BANACH SPACES WHICH ADMIT A NORM WITH THE UNIFORM KADEC-KLEE PROPERTY

S.J. Dilworth, Maria Girardi, Denka Kutzarova

Studia Math. 112 (1995) 267–277

Abstract. Several results are established about Banach spaces 𝔛 which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from 𝔛 to the Lebesgue-Bochner space L_{2}(𝔛) if and only if 𝔛 is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.

1. INTRODUCTION

A sequence {x_{n}} in a Banach space 𝔛 is separated (respectively, ε-separated) if inf{∥ x_{n} − x_{m} ∥ : n ≠ m} > 0 (respectively, ≥ ε). Recall that 𝔛 has the Kadec-Klee property if every separated weakly convergent sequence {x_{n}} in the closed unit ball of 𝔛 converges to an element of norm strictly less than one. We say that 𝔛 has the uniform Kadec-Klee (UKK) property (or that 𝔛 has a UKK norm) if for each ε > 0 there exists δ > 0 such that every ε-separated weakly convergent sequence {x_{n}} in the closed unit ball of 𝔛 converges to an element of norm less than 1−δ. This notion was introduced by Huff in [15]. Clearly, if 𝔛 has the Schur property (that is, if weak and norm sequential convergence are the same) or if 𝔛 is uniformly convex then 𝔛 has the UKK property. While uniformly convex spaces are necessarily reflexive, it turns out that many classical non-reflexive spaces, e.g. the Hardy spaces H_{1} of analytic functions on the ball or on the polydisk [1], the Lorentz spaces L_{p,1}(μ) [5,9], and the trace class C_{1} [13,25], all have UKK norms.

The question of characterizing the Banach spaces which are isomorphic to uniformly convex spaces has been studied intensively. This paper takes up the related question raised in [15]: under what conditions does a Banach space possess an

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 46B22, 46B20, 46B28, 46G99.

S.J. Dilworth was supported in part by the NSF Workshop in Linear Analysis and Probability

Maria Girardi was supported in part by NSF DMS-9306460

Denka Kutzarova was supported in part by the Bulgarian Ministry of Education and Science under contract MM-213/92

equivalent norm with the UKK property? We call this property UKK-ability, and a Banach space having this property is said to be UKK-able.

Recall that a Banach space 𝔛 has the complete continuity property (CCP) if every bounded linear operator from L_{1} into 𝔛 is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The CCP follows from the Radon-Nikodým property (RNP) because representable operators from L_{1} into 𝔛 are completely continuous. While there are examples of spaces [8] with the Schur (and hence the UKK) property but not the RNP, it is proved in Section 2 below that UKK-ability implies the CCP. In Section 3 it is proved that the Lebesgue-Bochner spaces L_{p}(𝔛) (p > 1) are UKK-able if and only if 𝔛 is super-reflexive (this is the isomorphic version of a theorem of Partington [28]), from which the existence of a B-convex space which fails to be UKK-able is deduced. In the fourth section UKK-ability is discussed for Banach spaces with a basis (or a finite-dimensional decomposition). Using ideas of Prus [31] a basis characterization for the weak-star version of UKK-ability is given for dual Banach spaces. It is also proved that the space constructed by Gowers [10] which does not contain c_{0}, ℓ_{1}, or any infinitedimensional reflexive subspace is UKK-able.

We wish to mention that UKK renormings were recently considered by Lancien, who earlier and independently proved Theorem 4 of this paper in his thesis [23].

Throughout the paper, 𝔛 denotes an arbitrary Banach space, 𝔛^{∗} the dual space, S(𝔛) the unit sphere, and Ba(𝔛) the closed unit ball of 𝔛. For any unexplained terminology the reader is referred to [8] or [26].

Finally, we wish to thank Haskell Rosenthal for several enlightening conversations about this subject matter during the conference on Banach space theory held in Ascona in September, 1993.

