DENTABILITY, TREES, AND

DUNFORD-PETTIS OPERATORS ON L_{1}

Maria Girardi

University of Illinois at Urbana-Champaign

Pacific J. Math. 148 (1991) 59–79

Abstract. If all bounded linear operators from L_{1} into a Banach space 𝔛 are \underline{Dunford-Pettis} (i.e. carry weakly convergent sequences onto norm convergent sequences), then we say that 𝔛 has the \underline{complete continuity property} (CCP). The CCP is a weakening of the Radon-Nikodým property (RNP). Basic results of Bourgain and Talagrand began to suggest the possibility that the CCP, like the RNP, can be realized as an internal geometric property of Banach spaces; the purpose of this paper is to provide such a realization. We begin by showing that 𝔛 has the CCP if and only if every bounded subset of 𝔛 is Bocce dentable, or equivalently, every bounded subset of 𝔛 is weak-norm-one dentable (Section 2). This internal geometric description leads to another; namely, 𝔛 has the CCP if and only if no bounded separated δ-trees grow in 𝔛, or equivalently, no bounded δ-Rademacher trees grow in 𝔛 (Section 3).

1 : Introduction

Throughout this paper, 𝔛 denotes an arbitrary Banach space, 𝔛^{∗} the dual space

of 𝔛, B(𝔛) the closed unit ball 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 other-

wise explained, are as in [DU]. For clarity, known results are presented as facts

while new results are presented as theorems, lemmas, and observations.

The following fact provides several equivalent formulations of the CCP.

1991 Mathematics Subject Classification. 46B20, 47B38, 46B22, 28B99. Key words and phrases. complete continuity property, Dunford-Pettis operators,

Radon-Nikodým property, dentability, tree, bush, martingale.

This paper is part of the Ph.D. work of the author, who is grateful for the guidance of her advisor Professor J. J. Uhl, Jr.

Typeset by AMS-TEX

MARIA GIRARDI 2

Fact 1.1. For a bounded linear operator T from L_{1} into 𝔛, the following statements are equivalent.

(1) T is Dunford-Pettis.

(2) T maps weak compact sets to norm compact sets.

(3) T (B(L_{∞})) is a relatively norm compact subset of 𝔛.

(4) The corresponding vector measure F : Σ → 𝔛 given by F (E) = T (χ_{E}) has a relatively norm compact range in 𝔛.

(5) The adjoint of the restriction of T to L_{∞} from 𝔛^{∗} into L_{∞}^{∗} is a compact operator.

(6) As a subset of L_{1}, T^{∗}(B(𝔛^{∗})) is relatively L_{1}-norm-compact.

(7) As a subset of L_{1}, T^{∗}(B(𝔛^{∗})) satisfies the Bocce criterion.

The equivalence of (2) and (3) follows from the fact that the subsets of L_{1} that are relatively weakly compact are precisely those subsets that are bounded and uniformly integrable, which in turn, are precisely those subsets that can be uniformly approximated in L_{1}-norm by uniformly-bounded subsets. As for the equivalence of (6) and (7), [G] presents the two definitions below and shows that a relatively weakly compact subset of L_{1} is relatively L_{1}-norm-compact if and only if it satisfies the Bocce criterion.

Definition 1.2. For f in L_{1} and A in Σ, the \underline{Bocce oscillation of f on A} is given by

Bocce-osc f|_{A} ≡ \frac{∫_{A} |f − ^{\frac{∫_{A} f dμ}{μ(A)}}| dμ}{μ(A)} ,

observing the convention that 0/0 is 0.

Definition 1.3. A subset K of L_{1} satisfies the \underline{Bocce criterion} if for each ϵ > 0 and B in Σ^{+} there is a finite collection F of subset of B each with positive measure such that for each f in K there is an A in F satisfying

Bocce-osc f|_{A} < ϵ .

The other implications in Fact 1.1 are straightforward and easy to verify. Because of (4), the CCP is also referred to as the compact range property (CRP).

