THE COMPLETE CONTINUITY PROPERTY AND

FINITE DIMENSIONAL DECOMPOSITIONS

Maria Girardi and William B. Johnson

Canad. Math. Bull. 38 (1995) 207–214

Abstract. A Banach space 𝔛 has the complete continuity property (CCP) if each bounded linear operator from L_{1} into 𝔛 is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP has a subspace with a finite dimensional decomposition which fails the CCP. If furthermore the space has some nice local structure (such as fails cotype or is a lattice), then the decomposition may be strengthened to a bases.

1. introduction

Given a property of Banach spaces which is hereditary, it is natural to ask whether a Banach space has the property if every subspace with a basis (or with a finite dimensional decomposition) has the property. The motivation for such questions is of course that it is much easier to deal with Banach spaces which have a basis (or at least a finite dimensional decomposition) than with general spaces. In this note we consider these questions for the complete continuity property (CCP), which means that each bounded linear operator from L_{1} into the space is completely continuous (i.e., carries weakly convergent sequences into norm convergent sequences).

The CCP is closely connected with the Radon-Nikodým property (RNP). Since a representable operator is completely continuous, the RNP implies the CCP; however, the Bourgain-Rosenthal space [BR] has the CCP but not the RNP. Bourgain [B1] showed that a space failing the RNP has a subspace with a finite dimensional decomposition which fails the RNP. Wessel [W] showed that a space failing the CCP has a subspace with a basis which fails the RNP. It is open whether a space has the RNP (respectively, CCP) if every subspace with a basis has the RNP (respectively, CCP).

Our main theorem shows that if 𝔛 fails the CCP, then there is an operator T: L_{1} → 𝔛 that behaves like the identity operator I: L_{1} → L_{1} on the Haar functions {h_{j}}. Specifically, there is a sequence {x_{n}^{∗}} in the unit ball of 𝔛^{∗} such that x_{n}^{∗}

Typeset by AMS-TEX

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

The first author was supported in part by NSF DMS-9204301 and NSF DMS-9306460.

The second author was supported in part by NSF DMS-9003550.

keeps the image of each Haar function along the n^{th}-level large (i.e., x_{n}^{∗}(T h_{2^{n}+k}) > δ > 0 ) and the natural blocking {sp(T h_{2^{n}+k}: k = 1, …, 2^{n})}_{n} of the images of the Haar functions is a finite dimensional decomposition for some subspace 𝔛_{0}. Note that 𝔛_{0} fails the CCP since T is not completely continuous (T keeps the Rachemacher functions larger than δ in norm). Thus a space failing the CCP has a subspace with a finite dimensional decomposition which fails the CCP. In the language of Banach space geometry, the theorem says that in any Banach space which fails the CCP grows a separated δ-tree with a difference sequence naturally blocking into a finite dimensional decomposition. If furthermore the space has some nice local structure (such as fails cotype or is a lattice), then modifications produce a separated δ-tree growing inside a subspace with a basis.

Throughout this paper, 𝔛 denotes an arbitrary Banach space, 𝔛^{∗} the dual space of 𝔛, and S(𝔛) the unit sphere of 𝔛. The triple (Ω, Σ, μ) refers to the Lebesgue measure space on [0, 1], Σ^{+} to the sets in Σ with positive measure, and L_{1} to L_{1}(Ω, Σ, μ). All notation and terminology, not otherwise explained, are as in [DU].

The authors are grateful to Michel Talagrand and Peter Casazza for helpful discussions.

2. operator view-point

A system A = {A_{k}^{n} ∈ Σ: n = 0, 1, 2, … and k = 1, …, 2^{n}} is a dyadic splitting of A_{1}^{0} ∈ Σ^{+} if each A_{k}^{n} is partitioned into the two sets A_{2k−1}^{n+1} and A_{2k}^{n+1} of equal measure for each admissible n and k . Thus the collection π_{n} = {A_{k}^{n}: k = 1, …, 2^{n}} of sets along the n^{th}-level partition A_{1}^{0} with π_{n+1} refining π_{n} and μ(A_{k}^{n}) = 2^{−n}μ(A_{1}^{0}). To a dyadic splitting corresponds a (normalized) Haar system {h_{j}}_{j≥1} where

h_{1} = \frac{1}{μ(A_{1}^{0})} 1_{A_{1}^{0}}  and  h_{2^{n}+k} = \frac{2^{n}}{μ(A_{1}^{0})} (1_{A_{2k−1}^{n+1}} − 1_{A_{2k}^{n+1}})

for n = 0, 1, 2, … and k = 1, …, 2^{n}.