2. UKK implies CCP

First we establish some notation. Let L_{1} denote the space of integrable functions on [0, 1] with respect to Lebesgue measure μ. Let

\{I_{k}^{n} = [\frac{k−1}{2^{n}}, \frac{k}{2^{n}}): n = 0, 1, 2, … and k = 1, …, 2^{n}\}

be the usual dyadic splitting of [0, 1]. The Haar functions {h_{j}}_{j≥1} are defined by

h_{1} = 1_{I_{1}^{0}}  and  h_{2^{n}+k} = 2^{n} (1_{I_{2k−1}^{n+1}} − 1_{I_{2k}^{n+1}})

for n = 0, 1, 2, … and k = 1, …, 2^{n}. The Rademacher functions {r_{n}}_{n≥0} are defined by r_{0} = h_{1} and r_{n} = 2^{1−n} ∑_{k=1}^{2^{n−1}} h_{2^{n−1}+k} for n ≥ 1.

Theorem 1. A UKK-able Banach space enjoys the CCP.

Proof. Let 𝔛 be a Banach space failing the CCP. By the main result of [11] there is a norm one operator T from L_{1} into 𝔛 along with an ε > 0 and a sequence {x_{n}^{∗}}_{n≥0} from S(𝔛^{∗}) such that the following conditions hold:

(a) x_{n}^{∗} (T h_{2^{n}+k}) > ε for each n ≥ 0 and k = 1, … 2^{n};

(b) the natural blocking 𝔛_{n} ≡ span(T h_{j}: 2^{n−1} < j ≤ 2^{n}) for n ≥ 0 of the images of the Haar functions forms a finite-dimensional decomposition with constant at most 2.

Fix δ > 0. Since

∥ T ∥ = sup \{\frac{T 1_{I_{k}^{n}}}{μ(I_{k}^{n})} : n = 0, 1, 2, … and k = 1, …, 2^{n}\} ,

there is a dyadic interval A = I_{k_{δ}}^{n_{δ}} such that 1 − δ < ∥ T (\frac{1_{A}}{μ(A)}) ∥. We now restrict attention to this interval A.

Consider the sequence {x_{n}}_{n>n_{δ}} from 𝔛 given by

x_{n} = T (\frac{(1 + r_{n})1_{A}}{μ(A)} ) .

Clearly, the sequence \{x_{n}\} is contained in Ba(𝔛) and converges weakly to T (\frac{1_{A}}{μ(A)}). Note that if n > n_{δ} then T (r_{n} 1_{A}) is in 𝔛_{n}. Thus condition (b) gives that for m > n > n_{δ},

∥ x_{n} − x_{m} ∥ ∥ T (\frac{(r_{n} − r_{m})1_{A}}{μ(A)} ) ∥ \frac{1}{2} ∥ T (\frac{r_{n} 1_{A}}{μ(A)} ) ∥ .

For n > n_{δ}, each r_{n}1_{A} has the form

r_{n}1_{A} = 2^{1−n} ∑_{j∈J} h_{2^{n−1}+j}

for an appropriate set J of cardinality 2^{n−1}μ(A). So condition (a) gives that

∥ T (r_{n}1_{A}) ∥ 2^{1−n} ∑_{j∈J} ε

Thus the sequence \{x_{n}\}_{n>n_{δ}} is \frac{ε}{2}-separated. Since δ > 0 is arbitrary, it follows that 𝔛 does not have the UKK property. But since the CCP is an isomorphic property, in fact it follows that 𝔛 is not UKK-able. □

Remark. Recall that a Banach space 𝔛 has the point of continuity property (PCP) if every bounded weakly-closed set A ⊂ 𝔛 has non-empty relatively weakly-open subsets of arbitrarily small diameter. Since the PCP is “separably determined” (see [2] or [32]), whenever 𝔛 fails the PCP there is a separable bounded weaklyclosed set A and an ϵ > 0 such that every a ∈ A belongs to the weak closure of A \ {x : ∥x − a∥ < 2ϵ}. If, in addition, 𝔛 does not contain ℓ_{1}, then by [27] and [3] the weak topology on A is sequentially determined, and so every a ∈ A is the weak-limit of a sequence {x_{n}} ⊂ A \ {x : ∥x − a∥ < 2ϵ}. Clearly, {x_{n}} contains a subsequence {x_{n_{k}}} which is ϵ-separated. Arguing now as in Theorem 1, it follows that 𝔛 is not UKK-able. Thus we have proved the following result.

Proposition 2. If 𝔛 is UKK-able and does not contain ℓ_{1} then 𝔛 has the PCP.

We thank Haskell Rosenthal for showing us this result and its proof. The Bourgain-Rosenthal space [4] has the Schur property and fails the PCP (see [33]), and so the requirement in Proposition 2 that 𝔛 does not contain ℓ_{1} cannot be eliminated.