Towards a martingale characterization of the CCP, fix an increasing sequence {π_{n}}_{n≥0} of finite positive interval partitions of Ω such that ∨ σ(π_{n}) = Σ and π_{0} = {Ω}. Let F_{n} denote the sub-σ-field σ(π_{n}) of Σ that is generated by π_{n}. For f in L_{1}(𝔛), let E_{n}(f) denote the conditional expectation of f given F_{n}. Definition 1.4. A sequence \{f_{n}\}_{n≥0} in L_{1}(𝔛) is an 𝔛-valued \underline{martingale} with respect to {F_{n}} if for each n we have that f_{n} is F_{n}-measurable and E_{n}(f_{n+1}) = f_{n} in L_{1}. The martingale \{f_{n}\} is \underline{uniformly bounded} provided that sup_{n} ∥ f_{n} ∥_{L_{∞}} is finite. Often the martingale is denoted by {f_{n}, F_{n}} in order to display both the functions and the sub-σ-fields involved.

There is a one-to-one correspondence between the bounded linear operators T from L_{1} into 𝔛 and the uniformly bounded 𝔛-valued martingales {f_{n}, F_{n}}. This correspondence is obtained by taking

T (g) = lim_{n→∞} ∫_{Ω} f_{n}(ω)g(ω) dμ(ω)  if {f_{n}} is the martingale,

and

f_{n}(ω) = ∑_{E∈π_{n}} \frac{T (χ_{E})}{μ(E)}χ_{E}(ω)  if T is the operator.

Fact 1.1.6 implies that a bounded linear operator T from L_{1} into 𝔛 is Dunford-Pettis if and only if

lim_{m,n→∞} sup_{x^{∗}∈B(𝔛^{∗})} ∥ E_{n}(T^{∗}x^{∗}) − E_{m}(T^{∗}x^{∗}) ∥_{L_{1}}= 0 .

Since E_{n}(T^{∗}x^{∗}) = x^{∗}f_{n} in L_{1}, we have the following martingale characterization of Dunford-Pettis operators, and thus of the CCP. Fact 1.5. A bounded linear operator from L_{1} into 𝔛 is Dunford-Pettis if and only if the corresponding martingale is Cauchy in the Pettis norm. Consequently, a Banach space 𝔛 has the CCP if and only if all uniformly bounded 𝔛-valued martingales are Pettis-Cauchy.

Recall that a bounded linear operator T : L_{1} → 𝔛 is \underline{(Bochner) representable} if there is g in L_{∞}(μ, 𝔛) such that for each f in L_{1}(μ)

T f = ∫_{Ω} fg dμ .

A Banach space 𝔛 has the \underline{Radon-Nikodým property} if all bounded linear operators from L_{1} into 𝔛 are Bochner representable. It is clear that a representable operator from L_{1} into 𝔛 is Dunford-Pettis. Thus, if 𝔛 has the RNP then 𝔛 has the CCP. Both the dual of the James tree space [J] and the Bourgain-Rosenthal space [BR] have the CCP yet fail the RNP.

DUNFORD-PETTIS OPERATORS 5

2 : Dentability

In this section, we examine in which Banach spaces bounded subsets have certain dentability properties.

Dentability characterizations of the RNP are well-known [cf. DU and GU].

Fact 2.1. The following statements are equivalent.

(1) 𝔛 has the RNP.

(2) Every bounded subset D of 𝔛 is dentable. Definition 2.2 D is \underline{dentable} if for each ϵ > 0 there is x in D such that x ∉ \overline{co} (D \ B_{ϵ}(x)) where B_{ϵ}(x) = \{y ∈ 𝔛 : ∥ x − y ∥< ϵ\}.

(3) Every bounded subset D of 𝔛 is σ-dentable. Definition 2.3 D is \underline{σ-dentable} if for each ϵ > 0 there is an x in D such that if x has the form x = ∑_{i=1}^{n} α_{i}z_{i} with z_{i} ∈ D, 0 ≤ α_{i}, and ∑_{i=1}^{n} α_{i} = 1, then ∥ x − z_{i} ∥< ϵ for some i .

The natural question to explore next is what dentability condition characterizes the CCP. Towards this, the next definition is a weakening of Definition 2.2.

Definition 2.4. A subset D of 𝔛 is \underline{weak-norm-one dentable} if for each ϵ > 0 there is a finite subset F of D such that for each x^{∗} in S(𝔛^{∗}) there is x in F satisfying

x ∉ \overline{co} \{z ∈ D : | x^{∗}(z − x) | ≥ ϵ\} ≡ \overline{co} (D \ V_{ϵ,x^{∗}}(x)) .

