Recall also that an Orlicz space L^{Φ}(μ) can be endowed with either of two equivalent norms: the Luxemburg norm ∥f∥_{Φ} = inf{r > 0 : ∫ Φ(f/r) dμ ≤ 1 } or the Orlicz norm N_{Φ}(·) = sup{∫ fg dμ : ∫ Ψ(g)dμ ≤ 1}. (Please note that notation in Orlicz spaces is not standard and our notation is the opposite of the notation in [41].) We shall also use that, if the measure space is finite and not purely atomic, the Young function Φ satisfies a Δ_{2}-condition for large values of its argument if and only if the Orlicz space L^{Φ}(μ) endowed with the Luxemburg norm contains no isometric copy of ℓ^{∞} ([45]). Since the exact nature of the isometry will prove useful in the following, let us briefly describe how ℓ^{∞} embeds in (L^{Φ}(μ), ∥·∥_{Φ}) when Φ fails to satisfy a Δ_{2}-condition for large values. In this setting, one can construct a sequence of norm-one functions (g_{n}) such that the g_{n}’s have disjoint support, ∫ Φ(g_{n}) dμ decreases to 0, and ∑_{n=1}^{∞} g_{n} is also norm-one. The map which sends a bounded sequence (α_{n}) to ∑ α_{n}g_{n} is then an isometry of ℓ^{∞} into L^{Φ}(μ). Finally, it is useful to note that ∥f∥_{Φ} ≤ N_{Φ}(f) for every f ∈ L^{Φ}(μ), and that if f ≠ 0, the inequality is strict. ([31], p. 78, [41], p. 73).

A connection between reflexivity and the fixed point property in Orlicz spaces can now be stated.

Theorem 2.17 Let (Ω, Σ, μ) be a finite measure space that is not purely atomic. The Orlicz space L^{Φ}(μ) endowed with the Orlicz norm has the fixed point property if and only if it is reflexive.

Proof of necessity. Suppose that (L^{Φ}(μ), N_{Φ}(·)) has the fixed point property but that L^{Φ}(μ) is not reflexive. Then, by Luxemburg’s result mentioned above, either the Young function Φ or its conjugate function Ψ fails to satisfy a Δ_{2}-condition for large values.

Suppose Φ fails to satisfy the Δ_{2}-condition for large values. Then ℓ^{∞} embeds isometrically into L^{Φ}(μ) with the Luxemburg norm and, since the Luxemburg and the Orlicz norms are equivalent, ℓ^{∞} embeds isomorphically into L^{Φ}(μ) with the Orlicz norm. But this would give a renorming of ℓ^{∞} with the fixed point property, a contradiction to Corollary 2.15.

The only other possibility is that Ψ fails to satisfy a Δ_{2}-condition for large values while Φ satisfies a Δ_{2}-condition for large values . Let (g_{n}) denote the functions in (L^{Ψ}, ∥ · ∥_{Ψ}) = (L^{Φ}(μ), N_{Φ}(·) )^{∗} giving rise to the isometry of ℓ^{∞} in (L^{Ψ}, || · ||_{Ψ}). Choose functions (f_{n}) in L^{Φ}(μ) such that N_{Φ}(f_{n}) = 1, the support of f_{n} is contained in the support of g_{n}, and ∫ g_{n}f_{n} dμ > 1 − ϵ_{n} where (ϵ_{n}) is a sequence decreasing to 0. Then, since ∥ ∑_{n}(sgn α_{n})g_{n}∥_{Ψ} = 1,

N_{Φ} (∑_{n} α_{n}f_{n}) ≥ ∫ (∑_{n}(sgn α_{n})g_{n}) (∑_{n} α_{n}f_{n}) dμ

= ∑_{n} |α_{n}| ∫ g_{n}f_{n} dμ

≥ ∑_{n}(1 − ϵ_{n})|α_{n}|.

Thus, since N_{Φ}(f_{n}) = 1,

∑_{n}(1 − ϵ_{n})|α_{n}| ≤ N_{Φ} (∑_{n} α_{n}f_{n}) ≤ ∑_{n} |α_{n}|.

Thus, if Ψ fails the Δ_{2}-condition for large values, (L^{Φ}(μ), N_{Φ}(·)) contains an asymptotically isometric copy of ℓ^{1} and thus fails to have the fixed point property . This contradicts the hypothesis and the proof that the fixed point property implies reflexivity is complete. ■

Since the proof of the sufficiency in Theorem 2.17 uses techniques unrelated to asymptotically isometric copies of ℓ^{1} or c_{0}, we refer the reader to [15] for the remaining half of the proof.