A set N in the unit sphere of the dual of a Banach space 𝔛 is said to norm a subspace 𝔛_{0} within τ > 1 if for each x ∈ 𝔛_{0} there is x^{∗} ∈ N such that ||x|| ≤ τx^{∗}(x). It is well known and easy to see that a sequence {𝔛_{j}} of subspaces of 𝔛 forms a finite dimensional decomposition with constant at most τ provided that for each n ∈ ℕ the space generated by {𝔛_{1}, …, 𝔛_{n}} can be normed by a set from S(𝔛_{n+1}^{⊥}) within τ_{n} > 1 where Πτ_{n} ≤ τ.

Theorem 1. If the bounded linear operator T : L_{1} → 𝔛 is not completely continuous and {τ_{n}}_{n≥0} is a sequence of numbers larger than 1, then there exists

(A) a dyadic splitting A = {A_{k}^{n}} (B) a sequence {x_{t_{n}}^{∗}}_{n≥0} in S(𝔛^{∗}) (C) a finite set {y_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) for each n ≥ 0

such that for the Haar system {h_{j}}_{j≥1} corresponding to A, for some δ > 0, and each n ≥ 0

(1) x_{t_{n}}^{∗}(T h_{2^{n}+k}) > δ for k = 1, …, 2^{n} (2) {y_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(T h_{j}: 1 ≤ j ≤ 2^{n}) within τ_{n}

(3) y_{n,i}^{∗}(T h_{2^{n}+k}) = 0 for k = 1, …, 2^{n} and i = 1, …, p_{n}. Note that if Πτ_{n} is finite, then conditions (2) and (3) guarantee that the natural blocking {sp(T h_{j}: 2^{n−1} < j ≤ 2^{n})}_{n≥0} forms a finite dimensional decomposition with constant at most Πτ_{n}.

The proof uses the following standard lemma which, for completeness, we shall prove later.

Lemma 2. If A ∈ Σ^{+} and {g_{i}}_{i=1}^{n} is a finite collection of L_{1} functions, then an extreme point u of the set C ≡ {f ∈ L_{1} : |f| ≤ 1_{A} and ∫_{A} fg_{i} dμ = 0 for i = 1, …, n} has the form |u| = 1_{A}.

Proof of Theorem 1. Let T: L_{1} → 𝔛 be a norm one operator that is not completely continuous. Then there is a sequence {r_{t}} in L_{1} and a sequence {x_{t}^{∗}} in S(𝔛^{∗}) satisfying:

(a) ||r_{t}||_{L_{∞}} ≤ 1

(b) r_{t} converges to 0 weakly in L_{1}

(c) 4δ ≤ x_{t}^{∗} T r_{t} for some δ > 0 . Consider T^{∗}x_{t}^{∗} ∈ L_{∞}. Since ||r_{t}(T^{∗}x_{t}^{∗})||_{L_{∞}} is at most 1, by passing to a subsequence we may assume that {r_{t}(T^{∗}x_{t}^{∗})} converges to some function h in the weak-star topology on L_{∞}. Since ∫ h dμ ≥ 4δ the set A_{1}^{0} ≡ [h ≥ 4δ] is in Σ^{+}. (Compare this with [B2, proposition 5]).

We shall construct, by induction on the level n, a dyadic splitting of A_{1}^{0} along with the desired functional. Fix n ≥ 0.

Suppose we are given a finite dyadic splitting {A_{k}^{m}: m = 0, …, n and k = 1, …, 2^{m}} of A_{1}^{0} up to n^{th}-level. This gives the corresponding Haar functions {h_{j}: 1 ≤ j ≤ 2^{n}}. For each 1 ≤ k ≤ 2^{n}, we shall partition A_{k}^{n} into 2 sets A_{2k−1}^{n+1} and A_{2k}^{n+1} of equal measure (thus finding h_{2^{n}+k}) and find x_{t_{n}}^{∗} ∈ S(𝔛^{∗}) and a sequence {y_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) such that conditions (1), (2), and (3) hold.