Petrakis and Uhl [PU] showed that if 𝔛 has the CCP then every bounded subset of 𝔛 is weak-norm-one dentable. For our characterization of the CCP, we introduce the following variations of definition 2.3 that are useful in showing the converse of the above implication of [PU].

Definition 2.5. A subset D of 𝔛 is \underline{Bocce dentable} if for each ϵ > 0 there is a finite

MARIA GIRARDI 6

subset F of D such that for each x^{∗} in S(𝔛^{∗}) there is x in F satisfying:

if x = ∑_{i=1}^{n} α_{i}z_{i} with z_{i} ∈ D, 0 ≤ α_{i}, and ∑_{i=1}^{n} α_{i} = 1 ,

then ∑_{i=1}^{n} α_{i} | x^{∗}(x − z_{i}) | < ϵ.

Definition 2.6. A subset D of 𝔛 is \underline{midpoint Bocce dentable} if for each ϵ > 0 there is a finite subset F of D such that for each x^{∗} in S(𝔛^{∗}) there is x in F satisfying:

if x = \frac{1}{2}z_{1} + \frac{1}{2}z_{2} with z_{i} ∈ D then | x^{∗}(x − z_{1}) | ≡ | x^{∗}(x − z_{2}) | < ϵ.

We obtain equivalent formulations of the above definitions by replacing S(𝔛^{∗}) with B(𝔛^{∗}).

The next theorem, this section’s main result, shows that these dentability conditions provide an internal geometric characterization of the CCP. Theorem 2.7. The following statements are equivalent.

(1) 𝔛 has the CCP.

(2) Every bounded subset of 𝔛 is weak-norm-one dentable.

(3) Every bounded subset of 𝔛 is midpoint Bocce dentable.

(4) Every bounded subset of 𝔛 is Bocce dentable.

The remainder of this section is devoted to the proof of Theorem 2.7. Because of its length and complexity and also for the sake of clarity of the exposition, we present the implications as separate theorems. It is clear from the definitions that (2) implies (3) and that (4) implies (3). [PU, Theorem II.7] shows that (1) implies (2) by constructing, in a bounded non-weak-norm-one dentable subset D, a (\overline{co} D)-valued martingale that is not Cauchy in the Pettis norm. Using Fact 1.1.7, Theorem 2.10 shows that (3) implies (1). That (1) implies (4) follows from Theorem 2.8 and the martingale characterization of the CCP (Fact 1.5).

DUNFORD-PETTIS OPERATORS 7

Theorem 2.8. If a subset D of 𝔛 is not Bocce dentable, then there is an increasing sequence {π_{n}} of partition of [0, 1) and a D-valued martingale {f_{n}, σ(π_{n})} that is not Cauchy in the Pettis norm. Moreover, {π_{n}} can be chosen so that ∨ σ(π_{n}) = Σ, π_{0} = {Ω} and each π_{n} partitions [0, 1) into a finite number of half-open intervals.

Proof. Let D be subset of 𝔛 that is not Bocce dentable. Accordingly, there is an ϵ > 0 satisfying:

for each finite subset F of D there is x_{F}^{∗} in S(𝔛^{∗}) such that each x in F has the form x = ∑_{i=1}^{m} α_{i}z_{i}

with ∑_{i=1}^{m} α_{i} | x_{F}^{∗} (x − z_{i}) | > ϵ

for a suitable choice of z_{i} ∈ D and α_{i} > 0 with ∑_{i=1}^{m} α_{i} = 1 .

(∗)

We shall use property (∗) to construct an increasing sequence {π_{n}}_{n≥0} of finite partitions of [0, 1), a martingale {f_{n}, σ(π_{n})}_{n≥0}, and a sequence {x_{n}^{∗}}_{n≥1} in S(𝔛^{∗}) such that for each nonnegative integer n:

(1) f_{n} has the form f_{n} = ∑_{E∈π_{n}} x_{E}χ_{E} where x_{E} is in D,

(2) ∫_{Ω} | x_{n+1}^{∗}(f_{n+1} − f_{n}) | dμ ≥ ϵ ,

(3) if E is in π_{n}, then E has the form [a, b) and μ(E) < 1/2^{n} and

(4) π_{0} = {Ω}. Condition (3) guarantees that ∨ σ(π_{n}) = Σ while condition (2) guarantees that {f_{n}} is not Cauchy in the Pettis norm.