As a further application of the result from [11], we give a characterization of the CCP for subspaces of Banach spaces with an unconditional basis: in this case the failure of the CCP is equivalent to isomorphic containment of c_{0}. Since L_{1}(0, 1) fails the CCP, this fact may be regarded as an extension of the theorem of Pełczyński [29] saying that L_{1}(0, 1) does not embed into a space with an unconditional basis. The result may also be obtained from a theorem of Wessel [35], although his methods are somewhat different. In fact, a stronger result is known: James [19, Theorem 4.5] has proved the same result for the PCP. He has also proved that there is no corresponding result for the RNP by constructing a Banach space 𝔛, which is a subspace of a space with an unconditional basis, such that 𝔛 fails the RNP and 𝔛 does not contain c_{0} [20].

Theorem 3. Suppose that 𝔛 embeds isomorphically into a Banach space with an unconditional basis. Then either 𝔛 contains c_{0} or 𝔛 has the CCP.

Proof. Suppose that 𝔛 fails the CCP and embeds into a space Y with an unconditional basis {y_{n}} with basis constant K. As in the proof of Theorem 1, there is a norm one operator T: L_{1} → 𝔛 along with an ε > 0 and a sequence {x_{n}^{∗}} from S(𝔛^{∗}) satisfying conditions (a) and (b) above. Fix a sequence {γ_{n}}_{n≥1} such that 0 < γ_{n} < ε(4K 2^{n})^{−1}.

First we construct, by induction on n, two increasing sequences {k_{n}}_{n≥1} and {p_{n}}_{n≥1} of positive integers such that for the sequence {f_{n}}_{n≥1} in L_{1} defined by

f_{n} = (f_{0} + … + f_{n−1}) r_{k_{n}}

(where f_{0} = r_{0}) and for the blocking {Y_{n}}_{n≥1} of Y defined by

Y_{n} = sp{y_{j}: p_{n−1} ≤ j < p_{n}}

(where p_{0} = 1) the following conditions hold:

(1) ∥ T f_{n} ∥ > ε;

(2) ∥ f_{0} + … + f_{n} ∥ < 2;

(3) d(T f_{n}, Y_{n}) < γ_{n}.

Put k_{1} = 1 so that f_{1} = r_{1}. Note that condition (a) gives (1), while (2) is clear. Select p_{1} such that d(T f_{1}, Y_{1}) < γ_{1}.

For m ≥ 1, assume that {k_{n}}_{n=1}^{m} and {p_{n}}_{n=1}^{m} have been chosen so that {f_{n}}_{n=1}^{m} satisfy (1), (2), and (3). As k tends to infinity, we have the following:

(i) ∥ (f_{0} + … + f_{m}) + (f_{0} + … + f_{m})r_{k} ∥_{L_{1}} → ∥ f_{0} + … + f_{m} ∥_{L_{1}}; (ii) (f_{0} + … + f_{m})r_{k} → 0 weakly in L_{1} .

So there exists k_{m+1} > k_{m} such that for f_{m+1} = (f_{0} + … + f_{m})r_{k_{m+1}} the following hold:

(iii) ∥ f_{0} + … + f_{m} + f_{m+1} ∥_{L_{1}} < 2;

(iv) T f_{m+1} is within \frac{1}{2} γ_{m+1} of some element from sp\{y_{j}: p_{m} ≤ j < ∞\}. Now choose p_{m+1} such that d(T f_{m+1}, Y_{m+1}) < γ_{m+1}.

To verify condition (1), note that f_{m+1} has the form

f_{m+1} = f_{0} [ ∏_{j<m+1}(1 + r_{k_{j}}) ] r_{m+1} .

The support of ∏_{j<m+1}(1 + r_{k_{j}}) has measure 2^{−m} and is the union of 2^{−m+k_{m}} dyadic intervals from {I_{i}^{k_{m}}: 1 ≤ i ≤ 2^{k_{m}}} of the k_{m}^{th} -level. Thus, for some subset J of integers with cardinality 2^{−m−1+k_{m+1}}, we have

f_{m+1} = 2^{m} 1_{A} r_{k_{m+1}} = 2^{m} ∑_{j∈J} 2^{1−k_{m+1}} h_{j+2^{−1+k_{m+1}}} .

