THE DUAL OF THE JAMES TREE SPACE IS ASYMPTOTICALLY UNIFORMLY CONVEX

MARIA GIRARDI

Studia Math. 147 (2001), no. 2, 119–130

Abstract. The dual of the James Tree space is asymptotically uniformly convex.

1. Introduction

In 1950, R. C. James [J1] constructed a Banach space which is now called the James space. This space, along with its many variants (such as the James tree space [J2]) and their duals and preduals, have been a rich source for further research and results (both positive ones and counterexamples), answering many questions, several of which date back to Banach [B, 1932]. See [FG] for a splendid survey of such spaces.

This paper’s main result, Theorem 5, shows that the dual JT^{∗} of the James tree space JT is asymptotically uniformly convex. (See Section 2 for definitions.)

Schachermayer [S, Theorem 4.1] showed that JT^{∗} has the Kadec-Klee property. It follows from Theorem 5 of this paper that JT^{∗} enjoys the uniform Kadec-Klee property. Of course, the same can be said about the (unique) predual JT_{∗} of JT. In fact, Theorem 3 shows that the modulus of asymptotic convexity of JT_{∗} is of power type 3.

Johnson, Lindenstrauss, Preiss, and Schechtman [JLPS] showed that an asymptotically uniformly convex space has the point of continuity property and thus asked whether an asymptotically uniformly convex space has the Radon-Nikodým property. It is well-known that both JT_{∗} and JT^{∗} have the point of continuity property yet fail the Radon-Nikodým property. It follows from Theorem 5 of this paper that JT^{∗} is an asymptotically uniformly convex (dual) space without the Radon-Nikodým property. Thus JT_{∗} is a separable asymptotically uniformly convex space without the Radon-Nikodým property. To the best of the author’s knowledge, these are the first known examples of asymptotically uniformly convex spaces without the Radon-Nikodým property.

Date: 4 March 2001.

1991 Mathematics Subject Classification. 46B20, 46B22, 46B99.

Key words and phrases. James tree space, asymptotically uniformly convex, RNP.

2. Definitions and Notation

Throughout this paper 𝔛 denotes an arbitrary (infinite-dimensional real) Banach spaces. If 𝔛 is a Banach space, then 𝔛^{∗} is its dual space, B(𝔛) is its (closed) unit ball, S(𝔛) is its unit sphere, î: 𝔛 → 𝔛^{∗∗} is the natural point-evaluation isometric embedding, x̂ = î(x) and 𝔛̂ = î (𝔛). If Y is a subset of 𝔛, then [Y] is the closed linear span of Y and

𝔑 (𝔛) = {[x_{i}^{∗}]_{1≤i≤n}^{⊤} : x_{i}^{∗} ∈ 𝔛^{∗} and n ∈ ℕ}
W (𝔛^{∗}) = {[x_{i}]_{1≤i≤n}^{⊥} : x_{i} ∈ 𝔛 and n ∈ ℕ} .

Thus 𝔑 (𝔛) is the collection of (norm-closed) finite codimensional subspaces of 𝔛 while W (𝔛^{∗}) is the collection of weak-star closed finite codimensional subspaces of 𝔛^{∗}. All notation and terminology, not otherwise explained, are as in [DU, LT1, LT2].

The modulus of convexity δ_{𝔛}: [0, 2] → [0, 1] of 𝔛 is

δ_{𝔛}(ε) = inf \{1 − ∥\frac{x + y}{2} ∥ : x, y ∈ S(𝔛) and ∥x − y∥ ≥ ε\}

and 𝔛 is uniformly convex (UC) if and only if δ_{𝔛}(ε) > 0 for each ε ∈ (0, 2]. The modulus of asymptotic convexity \overline{δ}_{𝔛}: [0, 1] → [0, 1] of 𝔛 is

\overline{δ}_{𝔛}(ε) = inf_{x∈S(𝔛)} sup_{Y∈𝔑(𝔛)} inf_{y∈S(Y)} [ ∥x + εy∥ − 1 ]

and 𝔛 is asymptotically uniformly convex (AUC) if and only if \overline{δ}_{𝔛}(ε) > 0 for each ε in (0, 1].

A space 𝔛 has the Kadec-Klee (KK) property provided the relative norm and weak topologies on B(𝔛) coincide on S(𝔛). A space 𝔛 has the uniform Kadec-Klee (UKK) property provided for each ε > 0 there exists δ > 0 such that every ε-separated weakly convergent sequence {x_{n}} in B(𝔛) converges to an element of norm less than 1 − δ.