One last note concerning Orlicz spaces: Although the the sufficiency in Theorem 2.17 holds for Orlicz spaces endowed with either the Orlicz norm or the Luxemburg norm, it is unclear if the necessity in Theorem 2.17 remains true if the Orlicz norm is replaced by the Luxemburg norm. The problem is in the case that the Young function Φ satisfies a Δ_{2}-condition for large values and its conjugate Ψ does not. If L^{Φ}(μ) is endowed with the Luxemburg norm, although the method of proof used above still shows that ℓ^{1} goes into (L^{Φ}(μ), ∥ · ∥_{Φ}) almost isometrically, these copies of ℓ^{1} need not be asymptotically isometric copies of ℓ^{1}. Indeed, if they were, the space (L^{Ψ}, N_{Ψ}(·)) = (L^{Φ}(μ), ∥ · ∥_{Φ})^{∗} would, by Theorem 2.10, contain an isometric copy of L^{1}[0, 1]. It is known however (Theorem 3.35, [6]) that, for classical Orlicz spaces, this never occurs. Thus we can ask the following question:

Question 2.18 If (Ω, Σ, μ) is a finite measure space that is not purely atomic and the Orlicz space L^{Φ}(μ) endowed with the Luxemburg norm has the fixed point property, is L^{Φ}(μ) reflexive?

There are other Banach spaces which fail to have the fixed point property because of the occurence of asymptotically isometric copies of ℓ^{1} or c_{0}. We state several theorems without proofs.

Theorem 2.19 ([7]) With mild restrictions on the weight function w, the Lorentz space L_{w,1}(0, ∞) contains asymptotically isometric copies of ℓ^{1}. With such weight functions w, a subspace of the Lorentz space L_{w,1}(0, ∞) has the fixed point property if and only if it is reflexive.

Theorem 2.20 ([12, 32]) Nonreflexive subspaces of the trace class C_{1} contain asymptotically isometric copies of ℓ^{1}. A subspace of the trace class C_{1} has the fixed point property if and only if it is reflexive.

Theorem 2.21 ([12]) Nonreflexive subspaces of the predual M_{∗} of a von Neumann algebra M with a faithful, normal and finite trace τ contain asymptotically isometric copies of ℓ^{1}. Thus, nonreflexive subspaces of the predual M_{∗} of a von Neumann algebra M with a faithful, normal, finite trace τ fail the fixed point property.

Theorem 2.22 ([4, 19]) Nonreflexive subspaces of the space K(H) of compact operators on a Hilbert space H contain asymptotically isometric copies of c_{0}. A subspace of the K(H) has the fixed point property if and only if it is reflexive.

Since there do exist Banach spaces which contain copies of ℓ^{1} or c_{0} but do not contain asymptotically isometric copies of ℓ^{1} or c_{0}, the above methods will not completely answer the general question:

Question 2.23 Can ℓ^{1} or c_{0} be renormed to have the fixed point property?

Given the usefulness of reformulating the definitions of a Banach space containing an asymptotically isometric copy of ℓ^{1} or an asymptotically isometric copy of c_{0}, it seems tempting to see if similar reformulations of a Banach space containing an isomorphic copy of ℓ^{1} or an isomorphic copy of c_{0} are useful in getting new fixed point theorems in the more general setting. Compare the following theorems to Theorems 2.12 and 2.13.

Theorem 2.24 Let X be a Banach space. The following conditions are equivalent:

(a) X contains an isomorphic copy of ℓ^{1}.

(b) There is a sequence (x_{n}) in X and constants 0 < m ≤ M < ∞ such that, for all (t_{n}) ∈ ℓ^{1} ,

m ∑_{n=1}^{∞} |t_{n}| ≤ ∥ ∑_{n=1}^{∞} t_{n}x_{n} ∥ ≤ M ∑_{n=1}^{∞} |t_{n}|

and

lim_{n→∞} ∥x_{n}∥ = M .

(c) There is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that, for all (t_{n}) ∈ ℓ^{1} and for all k ∈ ℕ,

(1 − ε_{k}) ∑_{n=k}^{∞} |t_{n}| ≤ ∥ ∑_{n=k}^{∞} t_{n}x_{n} ∥ ≤ ∑_{n=k}^{∞} |t_{n}| .