Find a finite set {y_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) that norms sp(T h_{j}: 1 ≤ j ≤ 2^{n}) within τ_{n} . Let

\begin{aligned}C_{k}^{n} & ≡ \\ & \{f ∈ L_{1} : |f| ≤ 1_{A_{k}^{n}}, ∫_{A_{k}^{n}} f dμ = 0 and ∫_{A_{k}^{n}}(T^{∗}y_{n,i}^{∗})f dμ = 0 for 1 ≤ i ≤ p_{n}\}. \end{aligned}

Note that each C_{k}^{n} is a convex weakly compact subset of L_{1}.

Since {r_{t}} tends weakly to 0, for large t there is a small perturbation r̃_{t} of r_{t} so that r̃_{t}1_{A_{k}^{n}} is in C_{k}^{n} for each k. To see this, put

F = sp ({1_{A_{k}^{n}}} ∪ {(T^{∗}y_{n,i}^{∗})1_{A_{k}^{n}}: k = 1, …, 2^{n} and i = 1, …, p_{n}}) ⊂ L_{1} .

Now pick t_{n} ≡ t so large that for k = 1, …, 2^{n} and i = 1, …, p_{n} (d) ∫_{A_{k}^{n}} r_{t}(T^{∗}x_{t}^{∗}) dμ ≥ 2δα_{n}

(e) |∫_{Ω} r_{t}f dμ| ≤ \frac{δ}{3}α_{n}||f|| for all f in F where α_{n} = 2^{−n} μ(A_{1}^{0}) ≡ μ(A_{k}^{n}). Condition (d) follows from the definition of A_{1}^{0} and the weak-star convergence of {r_{t}(T^{∗}x_{t}^{∗})} to h while condition (e) follows from (b) and the fact that F is finite dimensional.

Thus the L_{∞}-distance from r_{t} to F^{⊥} ≡ {g ∈ L_{∞}: ∫_{Ω} fg dμ = 0 for each f ∈ F } is at most \frac{δ α_{n}}{3} . So there is r̃_{t} ∈ F^{⊥} such that ||r̃_{t} − r_{t}||_{L_{∞}} is less than δα_{n} and, as with r_{t}, is of L_{∞}-norm at most 1. Clearly r̃_{t}1_{A_{k}^{n}} ∈ C_{k}^{n} for each k = 1, …, 2^{n}.

The functional T^{∗}x_{t}^{∗} ∈ L_{1}^{∗} attains its maximum on C_{k}^{n} at an extreme point u_{k}^{n} of C_{k}^{n}. By the lemma, u_{k}^{n} = 1_{A_{2k−1}^{n+1}} − 1_{A_{2k}^{n+1}} for 2 disjoint sets A_{2k−1}^{n+1} and A_{2k}^{n+1}

whose union is A_{k}^{n}. Furthermore, A_{2k−1}^{n+1} and A_{2k}^{n+1} are of equal measure since ∫_{A_{k}^{n}} u_{k}^{n} dμ = 0.

Condition (3) holds since for i = 1, …, p_{n} and k = 1, …, 2^{n}

y_{n,i}^{∗}(T h_{2^{n}+k}) = α_{n}^{−1} ∫_{A_{k}^{n}}(T^{∗}y_{n,i}^{∗})u_{k}^{n} dμ = 0 .

Condition (1) follows from the observations that

x_{t_{n}}^{∗}(T h_{2^{n}+k}) = α_{n}^{−1} (T^{∗}x_{t_{n}}^{∗})u_{k}^{n} ≥ α_{n}^{−1} (T^{∗}x_{t_{n}}^{∗})(r̃_{t}1_{A_{k}^{n}})

and

|(T^{∗}x_{t_{n}}^{∗})(r̃_{t}1_{A_{k}^{n}}) − (T^{∗}x_{t_{n}}^{∗})(r_{t}1_{A_{k}^{n}})| ≤ ||r̃_{t} − r_{t}||_{L_{1}} < δ α_{n}

and

(T^{∗}x_{t_{n}}^{∗})(r_{t}1_{A_{k}^{n}}) ≥ 2δα_{n} .

Proof of Lemma 2. Fix a function f of C such that |f| ≠ 1_{A}. Find a positive α and a subset B of A with positive measure such that |f1_{B}| < 1 − α.

Let Σ̃ = B ∩ Σ. Consider the measures λ_{i}: Σ̃ → ℝ given by λ_{i}(E) ≡ ∫_{E} g_{i} dμ. Define the measure λ: Σ̃ → ℝ^{n+1} by

λ(E) = (λ_{1}(E), …, λ_{n}(E), μ(E)) .