Related to the above geometric isometric properties are the following geometric isomorphic properties.

• 𝔛 has the Radon-Nikodým property (RNP) provided each bounded subset of 𝔛 has non-empty slices of arbitrarily small diameter.

• 𝔛 has the point of continuity property (PCP) provided each bounded subset of 𝔛 has non-empty relatively weakly open subsets of arbitrarily small diameter.

• 𝔛 has the complete continuity property (CCP) provided each bounded subset of 𝔛 is Bocce dentable.

JT^{∗} IS ASYMPTOTICALLY UNIFORMLY CONVEX 3

Implications between these various properties are summarized in the diagram below.

UC  → AUC → UKK → KK
↓ ↓ ↓
RNP → PCP  → CCP

Helpful notation is

\overline{δ}_{𝔛}(ε) = inf_{x∈S(𝔛)} \overline{δ}_{𝔛}(ε, x)

where

\overline{δ}_{𝔛}(ε, x) = sup_{Y∈𝔑(𝔛)} inf_{y∈S(Y)} [ ∥x + εy∥ − 1 ] .

Note that, for each x ∈ S(𝔛),

\overline{δ}_{𝔛}(ε, x) = sup_{Y∈𝔑(𝔛)} inf_{\begin{matrix}y∈Y \\ ∥y∥≥ε \end{matrix}} [ ∥x + y∥ − 1 ]

and so \overline{δ}_{𝔛}(ε, x) is a non-decreasing function of ε. Thus \overline{δ}_{𝔛} is a non-decreasing Lipschitz functions with Lipschitz constant at most one. For any space 𝔛 and ε ∈ [0, 1]

\overline{δ}_{𝔛}(ε) ≤ ε = \overline{δ}_{ℓ_{1}}(ε) ;

thus, ℓ_{1} is, in some sense, the most asymptotically uniformly convex space.

Uniform convexity, the KK property, and the UKK property have been extensively studied (for example, see [DGZ, LT2]). Asymptotic uniform convexity has been examined explicitly in [JLPS, M] and implicitly in [GKL, KOS]. The RNP, PCP, and CCP have also been extensively studied (for example, see [DU, GGMS, G1, G2]).

The JT space is construction on a (binary) tree

T = ∪_{n=0}^{∞} Δ_{n}

where Δ_{n} is the n^{th}-level of the tree; thus,

Δ_{0} = {∅} and Δ_{n} = {−1, +1}^{n}

for each n ∈ ℕ. The finite tree T_{N} up through level N ∈ ℕ ∪ {0} is

T_{N} = ∪_{n=0}^{N} Δ_{n}

The tree T is equipped with its natural (tree) ordering: if t_{1} and t_{2} are elements of T , then t_{1} < t_{2} provided one of the follow holds:

(1) t_{1} = ∅ and t_{2} ≠ ∅ (2) for some n, m ∈ ℕ

t_{1} = (ε_{1}^{1}, ε_{2}^{1}, …, ε_{n}^{1}) and t_{2} = (ε_{1}^{2}, ε_{2}^{2}, …, ε_{m}^{2})

with n < m and ε_{i}^{1} = ε_{i}^{2} for each 1 ≤ i ≤ n . A (finite) segment of T is a linearly order subset {t_{n}, t_{n+1}, …, t_{n+k}} of T where t_{i} ∈ Δ_{i} for each n ≤ i ≤ n + k. A branch of T is a linearly order subset {t_{0}, t_{1}, t_{2}, …} of T where t_{i} ∈ Δ_{i} for each i ∈ ℕ ∪ {0}.

The James-Tree space JT is the completion of the space of finitely supported functions x: T → ℝ with respect to the norm

∥x∥_{JT} =

sup \{[∑_{i=1}^{n} | ∑_{t∈S_{i}} x_{t}|^{2}]^{\frac{1}{2}} : S_{1}, S_{2}, …, S_{n} are disjoint segments of T \} .

By lexicographically ordering T , the sequence {η_{t}}_{t∈T} in JT, where

η_{t}(s) = \begin{cases}1 & if t = s \\ 0 & if t ≠ s , \end{cases}

forms a monotone boundedly complete monotone (Schauder) basis of JT with biorthogonal functions {η_{t}^{∗}}_{t∈T} in JT^{∗}. Thus ĴT_{∗} = [η_{t}^{∗}]_{t∈T}.