Condition (a) gives that

∥ T f_{m+1} ∥ > 2^{m} 2^{−m−1+k_{m+1}} 2^{1−k_{m+1}} ε = ε .

This completes the induction. Let x_{n} = T f_{n}. Condition (3) guarantees that {x_{n}} is equivalent to some (unconditional) block basic sequence of {y_{n}}. Condition (2) guarantees that ∥ x_{1} + … + x_{n} ∥ ≤ 3. Condition (1) just says that ∥ x_{n} ∥ > ε. Thus, {x_{n}} is equivalent to the standard unit vector basis of c_{0}. □

3. UKK-ability of Lebesgue-Bochner spaces

In this section L_{p}(𝔛) denotes the Lebesgue-Bochner space of strongly-measurable 𝔛-valued functions defined on a separable non-atomic probability space, equipped with the L_{p}-norm. In [28] Partington proved that if L_{2}(𝔛) is UKK (with its usual norm) then 𝔛 is uniformly convex. The isomorphic version of this result was recently proved by Lancien [23,24]. In this section we shall give a different proof of Lancien’s result. In order to prove the theorem we require the following necessary condition for 𝔛 to be UKK-able, which is due to Huff [15].

Fact. For ϵ > 0, let B_{ϵ}^{(0)} = Ba(𝔛), and for n ≥ 1 define B_{ϵ}^{(n)}(𝔛) inductively thus:

B_{ϵ}^{(n)}(𝔛) = {x ∈ 𝔛 : x = w − lim{x_{k}}, x_{k} ∈ B_{ϵ}^{(n−1)}, ∥x_{j} − x_{k}∥ ≥ ϵ (j ≠ k)}.

Suppose that 𝔛 admits an equivalent UKK norm. Then, for each ϵ > 0, B_{ϵ}^{(n)}(𝔛) = ∅ for all sufficiently large n.

Theorem 4. Let 1 < p < ∞. Then L_{p}(𝔛) admits an equivalent UKK norm if and only if 𝔛 is super-reflexive.

Proof. Suppose that 𝔛 is not super-reflexive. Then, by a result of James [16], there exists ϵ > 0 such that for every N ≥ 1 there is a dyadic martingale difference sequence {d_{k}}_{k=0}^{N}, with d_{0} = 0, adapted to the standard filtration of {0, 1}^{N}, such that the corresponding martingale takes its values in Ba(𝔛) and such that ∥d_{k}(ω)∥ ≥ ϵ for all k ≥ 1 and for all ω ∈ {0, 1}^{N}. For a fixed integer N ≥ 1, let Γ_{N} = ∪_{k=1}^{N}ℕ^{k}. We shall consider random variables defined on the Cantor group {0, 1}^{Γ_{N}} with its associated Haar probability measure μ. For each 1 ≤ k ≤ N and for each (n_{1}, n_{2}, …, n_{k}) ∈ ℕ^{k}, we define a random variable D_{(n_{1},...,n_{k})} thus:

D_{(n_{1},...,n_{k})}(ω) = d_{k}((ω_{(n_{1})}, ω_{(n_{1},n_{2})}, …, ω_{(n_{1},...,n_{k})}, 0, …, 0))

for each ω ∈ {0, 1}^{Γ_{N}}. Observe that for each N−tuple (n_{1}, n_{2}, …, n_{N}), the random variables D_{(n_{1})}, D_{(n_{1},n_{2})}, … D_{(n_{1},...,n_{N})} have the same joint distribution as d_{1}, d_{2}, …, d_{N}. Fix n_{1}, …, n_{N−1}. For k ≥ 1, let

X_{k} = (D_{(n_{1})} + ⋯ + D_{(n_{1},...,n_{N−1})}) + D_{(n_{1},n_{2},...,n_{N−1},k)}.