Liapounoff’s Convexity Theorem gives a subset B_{1} of B satisfying λ(B_{1}) = \frac{1}{2}λ(B) + \frac{1}{2}λ(∅). Set B_{2} = B \ B_{1}. Note that

λ_{i}(B_{1}) = \frac{1}{2}λ_{i}(B) = λ_{i}(B_{2})  and  μ(B_{1}) = \frac{1}{2}μ(B) = μ(B_{2})

for i = 1, …, n. Set

f_{1} = f + α (1_{B_{1}} − 1_{B_{2}})  and  f_{2} = f + α (1_{B_{2}} − 1_{B_{1}}) .

Clearly f_{1} and f_{2} are in C and f = \frac{1}{2}f_{1} + \frac{1}{2}f_{2}. Thus f is not an extreme point of C.

3. geometric view-point

Consider a non-completely-continuous operator T: L_{1} → 𝔛 along with the corresponding Haar system \{h_{j}\} from Theorem 1. Let \{I_{k}^{n} = [\frac{k−1}{2^{n}}, \frac{k}{2^{n}})\}_{n,k} be the usual

dyadic splitting of [0, 1] with corresponding Haar functions {h̃_{j}}_{j≥1}. Consider the map T̃ ≡ T ∘ S where S: L_{1} → L_{1} is the isometry that takes h̃_{j} to h_{j}. Theorem 1 gives that there is a sequence {x_{n}^{∗}}_{n≥0} in S(𝔛^{∗}) and a subspace 𝔛_{0} of 𝔛 such that

(1) x_{n}^{∗}(T̃h̃_{2^{n}+k}) > δ for some δ > 0

(2) {sp(T̃h̃_{j}: 2^{n−1} < j ≤ 2^{n})}_{n≥0} is a finite dimensional decomposition of 𝔛_{0}

with constant at most 1 + τ. The next corollary follows from the observation that T̃ is not completely continuous and T̃ L_{1} ⊂ 𝔛_{0}.

Corollary 3. A Banach space failing the CCP has a subspace with a finite dimensional decomposition (with constant arbitrarily close to 1) that fails the CCP.

A tree in 𝔛 is a system of the form {x_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}} satisfying

x_{k}^{n} = \frac{x_{2k−1}^{n+1} + x_{2k}^{n+1}}{2} .

Associated to a tree is its difference system {d_{j}}_{j≥1} where d_{1} = x_{1}^{0} and

d_{2^{n}+k} = \frac{x_{2k−1}^{n+1} − x_{2k}^{n+1}}{2} .

There is a one-to-one correspondence between the bounded linear operators T from L_{1} into 𝔛 and bounded trees {x_{k}^{n}} growing in 𝔛. This correspondence is realized by T (h̃_{j}) = d_{j}.

A tree is a δ-Rademacher tree if || ∑_{k=1}^{2^{n}} d_{2^{n}+k}|| ≥ 2^{n}δ. A tree is a separated δtree if there exists a sequence {x_{n}^{∗}}_{n≥0} in S(𝔛^{∗}) such that x_{n}^{∗}(d_{2^{n}+k}) > δ. Clearly, a separated δ-tree is also a δ-Rademacher tree. The operator corresponding to a δ-Rademacher tree is not completely continuous since the image of the Rademacher functions stay large in norm. Thus if a bounded δ-Rademacher tree (or separated δ-tree) grows in 𝔛, then 𝔛 fails the CCP.

In any Banach space failing the CCP, a bounded δ-Rademacher tree grows (see [G1] for a direct proof); in fact, even a bounded separated δ-tree grows (see [G2] for an indirect proof). The proof of Theorem 1 is a direct proof that if 𝔛 fails CCP then a bounded separated δ-tree, with a difference sequence naturally blocking into a finite dimensional decomposition, grows in 𝔛.

4. from decompositions to bases

As previously mentioned, we do not know whether a space failing the CCP necessarily contains a subspace with a basis which fails the CCP. However, if the space has some nice local properties, then the proof of Theorem 1 can be modified to show this is so.