Of course, the equivalence of (a) and (c) is just the stronger version of James’s distortion theorem for ℓ^{1} stated earlier. We state it again here for comparison to Theorem 2.12. The equivalence of (b) to the other two conditions is easy.

Theorem 2.25 Let X be a Banach space. The following conditions are equivalent:

(a) X contains an isomorphic copy of c_{0}.

(b) There is a sequence (x_{n}) in X and constants 0 < m ≤ M < ∞ such that, for all (t_{n}) ∈ ℓ^{1} ,

m sup_{n} |t_{n}| ≤ ∥ ∑_{n=1}^{∞} t_{n}x_{n} ∥ ≤ M sup_{n} |t_{n}|

and

lim_{n→∞} ∥x_{n}∥ = m .

(c) There is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that for all (t_{n}) ∈ c_{0} and for all k ∈ ℕ,

sup_{n≥k} |t_{n}| ≤ ∥∑_{n≥k} t_{n}x_{n} ∥ ≤ (1 + ε_{k}) sup_{n≥k} |t_{n}| .

Unlike Theorem 2.24, note that the equivalence of conditions (a) and (c) in Theorem 2.25 is an improvement over the corresponding statement in the stronger version of James’s distortion theorem for c_{0}. Once we have shown the equivalence of (a) and (c), their equivalence to condition (b) is immediate.

In order to prove the equivalence of conditions (a) and (c) in Theorem 2.25, we employ a beautiful result due to B. J. Cole, T. W. Gamelin, and W. B. Johnson.

Lemma 2.26 ([8]) Let (x_{n}) be a basic sequence in an infinite-dimensional Banach space X. Then there is a block basic sequence (y_{n}) of (x_{n}) and a sequence of functionals (y_{n}^{∗}) in X^{∗} which form a unit biorthogonal system of X. That is, for each n ∈ ℕ,

||y_{n}|| = ||y_{n}^{∗}|| = y_{n}^{∗}(y_{n}) = 1 and y_{n}^{∗}(y_{m}) = 0 for all m ≠ n.

Proof of Theorem 2.25. Suppose that (a) holds; i.e., assume that X contains an isomorphic copy of c_{0}. Then, by the stronger version of James’s distortion theorem, there is a decreasing null sequence (δ_{n}) in (0, 1) and a sequence (x_{n}) in X so that

(1 − δ_{k}) sup_{n≥k} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ (1 + δ_{k}) sup_{n≥k} |t_{n}|,

for all (t_{n}) ∈ c_{0} and for all k ∈ ℕ.

Since (x_{n}) is a basic sequence in X, there is a block basic sequence (y_{n}) of (x_{n}) and a sequence of functionals (y_{n}^{∗}) in X^{∗} which form a unit biorthogonal system of X by Lemma 2.26. Thus there is a strictly increasing sequence of integers (k_{n})_{n=0}^{∞}, with k_{0} = 0, and scalars t_{j}^{n}, where k_{n−1} < j ≤ k_{n} and n ∈ ℕ, so that

y_{n} = ∑_{j=k_{n−1^{+1}}}^{k_{n}} t_{j}^{n}x_{j} .

Since ||y_{n}|| = 1, we have

(1 − δ_{k_{n−1}+1}) max_{k_{n−1}+1≤j≤k_{n}} |t_{j}^{n}| ≤ 1.

Thus |t_{j}^{n}| ≤ (1 − δ_{k_{n−1}+1})^{−1} for all k_{n−1} + 1 ≤ j ≤ k_{n} and for all n ∈ ℕ.

Let (a_{n}) ∈ c_{0} and ℓ ∈ ℕ. Then, for each m ≥ ℓ,

∥∑_{n=ℓ}^{∞} a_{n}y_{n}∥ ≥ |y_{m}^{∗} (∑_{n=ℓ}^{∞} a_{n}y_{n})| = |a_{m}| .