It is easily seen that {D_{(n_{1},...,n_{N−1},k)}}_{k=1}^{∞} is weakly null in L_{p}(𝔛), and so {X_{k}} converges weakly to D_{(n_{1})} + ⋯ + D_{(n_{1},...,n_{N−1})}. Moreover, ∥X_{k}(ω)∥ ≤ 1 for all ω, and so X_{k} ∈ Ba(L_{p}(𝔛)). For j ≠ k, we have

μ\{X_{j} = X_{k}\} = μ\{∥X_{j} − X_{k}∥ ≥ 2ϵ\} = \frac{1}{2},

and so ∥X_{j} − X_{k}∥_{L_{p}(𝔛)} > ϵ. Hence B_{ϵ}^{(1)}(L_{p}(𝔛)) contains every random variable of the form D_{(n_{1})} + ⋯ + D_{(n_{1},...,n_{N−1})}. Repeating this argument a total of N times we see that B_{ϵ}^{(N)}(L_{p}(𝔛)) contains the zero random variable. Since this holds for every N ≥ 1, it follows from the Fact that L_{p}(𝔛) does not admit a UKK norm. Conversely, suppose that 𝔛 is super-reflexive. Then L_{p}(𝔛) is also super-reflexive, and so L_{p}(𝔛) admits an equivalent uniformly convex (and hence UKK) norm. □

Corollary 5. There exists a uniformly non-octahedral (in particular, a B-convex) space which does not admit an equivalent UKK norm.

Proof. Let 𝔛 be the non-reflexive uniformly non-octahedral space constructed by James [17]. It is well-known that the property of being uniformly non-octahedral lifts from 𝔛 to L_{2}(𝔛). On the other hand, since 𝔛 is non-reflexive, it follows from Theorem 4 that L_{2}(𝔛) does not admit an equivalent UKK norm. □

Remarks. 1. Similarly, if 𝔛 is non-reflexive and of type two, [18], then L_{2}(𝔛) is of type two but not UKK-able.

2. In [14] it is proved that if L_{2}(𝔛) has the CPCP (which is the PCP for convex sets) then 𝔛 has the RNP. Corollary 5 can also be proved by combining this theorem with Proposition 2 above.

3. Since (∑_{n=1}^{∞} ⨁ ℓ_{∞}^{n})_{2} has the UKK property [15] it follows that the UKK property does not imply any non-trivial super-property. On the other hand, every Banach space which is super-UKK-able is necessarily B-convex (since L_{1}(0, 1) is not UKK-able), and by Corollary 5 there exist Banach spaces which are B-convex but not super-UKK-able. Clearly, every super-reflexive space is super-UKK-able, and so one is led to pose the following question: is super-UKK-ability equivalent to super-reflexivity? A positive answer to this question would follow from a positive answer to the corresponding question for the super-CPCP, which was raised in [14].

4. UKK-ability for spaces with a basis

In [31] Prus characterizes reflexive UKK-able Banach spaces in terms of their basic sequences. While some results in [31] rely heavily on the weak compactness of the unit ball, using methods from [31] some partial information is obtained in this section concerning UKK-ability of non-reflexive Banach spaces with a basis. In particular, we prove that the space constructed by Gowers which is hereditarily non-reflexive and contains no copy of c_{0} or ℓ_{1} does admit an equivalent UKK norm.

First we recall some notation from [31]. Let {e_{n}} be a basic sequence in a Banach space 𝔛 with coefficient functional sequence {e_{n}^{∗}} in 𝔛^{∗}. An element x ∈ [e_{n}] ≡ span{e_{n}} is said to be a block if supp(x) = {n : e_{n}^{∗}(x) ≠ 0} is finite. A family {𝔛_{n}} of finite-dimensional subspaces of [e_{n}] is a blocking of {e_{n}} provided there exists an increasing sequence of integers {n_{k}}, n_{1} = 1, such that 𝔛_{k} = [e_{i}]_{i=n_{k}}^{n_{k+1}−1} for each k. We say that the blocks y_{1}, y_{2}, …, y_{n} are disjoint (with respect to the blocking {𝔛_{k}}) and write y_{1} < y_{2} < ⋯ < y_{n} if

min{m : y_{i} ∈ ∑_{j=1}^{m} 𝔛_{j}} < max{m : y_{i+1} ∈ ∑_{j=m}^{∞} 𝔛_{j}}

for i = 1, 2, … n − 1. Finally, we say that, for 1 ≤ p, q ≤ ∞, the blocking {𝔛_{n}} satisfies a (p, q)−estimate provided there exist positive constants c and C such that

c (∑_{i=1}^{n} ∥y_{i}∥^{p})^{1/p} ≤ ∥∑_{i=1}^{n} y_{i}∥ ≤ C (∑_{i=1}^{n} ∥y_{i}∥^{q})^{1/q}

for all disjoint blocks y_{1}, y_{2}, … y_{n}. The following result is essentially due to Prus. The reader is referred to [31, p.517] for the proof.