We now introduce some local properties. A Banach space is said to have the (K, n)–local basis property if each of its n-dimensional subspaces has a finite dimensional superspace which has a basis with basis constant at most K. A Banach space is said to have the (∗ − K) − property provided that each of its finite codimensional subspaces contains a finite codimensional subspace which has, for each n, the (K, n)–local basis property. A Banach space is said to have the (∗∗−K)−property provided that, for each n, each of its finite codimensional subspaces contains a finite codimensional subspace (depending on n) with the (K, n)–local basis property. Clearly, the (∗−K)−property implies the (∗∗−K)−property, but Szarek’s spaces [S] show that the properties are not equivalent. We do not know any example of a space which fails the (∗ ∗ −K) − property for all K.

Spaces failing cotype (i.e., containing ℓ_{∞}^{n} uniformly for all n), have the (∗ − K)- property. In fact, if 𝔛 fails cotype and Z is a finite codimensional subspace of 𝔛, then for any finite dimensional subspace W of 𝔛 there is a finite dimensional subspace Y of Z such that W + Y has a basis with basis constant less than, say,

10. To see this, use the fact ([P], [JRZ]) that W is (1 + ϵ)-complemented in a finite dimensional space which has a basis with basis constant less than 1 + ϵ and embed the complement to W in that space into Z ∩^{⊥}F, where F is a finite subset of 𝔛^{∗} which (1 + ϵ)-norms W. This is possible because finite codimensional subspaces of 𝔛 must contain ℓ_{∞}^{n} uniformly for all n and hence [J] contain even (1 + ϵ)-isomorphs of ℓ_{∞}^{n} for all n.

Banach lattices also enjoy the (∗ − K) − property. By the above observation, we need only consider lattices with cotype. Such a lattice X must be order continuous since it contains no copy of c_{0}. By a perturbation argument, it is enough to show that if F is a finite set of disjoint linear functionals, then F^{⊥} has the local basis property with uniform constant. To see this, consider F = {f_{1}, …, f_{n}}. Let X_{j} be the “support” of f_{j}; that is, let X_{j} be the complementary band to the band {x ∈ X : |f_{j}||x| = 0}. Notice that the X_{j}’s are disjoint since the f_{j}’s are disjoint. Thus, F^{⊥} is the disjoint sum of Y, Y_{1}, …, Y_{n}, where each Y_{j} is a one codimensional subspace of the band X_{j} and Y is the intersection of the bands {x ∈ X : |f_{j}||x| = 0}.

Corollary 4. If a Banach space 𝔛 fails the CCP and enjoys the (∗∗−K)−property, then 𝔛 has a subspace with a basis that fails the CCP.

To see this, it is enough by the argument for Corollary 3 to observe that when 𝔛 has the (∗ ∗ −K) − property Theorem 1 can be modified by adding:

(D) finite dimensional subspaces {G_{n}}_{n=0}^{∞} of 𝔛 changing (2) and (3) to:

(2′) {y_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(∪_{k=0}^{n}G_{k}) within τ_{n} (3′) G_{n+1} ⊂ ^{⊥}{y_{n,i}^{∗}}_{i=1}^{p_{n}}

and adding:

(4) {T h_{j}: 2^{n−1} < j ≤ 2^{n}} ⊂ G_{n}

(5) G_{n} has a basis with basis constant at most K. To achieve these modifications, at the first stage in the proof of Theorem 1, let G_{0} = sp(T h_{1}). Then in the inductive step in the proof, choose {y_{n,i}^{∗}}_{i=1}^{q_{n}} so that (2′) holds and, by appealing to the (∗∗−K)−property, enlarge the set to {y_{n,i}^{∗}}_{i=1}^{p_{n}} where p_{n} ≥ q_{n} so that ^{⊥}{y_{n,i}^{∗}}_{i=1}^{p_{n}} has the (K, 2^{n})–local basis property. Proceed as before and then, after selecting A_{k}^{n+1} (thereby defining h_{j} for j = 2^{n} +1, …, 2^{n+1}), choose a finite dimensional space G_{n+1} such that {T h_{j}: 2^{n} < j ≤ 2^{n+1}} ⊂ G_{n+1} ⊂ ⊥{y_{n,i}^{∗}}_{i=1}^{p_{n}} and G_{n+1} has a basis with basis constant at most K.

In the last years, geometric properties such as the CCP have allowed a deeper understanding of the RNP. Two such properties are the Point of Continuity property (PCP) and the Convex Point of Continuity property (CPCP). We refer the reader to [GGMS] for the definitions and a survey of these properties; here we merely recall that the RNP implies the PCP, which implies the CPCP, which in turn implies the CCP.