Hence ∥ ∑_{n=ℓ}^{∞} a_{n}y_{n}|| ≥ sup_{m≥ℓ} |a_{m}| . Also

∥∑_{n=ℓ}^{∞} a_{n}y_{n}∥ = ∥∑_{n=ℓ}^{∞} a_{n} ∑_{j=k_{n−1}+1}^{k_{n}} t_{j}^{n}x_{j}∥

≤ (1 + δ_{k_{ℓ−1}+1}) sup_{n≥ℓ^{k_{n−1}+1≤j≤kn}} |a_{n}t_{j}^{n}|

≤ (1 + δ_{k_{ℓ−1}+1}) (1 − δ_{k_{ℓ−1}+1})^{−1} sup_{n≥ℓ} |a_{n}|

= (1 + ε_{ℓ}) sup_{n≥ℓ} |a_{n}| ,

where ε_{ℓ} = (1 + δ_{k_{ℓ−1}+1}) (1 − δ_{k_{ℓ−1}+1})^{−1} − 1 .

Since δ_{n} → 0 as n → ∞, ε_{ℓ} → 0 as ℓ → ∞. Hence the proof of (a) implies (c) is complete. Since it is obvious that (c) implies (a), this also completes the proof of Theorem 2.25. ■

With this “improved” version of James’s distortion theorem for c_{0}, we are now in a position to improve on the result mentioned after Proposition 2.1 that c_{0} cannot be renormed to have the fixed point property for uniformly lipschitzian mappings. Recall that a mapping T : C → C is said to be asymptotically nonexpansive if ∥T^{n}x − T^{n}y∥ ≤ k_{n}∥x − y∥ for all x, y ∈ C and for all n ∈ ℕ, where (k_{n}) is a sequence of real numbers converging to 1.

Theorem 2.27 ([17]) If a Banach space X contains an isomorphic copy of c_{0}, then there exists a closed, bounded, convex subset C of X and an affine, asymptotically nonexpansive mapping T : C → C without a fixed point. In particular, c_{0} cannot be renormed to have the fixed point property for asymptotically nonexpansive mappings.

Proof. If X contains an isomorphic copy of c_{0}, then, by Theorem 2.25, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X so that

sup_{n≥k} |t_{n}| ≤ ∥ ∑_{n=k}^{∞} t_{n}x_{n} ∥ ≤ (1 + ε_{k}) sup_{n≥k} |t_{n}| ,

for all (t_{n}) ∈ c_{0} and for all n ∈ ℕ.

Define C = {∑_{n=1}^{∞} t_{n}x_{n} : 0 ≤ t_{n} ≤ 1 and (t_{n}) ∈ c_{0}}. Clearly, C is a closed bounded convex subset of X. Define T : C → C by T (∑_{n=1}^{∞} t_{n}x_{n}) = x_{1} + ∑_{n=1}^{∞} t_{n}x_{n+1}, for all ∑_{n=1}^{∞} t_{n}x_{n} ∈ C. It is easily seen that T is affine and has no fixed point in C. Let z = ∑_{n=1}^{∞} t_{n}x_{n} and w = ∑_{n=1}^{∞} s_{n}x_{n} be elements of C and let k ∈ ℕ. Then

||T^{k}z − T^{k}w|| = ∥ ∑_{n=1}^{∞}(t_{n} − s_{n})x_{n+k} ∥

≤ (1 + ε_{k+1}) sup_{n} |t_{n} − s_{n}|

≤ (1 + ε_{k+1}) ∥ ∑_{n=1}^{∞}(t_{n} − s_{n})x_{n} ∥

= (1 + ε_{k+1}) ||z − w||.

Since 1 + ε_{k+1} → 1 as k → ∞, this shows that T is an asymptotically nonexpansive mapping on C and the proof is complete. ■

Whether the analogous result for ℓ^{1} holds is still unknown.

Question 2.28 If a Banach space X contains an isomorphic copy of ℓ^{1}, does X fail the fixed point property for asymptotically nonexpansive mappings or for affine asymptotically nonexpansive mappings on closed bounded convex subsets of X?

Theorem 2.25 can also be applied to relate the occurence of isomorphic copies of c_{0} in a Banach space and the absence of a geometric condition which implies weak normal structure. A Banach space X has the generalized Gossez-Lami Dozo property (GGLD) if, whenever (x_{n}) is a weakly null sequence in X that is not norm null, then lim inf_{n} ∥x_{n}∥ < lim sup_{n} lim sup_{m} ∥x_{n} − x_{m}∥ . A Banach space X has property asymptotic (P) if, whenever (x_{n}) is a weakly null sequence in X that is not norm null, then lim inf_{n} ∥x_{n}∥ < diam_{a}{x_{n}}, where diam_{a}{x_{n}} = lim_{n} diam{x_{k} : k ≥ n} is the asymptotic diameter of the sequence (x_{n}). B. Sims and M. Smyth [43] have shown that the generalized Gossez-Lami Dozo property and property asymptotic (P) are equivalent. The next theorems appear in [20].