For N, M ∈ ℕ ∪ {0} with N ≤ M, the restriction maps from JT to JT given by

π_{N}(x) = ∑_{t∈Δ_{N}} η_{t}^{∗}(x)η_{t}
π_{[N,M]}(x) = ∑_{t∈∪_{i=N}^{M} Δ_{i}} η_{t}^{∗}(x)η_{t}

π_{[N,ω)}(x) = ∑_{t∈∪_{i=N}^{∞} Δ_{i}} η_{t}^{∗}(x)η_{t}

are each contractive projections (by the nature of the norm on JT); thus, so are their adjoints.

Let Γ be the set of all branches of T . Then [LS, Theorem 1] the mapping

π_{∞}: JT^{∗} → ℓ_{2}(Γ)

given by

π_{∞}(x^{∗}) = {lim_{t∈B} x^{∗}(η_{t})}_{B∈Γ}

is an isometric quotient mapping with kernel ĴT_{∗}. Also, for each x^{∗} ∈ JT^{∗},

∥x^{∗}∥ = lim_{N→∞} ∥π_{[0,N]}^{∗} x^{∗}∥

∥π_{∞}x^{∗}∥ = lim_{N→∞} ∥π_{[N,ω)}^{∗} x^{∗}∥ = lim_{N→∞} ∥π_{N}^{∗}x^{∗}∥

by the weak-star lower semicontinuity of the norm on JT^{∗}.

To show that JT^{∗} has the Kadec-Klee property, Schachermayer calculated the below two quantitative bounds.

Fact 1. [S, Lemma 3.8] Let

f_{1} : (0, 1) → (0, ∞)

be a continuous strictly increasing function satisfying f_{1}(t) < 2^{−10}t^{3} for each t ∈ (0, 1). Let N ∈ ℕ and z^{∗} ∈ JT^{∗}. If

[1 − f_{1}(t)] ∥z^{∗}∥ < ∥π_{[0,N]}^{∗} z^{∗}∥

then

∥π_{[N,ω)}^{∗} z^{∗}∥ < ∥π_{N}^{∗} z^{∗}∥ + t ∥z^{∗}∥ .

Fact 2. [S, Lemma 3.11] Let

f_{2} : (0, 1) → (0, ∞)

be a continuous strictly increasing function satisfying f_{2}(t) < 2^{−26}t^{5} for each t ∈ (0, 1). Let N ∈ ℕ and ε_{0} ∈ (0, 1) and x̃^{∗}, ũ^{∗} ∈ JT^{∗}. If

(2.1) ∥π_{[N,ω)}^{∗} x̃^{∗}∥ ≤ 1 (2.2) ∥π_{N}^{∗} x̃^{∗}∥ > 1 − f_{2}(ε_{0}) (2.3) ∥π_{∞} x̃^{∗}∥ > 1 − f_{2}(ε_{0}) (2.4) ∥π_{[N,ω)}^{∗} (x̃^{∗} + ũ^{∗})∥ ≤ 1 (2.5) ∥π_{N}^{∗} ũ^{∗}∥ < f_{2}(ε_{0}) (2.6) ∥π_{∞} ũ^{∗}∥ < f_{2}(ε_{0}) . Then (2.7) ∥π_{[N,ω)}^{∗} ũ^{∗}∥ < ε_{0} .

3. Results

Theorem 3 shows that the modulus of asymptotic convexity of JT_{∗} is of power type 3. Its proof uses Fact 1.

Theorem 3. There exists a positive constant k so that

\overline{δ}_{JT_{∗}}(ε) ≥ kε^{3}

for each ε ∈ (0, 1]. Thus JT_{∗} is asymptotically uniformly convex.

Proof. Fix c ∈ (0, 2^{−10}) and find k so that

0 < k(1 + k)^{2} ≤ c .  (1)

Fix ε ∈ (0, 1) and a finitely supported x_{∗} ∈ S (JT_{∗}). It suffices to show that

\overline{δ}_{JT_{∗}}(ε, x_{∗}) ≥ kε^{3} .  (2)

Find N ∈ ℕ so that

π_{[0,N−1]}^{∗} x̂_{∗} = x̂_{∗}

and let

Y = [η_{t}]_{t∈T_{N}}^{⊤} .