Theorem 6. Let 𝔛 be a Banach space with a basis {e_{n}}. If there exists a blocking {𝔛_{n}} of the basis which satisfies a (p, 1)−estimate for some p < ∞ then 𝔛 admits an equivalent UKK norm.

In fact, the equivalent norm constructed on 𝔛 in Theorem 6 makes the blocking {𝔛_{n}} monotone and boundedly complete; and so, under this renorming 𝔛 is isometric to the dual space of Y = [e_{n}^{∗}] ⊂ 𝔛^{∗}. Moreover, 𝔛 has the weak-star UKK property with respect to its predual Y. (The weak-star UKK is in general stronger than the UKK property: it is defined in the obvious way by considering all ϵ-separated weak-star convergent sequences.) Thus, in looking for a converse to Theorem 6, it is appropriate to consider dual Banach spaces with the weak-star UKK property. To that end, say that a basis {e_{n}} of a Banach space 𝔛 is weakly nearly uniformly smooth (WNUS) if there exists c > 0 such that for every normalized block basic sequence {x_{n}} there exists k > 1 such that ∥x_{1} + x_{k}∥ < 2 − c. We shall use the following result, which is again essentially due to Prus. The reader is referred to [31, pp.512-513] for the proof.

Theorem 7. Let {e_{n}} be a WNUS basis for a Banach space 𝔛. Then {e_{n}} has a blocking {𝔛_{n}} which satisfies an (∞, q)−estimate for some q > 1.

We can now prove the converse of Theorem 6.

Theorem 8. Suppose that 𝔛^{∗} has a basis and that 𝔛^{∗}, with its usual dual norm, has the weak-star UKK property. Then 𝔛^{∗} has a basis which admits a blocking satisfying a (p, 1)−estimate for some p < ∞.

Proof. By [21] we may suppose that 𝔛 has a shrinking basis, so that {e_{n}^{∗}} is a basis for 𝔛^{∗}. We shall show that {e_{n}} has a blocking which satisfies an (∞, q)−estimate for some q > 1, whence by a duality argument [30] the basis {e_{n}^{∗}} of 𝔛^{∗} has a blocking which satisfies a (p, 1)− estimate, where \frac{1}{p} + \frac{1}{q} = 1. By Theorem 7, it suffices to prove that \{e_{n}\} is WNUS. Choose δ > 0 corresponding to ϵ = \frac{1}{4} in the definition of the weak-star UKK property; clearly, we may assume that δ < \frac{1}{4}. Let {x_{k}} be a normalized block basis with respect to {e_{n}}. Select x_{k}^{∗} ∈ Ba(𝔛^{∗}) such that x_{k}^{∗}(x_{1} +x_{k}) = ∥x_{1} +x_{k}∥. By passing to a subsequence and using the weak-star sequential compactness of Ba(𝔛^{∗}) we may assume that {x_{k}^{∗}} converges weak-star to x^{∗} ∈ Ba(𝔛^{∗}). Hence there exists k_{0} such that |(x_{k}^{∗} − x^{∗})(x_{1})| < δ/2 for all k > k_{0}. The proof now divides into two cases.

Case 1. There exists k > k_{0} such that |x^{∗}(x_{k})| < δ/2 and ∥x_{k}^{∗} − x^{∗}∥ < \frac{1}{2}. Then

∥x_{1} + x_{k}∥ = x_{k}^{∗}(x_{1} + x_{k})

= x^{∗}(x_{1} + x_{k}) + (x_{k}^{∗} − x^{∗})(x_{1} + x_{k})

≤ |x^{∗}(x_{1})| + |x^{∗}(x_{k})| + |(x_{k}^{∗} − x^{∗})(x_{1})| + ∥x_{k}^{∗} − x^{∗}∥

< 1 + \frac{δ}{2} + \frac{δ}{2} + \frac{1}{2} ≤ \frac{7}{4}.