Theorem 2.29 If X contains an isomorphic copy of c_{0}, then X fails to have the generalized Gossez-Lami Dozo property.

Proof. Suppose that X contains an isomorphic copy of c_{0}. By Theorem 2.25, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X so that

sup_{n≥k} |t_{n}| ≤ ∥∑_{n=k}^{∞} t_{n}x_{n}∥ ≤ (1 + ε_{k}) sup_{n≥k} |t_{n}|,  (**)

for all (t_{n}) ∈ c_{0} and for all k ∈ ℕ. By passing to subsequences if necessary, we assume that the sequence (ε_{n}) is decreasing. Since (x_{n}) is equivalent to the unit vector basis of c_{0}, (x_{n}) is weakly null. Clearly, from (∗∗), the sequence (x_{n}) is not norm null.

Also 1 ≤ ∥x_{k} −x_{n}∥ ≤ 1+ε_{k} for n ≥ k implies that 1 ≤ diam{x_{n} : n ≥ k} ≤ 1+ε_{k} for all k ∈ ℕ which, in turn, implies that diam_{a}{x_{n}} = 1. Thus X fails to have property asymptotic (P), and thus fails to have the GGLD. This completes the proof of Theorem 2.29. ■

Although Banach spaces containing isomorphic copies of c_{0} fail the generalized Gossez-Lami Dozo property, they may have weak normal structure. In fact, since every separable Banach space can be renormed to be uniformly rotund in every direction and Banach spaces that are uniformly rotund in every direction have weak normal structure [46], there exists an equivalent norm on c_{0} which has weak normal structure and fails the generalized Gossez-Lami Dozo property. The next result shows that such renormings of c_{0} also fail to contain asymptotically isometric copies of c_{0}.

Theorem 2.30 If a Banach space X contains an asymptotically isometric copy of c_{0}, then X fails to have weak normal structure.

Proof. Suppose that X contains an asymptotically isometric copy of c_{0}. Then there is null sequence (ε_{n}) in (0, 1), which we may assume is decreasing, and a sequence (x_{n}) in X such that

sup_{n} (1 − ε_{n})|t_{n}| ≤ ∥∑_{n=1}^{∞} t_{n}x_{n}∥ ≤ sup_{n} |t_{n}| , for all (t_{n}) ∈ c_{0}.

Since the sequence (x_{n}) is equivalent to the unit vector basis of c_{0} and the unit vector basis of c_{0} is weakly null, (x_{n}) is weakly null. Hence the set {x_{n} : n ∈ ℕ} is a relatively weakly compact set in X and so, by the Krein-Šmulian Theorem, K = \overline{co}(\{x_{n} : n ∈ ℕ\}), the closed convex hull of {x_{n} : n ∈ ℕ}, is a weakly compact convex subset of X.

Since ∥x_{n} − x_{m}∥ ≤ 1 for all n, m ∈ ℕ, diam K ≤ 1. Since ∥x_{n} − x_{m}∥ ≥ 1 − ε_{n} for all n, m ∈ ℕ with n > m, and (ε_{n}) is a null sequence in (0, 1), diam K ≥ 1. Thus diam K = 1.

Consider an element x ∈ co({x_{n} : n ∈ ℕ}). Then x = ∑_{j=1}^{N} t_{j}x_{j} where t_{j} ≥ 0 for all

1 ≤ j ≤ N and ∑_{j=1}^{N} t_{j} = 1. For each n > N, ∥x − x_{n}∥ = ∥ ∑_{j=1}^{N} t_{j}x_{j} − x_{n}∥ ≥ 1 − ε_{n}. Since

(ε_{n}) is a null sequence, this implies that sup{∥x − y∥ : y ∈ K} ≥ 1. Since diam K = 1, we get that sup{∥x − y∥ : y ∈ K} = 1. Hence each x in co({x_{n} : n ∈ ℕ}), and each x in the weakly compact set K, is a diametral point of K. Therefore X fails to have weak normal structure. This completes the proof of Theorem 2.30. ■

Combining this with previous results yields:

Corollary 2.31 Every infinite-dimensional subspace of (c_{0}, ∥·∥_{∞}) and every renorming of c_{0}(Γ), where Γ is uncountable, fails to have weak normal structure.