Fix y_{∗} ∈ S(Y).

Assume that

∥x_{∗} + εy_{∗}∥ − 1 < k ε^{3} .

Then

[1 − \frac{kε^{3}}{1 + kε^{3}} ] ∥x̂_{∗} + εŷ_{∗}∥ < 1 = ∥π_{[0,N]}^{∗} (x̂_{∗} + εŷ_{∗})∥ .

Thus by Fact 1, with f_{1}(t) = ct^{3},

∥π_{[N,ω)}^{∗} (x̂_{∗} + εŷ_{∗})∥ < ∥π_{N}^{∗} (x̂_{∗} + εŷ_{∗})∥ + f_{1}^{−1} (\frac{kε^{3}}{1 + kε^{3}} ) ∥ (x̂_{∗} + εŷ_{∗})∥

and so

ε < [1 + kε^{3}] f_{1}^{−1} (\frac{kε^{3}}{1 + kε^{3}} ) .  (3)

But inequality (3) is equivalent to

c^{1/3} < k^{1/3} (1 + kε^{3})^{2/3} ,

which contradicts (1). Thus ∥x_{∗} + εy_{∗}∥ − 1 ≥ k ε^{3} and so (2) holds. □

A modification of the proof of Theorem 3 shows that, for each ε ∈ (0, 1), the \overline{δ}_{JT^{∗}}(ε, x^{∗}) stays uniformly bounded below from zero for x^{∗} ∈ S(JT^{∗}) whose ∥π_{∞} x^{∗}∥ is small. Recall that if x_{∗} ∈ JT_{∗} then ∥π_{∞} x̂_{∗}∥ = 0.

Lemma 4. For each ε ∈ (0, 1) there exists η = η(ε) > 0 so that

inf_{\begin{matrix}x^{∗}∈S(JT^{∗}) \\ ∥π_{∞}x^{∗}∥≤η \end{matrix}} sup_{Y∈W(JT^{∗})} inf_{y^{∗}∈S(Y)} [ ∥x^{∗} + εy^{∗}∥ − 1 ] > 0 .

Proof. Fix ε ∈ (0, 1). Keeping with the notation in Fact 1, find δ, η_{2} > 0 so that

4η_{2} + \frac{δ}{1 − f_{1}(δ)} < ε .

Fix x^{∗} ∈ S(JT^{∗}) with

∥π_{∞}x^{∗}∥ ≡ b ≤ η_{2} .

It suffices to show that

sup_{Y∈W(JT^{∗})} inf_{y^{∗}∈S(Y)} ∥x^{∗} + εy^{∗}∥ ≥ \frac{1}{1 − f_{1}(δ)} .  (4)

Fix η_{1} ∈ (0, 1). Find N ∈ ℕ so that

1 − η_{1} ≤ ∥π_{[0,N]}^{∗} x^{∗}∥ and ∥π_{[N,ω)}^{∗} x^{∗}∥ < b + η_{2}

and let

Y = [η_{t}]_{t∈T_{N}}^{⊥} .

Fix y^{∗} ∈ S(Y).

Assume that

∥x^{∗} + εy^{∗}∥ < \frac{1 − η_{1}}{1 − f_{1}(δ)} .

Then

[1 − f_{1}(δ)] ∥x^{∗} + εy^{∗}∥ < ∥π_{[0,N]}^{∗} x^{∗}∥ = ∥π_{[0,N]}^{∗} (x^{∗} + εy^{∗})∥ .

Thus by Fact 1

∥π_{[N,ω)}^{∗} (x^{∗} + εy^{∗})∥ < ∥π_{N}^{∗} (x^{∗} + εy^{∗})∥ + δ ∥(x^{∗} + εy^{∗})∥

and so

ε − (b + η_{2}) < (b + η_{2}) + \frac{δ}{1 − f_{1}(δ)} .

But b ≤ η_{2} and so

ε < 4η_{2} + \frac{δ}{1 − f_{1}(δ)} .

A contradiction, thus

∥x^{∗} + εy^{∗}∥ ≥ \frac{1 − η_{1}}{1 − f_{1}(δ)} .

Since η_{1} > 0 was arbitrary, inequality (4) holds. □

Thus to show that JT^{∗} is asymptotically uniformly convex, one just needs to examine \overline{δ}_{JT^{∗}}(ε, x^{∗}) for x^{∗} ∈ S(JT^{∗}) whose ∥π_{∞}x^{∗}∥ is not small. Fact 2 is used for this case.