Relevant for this paper is Bourgain’s result [B3, prop. 5.4] that a space failing the PCP has a subspace with a finite dimensional decomposition which fails the PCP. Similar to the situation with the CCP, additional local structure on the space can help to sharpen the decomposition to a basis.

Proposition 5. If a Banach space 𝔛 fails the PCP and enjoys the (∗ − K) − property, then 𝔛 has a subspace with a basis which fails the PCP.

To see this, it is convenient for us to Rosenthal’s exposition of Bourgain’s result [R, Remark, pg. 315]. In a space 𝔛 failing the PCP, Rosenthal finds a “bad” bounded subset U of 𝔛 and δ > 0 and then constructs by induction on n, for a given sequence {τ_{n}} of numbers larger than one with finite product

(A) finite subsets {D_{n}}_{n=1}^{∞} of U (B) finite dimensional subspaces {F_{n}}_{n=1}^{∞} of 𝔛 (C) a finite set {x_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) for each n ≥ 1

such that, for H_{n} ≡ sp(x_{n,i}^{∗})_{i=1}^{p_{n}}, (1) D_{n} ⊂ D_{n+1}

(2) D_{n} ⊂ F_{1} + … + F_{n}

(3) {x_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(∪_{j=1}^{n}F_{j}) within τ_{n}

(4) F_{n+1} ⊂ ^{⊥}H_{n}

(5) for every d ∈ D_{n} and n — \frac{1}{n} neighborhood V of d, there is a d′ ∈ D_{n+1} ∩ V

such that ||d − d′|| > δ. He then considers the set D ≡ ∪_{n=1}^{∞}D_{n}. By construction, each relatively weakly open neighborhood of \overline{D} has diameter at least δ and \{F_{n}\}_{n=1}^{∞} forms a finite dimensional decomposition (with constant at most Πτ_{n}) of a subspace which contains \overline{D}.

If 𝔛 also enjoys the (∗ − K) − property, then Rosenthal’s construction can be modified by adding:

(D) finite dimensional subspaces {G_{n}}_{n=1}^{∞} of 𝔛 and changing (3) and (4) to:

(3′) {x_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(∪_{j=1}^{n}G_{j}) within τ_{n} (4′) G_{n+1} ⊂ ^{⊥}H_{n}

and adding:

(6) F_{n} ⊂ G_{n}

(7) G_{n} has a basis with constant at most K. To accomplish this, at the first stage of his construction, let G_{1} = F_{1}. Then, in the inductive step, when given D_{n}, {F_{j}}_{j=1}^{n}, {G_{j}}_{j=1}^{n}, and {x_{n,i}^{∗}}_{i=1}^{q_{n}} satisfying (2), (3′), and (6), appeal to the (∗ − K) − property to find {x_{n,i}^{∗}}_{i=1}^{p_{n}} with p_{n} ≥ q_{n} such that ⊥{x_{n,i}^{∗}}_{i=1}^{p_{n}} has the (K, m)-local basis property for all m. Put H_{n} = sp{x_{n,i}^{∗}}_{i=1}^{p_{n}}. Proceed as in Rosenthal’s argument to find the finite dimensional subspace F_{n+1} of ^{⊥}H_{n}. The (∗ − K) − property then provides the desired G_{n+1}. Clearly this is sufficient.

Bourgain [B3, thm. 5.7; B1, thm. 1] also showed that a space failing the RNP has a subspace with a finite dimensional decomposition which fails the RNP. The argument is split into two cases. In the first case, Bourgain shows that a space failing not only the RNP but also the CPCP has a subspace with a finite dimensional decomposition which fails the RNP. It immediately follows from the last proposition that if such a space also enjoys the (∗ − K) − property, then it has a subspace with a basis which fails the RNP. In the second case, Bourgain shows that a space which fails the RNP but has the CPCP contains a subspace with a finite dimensional decomposition which fails the RNP. His argument is rather delicate; the above technique for passing from a finite dimensional decomposition to a basis seems not to work.

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University of South Carolina, Department of Mathematics, Columbia, SC 29208. E-mail: girardi@math.scarolina.edu

Texas A&M University, Department of Mathematics, College Station, TX 77843. E-mail: WBJ7835@venus.tamu.edu