Note that the second part of this corollary is a generalization of the result of M. M. Day, R. C. James, and S. Swaminathan [9] that c_{0}(Γ), Γ uncountable, cannot be renormed to be uniformly rotund in every direction.

Although there are many other results of a geometric nature concerning Banach spaces containing asymptotically isometric copies of ℓ^{1} or c_{0}, we end with a result concerning the duality of these conditions. Statements and references of other results will be given in the Notes and Remarks section to follow.

Theorem 2.32 Let X be a Banach space containing an asymptotically isometric copy of c_{0}. Then X^{∗}, with its dual norm, contains an asymptotically isometric copy of ℓ^{1}.

Proof. Since X contains an asymptotically isometric copy of c_{0}, there is a null sequence (ε_{n}) in (0, 1) and a sequence (x_{n}) in X such that

max_{1≤n≤N}(1 − ε_{n})|a_{n}| ≤ ∥∑_{n=1}^{N} a_{n}x_{n}∥ ≤ max_{1≤n≤N} |a_{n}| ,

for all scalars a_{1}, ..., a_{N} and all N ∈ ℕ. Let (x_{n}^{∗}) be the Hahn-Banach extensions to elements of X^{∗} of the linear functionals on the span of (x_{n}) that are biorthogonal to (x_{n}).

Fix m ∈ ℕ. Then, for all vectors x of the form ∑_{n=1}^{N} a_{n}x_{n} with N ≥ m, we have

|x_{m}^{∗}(x)| = |a_{m}| = (1 − ε_{m})^{−1}(1 − ε_{m})|a_{m}|

≤ (1 − ε_{m})^{−1} max_{1≤n≤N}(1 − ε_{n})|a_{n}| ≤ (1 − ε_{m})^{−1}∥x∥.

Therefore ||x_{m}^{∗}|| ≤ (1 − ε_{m})^{−1}.

Define y_{n}^{∗} = x_{n}^{∗}∥x_{n}^{∗}∥^{−1} for each n ∈ ℕ. Fix scalars a_{1}, a_{2}, ..., a_{N} and let b_{n} = sign a_{n} for all 1 ≤ n ≤ N. Then, since || ∑_{n=1}^{N} b_{n}x_{n}|| ≤ max_{1≤n≤N} |b_{n}| = 1, we have

∑_{n=1}^{N} |a_{n}| ≥ || ∑_{n=1}^{N} a_{n}y_{n}^{∗}||

≥ (∑_{n=1}^{N} a_{n}y_{n}^{∗}) (∑_{n=1}^{N} b_{n}x_{n})

= ∑_{n=1}^{N} ||x_{n}^{∗}||^{−1}|a_{n}|

≥ ∑_{n=1}^{N}(1 − ε_{n})|a_{n}| .

Thus X^{∗} contains an asymptotically isometric copy of ℓ^{1}. ■

The converse of Theorem 2.32 does not hold as is easily seen by considering X = ℓ^{1} with its usual norm.

Finally, as an immediate corollary of Theorem 2.32 and previous results, we have:

Corollary 2.33 If a Banach space X contains an asymptotically isometric copy of c_{0}, then neither X nor X^{∗} has the fixed point property, and X^{∗∗} cannot even be renormed to have the fixed point property.

3. Notes and Remarks

The notion of an asymptotically isometric copy of ℓ^{1} in a Banach space was introduced in [14] in order to show that the nonreflexive subspaces of L^{1}[0, 1] fail to have the fixed point property. Earlier, J. Hagler, in his Ph.D. thesis [27], Remark 2.3, had noted that a real Banach space contains an asymptotically isometric copy of ℓ^{1} if and only if its dual space contains an isometric copy of L^{1}[0, 1]. Hagler did not publish this result from his thesis until recently when he, S. Dilworth, and M. Girardi [11] showed Theorem 2.10 (and more) for both real and complex Banach spaces. Unfortunately, until then, this particular result in Hagler’s thesis was not widely known.

The corresponding notion of a Banach space containing an asymptotically isometric copy of c_{0} was introduced in [15] where Theorem 2.4 was shown and in [13] as a condition for a Banach space X to be in partial duality with the dual space X^{∗} containing an asymptotically isometric copy of ℓ^{1}.