Case 2. ∥x_{k}^{∗} − x^{∗}∥ ≥ \frac{1}{2} for all k > k_{0}. It follows from the weak-star lower semicontinuity of the norm and from the weak-star convergence of {x_{k}^{∗}} to x^{∗} that {x_{k}^{∗}} has a subsequence which is \frac{1}{4}-separated. Hence, by the weak-star UKK property, ∥x^{∗}∥ < 1 − δ. So, provided k > k_{0}, we have

∥x_{1} + x_{k}∥ = x_{k}^{∗}(x_{1} + x_{k})

≤ |x^{∗}(x_{1})| + |(x_{k}^{∗} − x^{∗})(x_{1})| + |x_{k}^{∗}(x_{k})|

≤ (1 − δ) + \frac{δ}{2} + 1

≤ 2 − \frac{δ}{2}.

So in both cases we have shown that 𝔛 is WNUS. □

Remark. With only straightforward modifications to the proofs, Theorems 6, 7 and 8 remain valid if “basis” is replaced by “finite-dimensional decomposition” throughout.

Next we show that the space discovered by Gowers [10], which contains no copy of c_{0}, ℓ_{1}, or any infinite-dimensional reflexive Banach space, admits a UKK norm. To see this, we require the following criterion for a basis to admit a (p, 1)−estimate (cf. [22]).

Proposition 9. Let 𝔛 be a Banach space with a basis {e_{n}}. Suppose that there exists 0 < c < 2 such that c∥x + y∥ ≥ ∥x∥ + ∥y∥ for all disjoint blocks x, y ∈ 𝔛. Then {e_{n}} satisfies a (p, 1)−estimate for some p < ∞.

Proof. Consider the basic sequence {e_{n}^{∗}} in 𝔛^{∗}. First we show that if x^{∗}, y^{∗} are disjoint blocks in Ba([e_{n}^{∗}]), then ∥x^{∗} + y^{∗}∥ ≤ c. Indeed, to derive a contradiction, suppose that there exist such blocks with ∥x^{∗} + y^{∗}∥ > c. We may suppose that supp(x^{∗})≤ n < supp(y^{∗}). Select z ∈ Ba(𝔛) with (x^{∗} + y^{∗})(z) > c. Write z = x + y, where supp(x)≤ n < supp(y). Then

c < (x^{∗} + y^{∗})(z)
= x^{∗}(x) + y^{∗}(y)
≤ ∥x∥ + ∥y∥
≤ c∥x + y∥
≤ c,

which is a contradiction. By an argument of Gurarii and Gurarii [12] (see e.g. [7]) it now follows that {e_{n}^{∗}} satisfies an (∞, q)−estimate for some q > 1, and hence by duality that {e_{n}} satisfies a (p, 1)−estimate for some p < ∞. □

Corollary 10. The space constructed by Gowers (without c_{0}, ℓ_{1}, or a reflexive subspace) admits a UKK norm.

Proof. Let f(x) = \sqrt{log_{2}(x + 1)}. From the definition of the norm [10] it is clear that for disjoint blocks x_{1}, x_{2}, …, x_{n} with respect to the basis {e_{n}} of the space one has

∥∑_{i=1}^{n} x_{i}∥ ≥ \frac{1}{f(n)} ∑_{i=1}^{n} ∥x_{i}∥.

In particular, if x_{1} < x_{2}, then

c∥x_{1} + x_{2}∥ ≥ ∥x_{1}∥ + ∥x_{2}∥

for c = \sqrt{log_{2} 3} < 2. So by Proposition 9 \{e_{n}\} satisfies a (p, 1)−estimate for some p < ∞, whence by Theorem 6 the space admits a UKK norm. □

Remarks. 1. It can be shown (see e.g. [6]) that without passing to a blocking the basis of the Gowers space satisfies a (p, 1)−estimate for every p > 1.

2. The original Tsirelson space [34] is reflexive and has the property that ℓ_{∞}^{n} is representable on blocks in every infinite-dimensional subspace. So, by the results of [30], no infinite-dimensional subspace is UKK-able.

References

1. M. Besbes, S. J. Dilworth, P. N. Dowling and C. J. Lennard, New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces, J. Funct. Anal. 119 (1994), 340–357.

2. J. Bourgain, La propriété de Radon-Nikodým, Publ. Math. Univ. Pierre et Marie Curie 36 (1979).

3. J. Bourgain, D.H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1979), 845–886.

4. J. Bourgain and H.P. Rosenthal, Martingales valued in certain subsets of L_{1}, Israel J. Math. 37 (1980), 54–75.

5. N. L. Carothers, S. J. Dilworth, C. J. Lennard and D. A. Trautman, A fixed point property for the Lorentz space L_{p,1}(μ), Indiana Univ. Math. J. 40 (1991), 345–352.