Theorem 5. JT^{∗} is asymptotically uniformly convex.

Proof. Fix ε ∈ (0, 1) and let ε_{0} = ε/4. Let f_{1} : (0, 1) → (0, 2^{−12}) be given by f_{1}(t) = 2^{−12}t^{3} and f_{2} be a function satisfying the hypothesis in Fact 2. Find δ, η_{2} > 0 so that

4η_{2} + \frac{δ}{1 − f_{1}(δ)} < ε .

Next find γ_{i} > 0 and τ > 1 so that

γ_{3} < γ_{2} < \frac{1}{2}  (5)
τ ≤ \frac{(1 − γ_{1}) (1 − γ_{2})}{1 − f_{2}(ε_{0})}  (6)
τ < \frac{1 − γ_{2}}{\sqrt{1 − f_{2}^{2}(ε_{0})}}  (7)

τ ≤ \frac{η_{2}^{3}γ_{3}^{3}}{2^{15}(1 − γ_{2})^{3}} − γ_{4} + 1  (8)
\frac{τ − 1 + γ_{4}}{τ} < f_{1}(1)  (9)
τ ≤ \frac{1}{1 − f_{1}(δ)} .  (10)

Fix x^{∗} ∈ S(JT^{∗}). It suffices to show that

sup_{Y∈𝔑(JT^{∗})} inf_{y∈S(Y)} ∥x^{∗} + εy^{∗}∥ ≥ τ .  (11)

Let

∥π_{∞}x^{∗}∥ ≡ b .

If b ≤ η_{2}, then by the proof of Lemma 4 and (10), inequality (11) holds. So let b > η_{2}. Find N ∈ ℕ so that

(1 − γ_{1}) b < ∥π_{N}^{∗} x^{∗}∥ ≤ ∥π_{[N,ω)}^{∗} x^{∗}∥ < b (\frac{1 − γ_{3}}{1 − γ_{2}} ) < \frac{b}{1 − γ_{2}} (12)
1 − γ_{4} < ∥π_{[0,N]}^{∗} x^{∗}∥ .  (13)

Let g_{x^{∗}} ∈ JT^{∗∗} be the functional given by

g_{x^{∗}}(z^{∗}) = ⟨π_{∞}z^{∗}, π_{∞}x^{∗}⟩_{H_{2}}

where the inner product in the natural inner product on ℓ_{2}(Γ). Let

Y = [η_{t}]_{t∈T_{N}}^{⊥} ∩ [g_{x^{∗}}]^{⊤}

and fix y^{∗} ∈ S(Y).

Assume that

∥x^{∗} + εy^{∗}∥ < τ .  (14)

It suffices to find a contradiction to (14). Towards this, let

x̃^{∗} = \frac{1 − γ_{2}}{τb} x^{∗} and ỹ^{∗} = \frac{1 − γ_{2}}{τb} y^{∗} .

It suffices to show (keeping with the same notation but with ũ^{∗} = εỹ^{∗}) that conditions (2.1) through (2.6) of Fact 2 hold; for then condition (2.7) holds and so by (5)

ε_{0} > ∥π_{[N,ω)}^{∗} εỹ^{∗}∥ = \frac{1 − γ_{2}}{τb} ε ≥ \frac{ε}{4} = ε_{0} .

Condition (2.1) follows from (12) since

∥π_{[N,ω)}^{∗} x̃^{∗}∥ ≤ \frac{1 − γ_{2}}{τb} \frac{b}{1 − γ_{2}} ≤ 1 .

Condition (2.2) follows from (12) and (6) since

∥π_{N}^{∗} x̃^{∗}∥ > \frac{1 − γ_{2}}{τb} (1 − γ_{1}) b = \frac{(1 − γ_{1}) (1 − γ_{2})}{τ} ≥ 1 − f_{2}(ε_{0}) .

Towards condition (2.3), note that by (7)

∥π_{∞} x̃^{∗}∥ = \frac{1 − γ_{2}}{τb} b = \frac{1 − γ_{2}}{τ} > \sqrt{1 − f_{2}^{2}(ε_{0})}  (15)

and so

∥π_{∞} x̃^{∗}∥ > 1 − f_{2}(ε_{0}) .