Theorem 2.5, which appeared in [16], can be viewed as optimal. D. Alspach [2] and E. Behrends [3] have noted that there exist closed subspaces of c_{0} which do not contain subspaces isometric to c_{0} and, in fact, do not even contain isometric copies of two-dimensional ℓ_{2}^{∞}. V. P. Fonf and M. I. Kadec [23] have constructed subspaces of ℓ^{1} which are rotund and thus contain no further subspaces isometric to ℓ^{1}.

Corollary 2.15 appeared as Corollary 3 in [15] where the Banach space was assumed to be separable. Since, in Theorem 2.9, condition (a) always implies condition (d) independent of the separability of the Banach space X, this extra assumption in [15] was unnecessary.

Theorems 2.17, 2.19, 2.20, 2.21, and 2.22 all consider connections between Banach spaces with the fixed point property and reflexivity. One natural inclination would be to hope to connect weakly compact convex sets C with the existence of fixed points for all nonexpansive self-maps of C. However Alspach’s [1] example of an isometry on a weakly compact convex subset of L^{1}[0, 1] without a fixed point shows that weak compactness is not sufficient for a set to have the fixed point property. For examples of non-weakly compact sets such that all nonexpansive self-maps of those sets have fixed points, see Soardi [42] where it is shown that the unit ball of L^{∞} has the fixed point property or the book of Goebel and Kirk [24], page 31, where a certain non-weak^{∗} compact subset of ℓ^{1} is shown to have the fixed point property. In fact, the example in [24] comes from [26] where a decreasing sequence of closed convex subsets of ℓ^{1} is constructed such that the even-indexed terms have the fixed point property and the odd-indexed terms fail the fixed point property. Inspired by their example, Dowling, Lennard, and Turett [18] showed that, if a closed bounded convex subset K of a Banach space contains an asymptotically isometric ℓ^{1}-basis, then there exists a nonempty closed bounded convex subset C of K and an affine nonexpansive self-map of C without a fixed point. One corollary of this theorem is that a closed bounded convex subset C of ℓ^{1} is norm compact if and only if every closed bounded subset of C has the fixed point property for nonexpansive maps. The theorem also combines with a result of Dowling’s [21] to show that a closed bounded convex subset K of L^{1}[0, 1] is weakly compact if and only if every closed bounded convex subset of K has the fixed point property for nonexpansive \underline{affine} self-maps. Obviously, given Alspach’s example, the word “affine” cannot be omitted.

Finally, recent articles [37, 38, 39] by H. Pfitzner show the existence of asymptotically isometric copies of ℓ^{1} in nonreflexive subspaces of L-embedded Banach spaces and asymptotically isometric copies of c_{0} in nonreflexive subspaces of M-embedded Banach spaces and consider asymptotically isometric copies of ℓ^{1} in preduals of von Neumann algebras. Moreover, a recent paper of N. Randrianantoanina [40] constructs asymptotically isometric copies of ℓ^{1} in any nonreflexive subspace of the dual of an arbitrary C^{∗}-algebra or Jordan triple.

References

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[2] D. Alspach, private communication.

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Index

Adjoint, 13 Affine, 25 Algebra

von Neumann, 19 Almost isometric, 2 Ball

closed unit, 8 unit, 14, 25

Banach lattice, 2–3 Banach space

containing an asymptotically isometric copy of c_{0},

10, 20, 24–25 containing an asymptotically isometric copy of ℓ^{1},

10, 16, 20 containing an isomorphic copy of c_{0}, 20 containing an isomorphic copy of ℓ^{1}, 3, 20 containing asymptotically isometric copies of ℓ^{1}, 9 containing isomorphic copies of ℓ^{1}, 9 contains an asymptotically isometric copy of c_{0},

3–4, 23–24 contains an asymptotically isometric copy of ℓ^{1},

3–4, 10, 25 contains an isomorphic copy of c_{0}, 2, 22 contains an isomorphic copy of ℓ^{1}, 2, 16, 22 having an asymptotically isometric copy of ℓ^{1} or c_{0},