6. J. Castillo and F. Sanchez, Weakly p−compact, p−Banach Saks and super-reflexive Banach spaces, J. Math. Anal. Appl., to appear.

7. J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, New York, 1983.

8. J. Diestel and J.J. Uhl, Vector Measures, American Mathematical Society, 1977.

9. S.J. Dilworth and Yu-Ping Hsu, The uniform Kadec-Klee property for the Lorentz spaces L_{w,1}, J. Austral. Math. Soc., to appear.

10. W.T. Gowers, A space not containing c_{0}, ℓ_{1}, or a reflexive subspace, preprint, 1992.

11. Maria Girardi and W.B. Johnson, The Complete Continuity Property and Finite-Dimensional Decompositions, Canad. Math. Bull., to appear.

12. V.I. Gurarii and N.I. Gurarii, On bases in uniformly convex and uniformly smooth Banach spaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 210–215.

13. Yu-Ping Hsu, The lifting of the UKK property from E to C_{E}, Proc. Amer. Math. Soc., to appear.

14. Z. Hu and B-L Lin, RNP and CPCP in Lebesgue-Bochner function spaces, Illinois J. Math. 37 (1993), 329–347.

15. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743–749.

16. R.C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite Dimensional Topology, Ann. of Math. Stud. 69 (1972), 159–175.

17. R.C. James, A non-reflexive Banach space that is uniformly non-octahedral, Israel J. Math. 18 (1974), 145–155.

18. R.C. James, Non-reflexive spaces of type 2, Israel J. Math. 30 (1978), 1–13.

19. R.C. James, KMP, RNP and PCP for Banach spaces, Contemp. Math. 85 (1989), 281–317.

20. R.C. James, Unconditional bases and the Radon-Nikodým property, Studia Math. 95 (1990), 255–262.

21. W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions, and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506.

22. Denka Kutzarova and T. Zachariades, On orthogonal convexity and related properties, preprint, 1992.

23. Gilles Lancien, Applications de la théorie de l’indice en géometrie des espaces de Banach, Ph.D. thesis, Université de Paris VI, Paris, 1992.

24. Gilles Lancien, On uniformly convex and uniformly Kadec-Klee renormings, preprint.

25. C. J. Lennard, C_{1} is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), 71–77.

26. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces 1, Sequence spaces, Springer-Verlag, Berlin-Heidelberg, 1977.

27. E.W. Odell and Haskell P. Rosenthal, A double-dual characterization of Banach spaces containing ℓ_{1}, Israel J. Math. 22 (1978), 290–294.

28. J.R. Partington, On nearly uniformly convex Banach spaces, Math. Proc. Cambridge Philos. Soc. 93 (1983), 127–129.

29. A. Pełczyński, On the impossibility of embedding of the space L in certain Banach spaces, Colloq. Math. 8 (1961), 199–203.

30. Stanislaw Prus, Finite-dimensional decompositions of Banach spaces with (p, q)-estimates, Dissertationes Math. 263 (1987), 1–41.

31. Stanislaw Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. 7 (1989), 507– 521.

32. H.P. Rosenthal, Weak^{∗} Polish Banach spaces, J. Funct. Anal. 76 (1988), 267–316.

33. H.P. Rosenthal, On the structure of non-dentable closed bounded convex sets, Adv. Math. 70 (1988), 1–58.

34. B. Tsirelson, Not every Banach space contains an imbedding of ℓ_{p} or c_{0}, Functional Anal. Appl. 8 (1974), 138–141.

35. Alan Wessel, Some results on Dunford-Pettis operators, strong regularity and the Radon-Nikodým property, Séminaire d’Analyse Fonctionelle, Paris VI-VII, 1985-86, Publ. Math. Univ. Paris VII, Paris.

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. Current address: Department of Mathematics, Texas A&M University, College Station, Texas 77843, U.S.A.

E-mail address: dilworth@math.scarolina.edu

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

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

Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria Current address: Department of Mathematics, University of Missouri, Columbia, Missouri 65211, U.S.A.

E-mail address: Denka@bgearn.bitnet