Towards condition (2.4), note that by (14) and (13)

∥x^{∗} + εy^{∗}∥ < \frac{τ}{1 − γ_{4}} ∥π_{[0,N]}^{∗} (x^{∗} + εy^{∗})∥ .

Thus by Fact 1 and (9)

∥π_{[N,ω)}^{∗} (x^{∗} + εy^{∗})∥

< ∥π_{N}^{∗} (x^{∗} + εy^{∗})∥ + f_{1}^{−1} (\frac{τ − 1 + γ_{4}}{τ} ) ∥(x^{∗} + εy^{∗})∥

≤ b \frac{1 − γ_{3}}{1 − γ_{2}} + τ2^{4} (\frac{τ − 1 + γ_{4}}{τ} )^{1/3} .

Thus condition (2.4) holds provided

b \frac{1 − γ_{3}}{1 − γ_{2}} + τ2^{4} (\frac{τ − 1 + γ_{4}}{τ} )^{1/3} ≤ \frac{τb}{1 − γ_{2}} ,

or equivalently

τ^{2/3} (τ − 1 + γ_{4})^{1/3} ≤ \frac{b (τ − 1 + γ_{3})}{2^{4} (1 − γ_{2})} .

But by (8) and that b > η_{2}

τ^{2/3} (τ − 1 + γ_{4})^{1/3} ≤ 2 (τ − 1 + γ_{4})^{1/3} ≤ \frac{2η_{2}γ_{3}}{2^{5}(1 − γ_{2})}
≤ \frac{bγ_{3}}{2^{4}(1 − γ_{2})}  ≤ \frac{b (τ − 1 + γ_{3})}{2^{4} (1 − γ_{2})} .

Thus condition (2.4) holds.

Condition (2.5) follows from the fact that y^{∗} ∈ [η_{t}]_{t∈T_{N}}^{⊥}. Towards condition (2.6), since y^{∗} ∈ [g_{x^{∗}}]^{⊤}, the vectors π_{∞}ỹ^{∗} and π_{∞}x̃^{∗} are orthogonal in ℓ_{2}(Γ) and so

∥π_{∞}εỹ^{∗}∥^{2} = ∥π_{∞}(x̃^{∗} + εỹ^{∗})∥^{2} − ∥π_{∞}x̃^{∗}∥^{2} ;

but π_{∞} = π_{∞}π_{[N,ω)}^{∗} and so by condition (2.4) and (15)

∥π_{∞}εỹ^{∗}∥^{2} ≤ ∥π_{[N,ω)}^{∗}(x̃^{∗} + εỹ^{∗})∥^{2} − ∥π_{∞}x̃^{∗}∥^{2}
< 1 − [1 − f_{2}^{2}(ε_{0})] = f_{2}^{2}(ε_{0}).

Thus condition (2.6). □

The proof in [JLPS] that an asymptotically uniformly convex space has the PCP show that if \overline{δ}_{𝔛}(ε) > 0 for each ε ∈ (0, 1] then 𝔛 has the PCP. A bit more can be said.

Proposition 6. If \overline{δ}_{𝔛} (\frac{1}{2}) > 0 then 𝔛 has the PCP.

The proof of Proposition 6 uses the following (essentially known) lemma.

Lemma 7. Let 𝔛 be a space without the PCP and 0 < ε < 1. Then there is a closed subset A of 𝔛 so that

(1) each (nonempty) relatively weakly open subset of A has diameter larger than 1 − ε

(2) sup{ ∥a∥ : a ∈ A} = 1.

Proof of Lemma 7. Let 𝔛 fail the PCP and 0 < ε < 1. By a standard argument (e.g., see [SSW, Prop. 4.10]), there is a closed subset Ã of 𝔛 of diameter one such that each (nonempty) relatively weakly open subset of Ã has diameter larger than 1 − ε. Without loss of generality 0 ∈ Ã (just consider a translate of Ã). Let

b = sup \{ ∥x∥ : x ∈ Ã\} and A = \frac{Ã}{b} .

Note that 0 < b ≤ 1. If V is (nonempty) relatively weakly open subset of A, then bV is a relatively weakly open subset of Ã and so

diam V = \frac{1}{b} diam bV > 1 − ε .

Thus A does the job. □

Proof of Proposition 6. Let 𝔛 be a Banach space without the PCP. Fix t ∈ (0, \frac{1}{2}) and δ ∈ (0, t). It suffices to show that \overline{δ}_{𝔛}(t) ≤ 2δ.