15 reflexive, 1 separable, 13, 23 with unconditional basis, 2–3

Banach spaces L-embedded, 25 M-embedded, 26 containing asymptotically isometric copies of ℓ^{1} or

c_{0}, 5, 24 containing asymptotically isometric copies of ℓ^{1}, 10 containing isomorphic copies of c_{0}, 23 nonreflexive, 1–2 which contain copies of ℓ^{1} or c_{0}, 19 which fail to have the fixed point property, 15 with the fixed point property, 25

Basis canonical, 8 standard unit vector, 6 unit vector, 23

Canonical image, 13 Class

of nonexpansive self-maps, 3 of uniformly lipschitzian maps, 3 trace, 19

Classes isomorphism, 2 of Banach spaces, 2

Compactness

weak, 25 Condition

Δ_{2}-, 18–19 Conditions

geometric, 1 Convexity, 17 Dual space

contains an asymptotically isometric copy of c_{0}, 10 contains an asymptotically isometric copy of ℓ^{1}, 24 contains an isometric copy of L^{1}[0,1], 25

Dual

of an arbitrary C^{∗}-algebra, 26 Duality

partial, 25 Embeds

ℓ^{∞} in (L^{Φ}(μ), ∥ · ∥_{Φ}), 18

isometrically, 18 isomorphically, 18

Extensions

Hahn-Banach, 24 Fixed point property, 1–3, 5–8, 10, 15–16, 18–20,

24–25 for k-uniformly lipschitzian mappings, 3 for affine asymptotically nonexpansive mappings, 22 for asymptotically nonexpansive mappings, 22 for nonexpansive affine self-maps, 25 for nonexpansive mappings, 4 for nonexpansive maps, 25 for uniformly lipschitzian mappings, 3, 22

Function conjugate Young, 18–19 Rademacher, 14 Rademacher-like, 14 weight, 19 Young, 18–19

Functions Rademacher, 14 weight, 19 GGLD, 23 Jordan triple, 26 Lipschitz constant uniform, 3

Map

affine, uniformly lipschitzian, 3 Mapping

asymptotically nonexpansive, 22

affine, 22 uniformly lipschitzian, 3

Measure space finite and not purely atomic, 18–19 finite nonatomic, 18

Nonreflexive, 7, 25

Norm dual, 24 equivalent, 15, 17, 23 Luxemburg, 18–19 Orlicz, 18–19

Point

diametral, 23 Predual, 8, 19 Preduals

of von Neumann algebras, 26 Property

asymptotic (P), 22–23 generalized Gossez-Lami Dozo (GGLD), 22 generalized Gossez-Lami Dozo, 23 Hahn-Banach extension, 13

Reflexive, 2, 6, 8, 18–19 Reflexivity, 1–2, 18–19, 25 Renorming

of c_{0}, 10 of c_{0}(Γ), 15, 23 of ℓ^{1}, 10 of ℓ^{1}(Γ), 15 of ℓ^{∞}, 16, 18

Renormings of c_{0} or ℓ^{1}, 2 of c_{0}, 8, 23 of ℓ^{1} and c_{0}, 2, 8 of ℓ^{1} or c_{0}, 5 of ℓ^{1}, 8, 10

Result Kadec-Pełczyński, 7 Luxemburg’s, 18 Maurey’s, 8 of Pełczyński and Hagler, 16

Rotund dual renorming

of ℓ^{∞}, 10 Rotund renorming

of ℓ^{∞}, 10 Rotund, 25 Self-map

affine nonexpansive, 25 Self-maps

nonexpansive, 25 Separability, 16, 25 Sequence

disjointly supported, 8 of norm-one functions, 18

Space Lorentz, 19 of compact operators on a Hilbert space, 19 Orlicz, 18

Spaces classical Orlicz, 19 Orlicz, 18–19

Sphere

unit, 13 Structure normal, 1 weak normal, 22–23

System

unit biorthogonal, 21 Techniques

ultrapower, 8 Theorem

Bessaga-Pełczyński, 8

Goldstines’s, 14 Theorem

James’s distortion improved version for c_{0}, 21 stronger version for c_{0}, 20 stronger version for ℓ^{1}, 20 stronger version, 21

Theorem James’s

stronger version for ℓ^{1}, 3 Krein-Šmulian, 23

Theorems fixed point, 3 James’s Distortion, 2 Stronger versions, 2

James’s, 3 Trace

faithful, normal and finite, 19 Uniform rotundity, 1 Uniform smoothness, 1 Uniformly rotund

in every direction, 23–24 Weak fixed point property, 10