Find a subset A of 𝔛 which satisfies the conditions of Lemma 7 with ε = 1 − 2t and find a ∈ A so that

∥\frac{a}{∥a∥} − a∥ < δ .

Let Y ∈ 𝔑 (𝔛). It suffices to show that

inf_{\begin{matrix}y∈Y \\ ∥y∥≥t \end{matrix}} [ ∥\frac{a}{∥a∥} + y∥ − 1 ] ≤ 2 δ .

By condition (1) of Lemma 7 there exists x ∈ A so that ∥x − a∥ ≥ t and x − a is almost in Y; thus, by a standard perturbation argument (e.g., see [GJ, Lemma 2]) there exists y ∈ Y so that

∥y∥ ≥ t and ∥y − (x − a)∥ < δ .

Thus

∥\frac{a}{∥a∥} + y∥ ≤ ∥\frac{a}{∥a∥} − a∥ + ∥y − x + a∥ + ∥x∥ < 1 + 2δ .

Thus \overline{δ}_{𝔛}(\frac{1}{2}) = 0. □

The observation below formalizes an essentially known fact, which to the best of the author’s knowledge, has not appeared in print as such. Recall that the modulus of asymptotic smoothness \overline{ρ}_{𝔛}: [0, 1] → [0, 1] of 𝔛 is

\overline{ρ}_{𝔛}(ε) = sup_{x∈S(𝔛)} inf_{Y∈𝔑(𝔛)} sup_{y∈S(Y)} [ ∥x + εy∥ − 1 ]

and 𝔛 is asymptotically uniformly smooth if and only if lim_{ε→0^{+}} ρ_{𝔛}(ε)/ε = 0. Also, L_{p}(𝔛) is the Lebesgue-Bochner space of strongly measurable 𝔛-valued functions defined on a separable non-atomic probability space, equipped with its usual norm.

Observation 8. Let 1 < p < ∞. For a Banach space 𝔛, the following are equivalent.

(1) 𝔛 is uniformly convexifiable.

(2) L_{p}(𝔛) is uniformly convexifiable.

(3) L_{p}(𝔛) is asymptotically uniformly convexifiable.

(4) L_{p}(𝔛) admits an equivalent UKK norm.

(5) L_{p}(𝔛) is asymptotically uniformly smoothable.

Proof. Let 1 < p < ∞ and 𝔛 be a Banach space.

That (1) though (4) are equivalent and that (2) implies (5) follows easily from the below known facts about a Banach space Y.

(i) Y is uniformly convex if and only if L_{p}(Y) is [Mc].

(ii) Y is uniformly convexifiable if and only if L_{p}(Y) admits an equivalent

UKK norm [DGK, Theorem 4]. (iii) Y is uniformly convexifiable if and only if Y is uniformly smoothable

(cf. [DU, page 144]).

Towards showing that (5) implies (1), let L_{p}(𝔛) be asymptotically uniformly smoothable and 𝔛_{0} be a separable subspace of 𝔛. It suffices to show that 𝔛_{0} is uniformly convexifiable (cf. [DGZ, Remark IV.4.4]).

It follows from [GKL, Proposition 2.6] that if Y is separable, then Y is asymptotically uniformly smooth if and only if Y^{∗} has the UKK^{∗} property. Thus [L_{p}(𝔛_{0})]^{∗} admits an equivalent UKK^{∗} norm. But ℓ_{1} cannot embed into L_{p}(𝔛_{0}) since L_{p}(𝔛_{0}) is asymptotically uniformly smoothable and so [L_{p}(𝔛_{0})]^{∗} is asymptotically weak^{∗} uniformly convexifiable and so is also asymptotically uniformly convexifiable. Thus L_{q}(𝔛_{0}^{∗}) is asymptotically uniformly convexifiable where 1/p + 1/q = 1. From (3) implies (1) it follows that 𝔛_{0}^{∗} is uniformly convexifiable and so so is 𝔛_{0}. □

Acknowledgement. The author thanks William B. Johnson and Thomas Schlumprecht for their fruitful discussions on asymptotic uniform convexity at the NSF Workshop in Linear Analysis and Probability, Texas A&M University, during the Summer of 1999.

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Dept. of Math., University of South Carolina, Columbia, SC 29208, U.S.A.

E-mail address: girardi@math.sc.edu