Towards the construction, pick an arbitrary x in D. Set π_{0} = {Ω} and f_{0} = x χ_{Ω}. Fix n ≥ 0. Suppose that a partition π_{n} of Ω consisting of intervals of length at most 1/2^{n} and a function f_{n} = ∑_{E∈π_{n}} x_{E}χ_{E} with x_{E} ∈ D have been constructed. We now construct f_{n+1}, π_{n+1} and x_{n+1}^{∗} satisfying conditions (1), (2) and (3).

MARIA GIRARDI 8

Apply (∗) to F = {x_{E} : E ∈ π_{n}} and find the associated x_{F}^{∗} = x_{n+1}^{∗} in S(𝔛^{∗}). Fix an element E = [a, b) of π_{n}. We first define f_{n+1}χ_{E}. Property (∗) gives that x_{E} has the form

x_{E} = ∑_{i=1}^{m} α_{i}x_{i}  with  ∑_{i=1}^{m} α_{i} | x_{n+1}^{∗}(x − x_{i}) | > ϵ

for a suitable choice of x_{i} ∈ D and positive real numbers α_{1}, …, α_{m} whose sum is one. Using repetition, we arrange to have α_{i} < 1/2^{n+1} for each i. It follows that there are real numbers d_{0}, d_{1}, …, d_{m} such that

a = d_{0} < d_{1} < … < d_{m−1} < d_{m} = b

and

d_{i} − d_{i−1} = α_{i} (b − a) for i = 1, …, m .

Set

f_{n+1}χ_{E} = ∑_{i=1}^{m} x_{i} χ_{[d_{i−1},d_{i})} .

Define f_{n+1} on all of Ω similarly. Let π_{n+1} be the partition consisting of all the intervals [d_{i−1}, d_{i}) obtained from letting E range over π_{n}.

Clearly, f_{n+1} and π_{n+1} satisfy conditions (1) and (3). Condition (2) is also satisfied since for each E = [a, b) in π_{n} we have, using the above notation,

∫_{E} | x_{n+1}^{∗}(f_{n+1} − f_{n}) | dμ = ∑_{i=1}^{m} ∫_{d_{i−1}}^{d_{i}} | x_{n+1}^{∗}(x_{i} − x_{E}) | dμ

= (b − a) ∑_{i=1}^{m} α_{i} | x_{n+1}^{∗}(x_{i} − x_{E}) |

≥ μ(E) ϵ .

To insure that {f_{n}} is indeed a martingale, we need to compute E_{n}(f_{n+1}). Fix E = [a, b) in π_{n}. Using the above notation, we have for almost all t in E,

E_{n}(f_{n+1})(t) = \frac{1}{b − a} ∫_{a}^{b} f_{n+1} dμ

= \frac{1}{b − a} ∑_{i=1}^{m} ∫_{d_{i−1}}^{d_{i}} f_{n+1} dμ

= ∑_{i=1}^{m} \frac{d_{i} − d_{i−1}}{b − a} x_{i}

= ∑_{i=1}^{m} α_{i} x_{i} = x_{E}

= f_{n}(t) .

Thus E_{n}(f_{n+1}) = f_{n} a.e., as needed.

This completes the necessary constructions.

We need the following lemma which we will prove after the proof of Theorem 2.10. Lemma 2.9. If A is in Σ^{+} and f in L_{∞}(μ) is not constant a.e. on A, then there is an increasing sequence {π_{n}} of positive finite measurable partitions of A such that ∨ σ(π_{n}) = Σ ∩ A and for each n

μ ( ⋃ \{E : E ∈ π_{n}, \frac{∫_{E} f dμ}{μ(E)} ≥ \frac{∫_{A} f dμ}{μ(A)} \}) = \frac{μ(A)}{2} ,

and so

μ ( ⋃ \{E : E ∈ π_{n}, \frac{∫_{E} f dμ}{μ(E)} < \frac{∫_{A} f dμ}{μ(A)} \}) = \frac{μ(A)}{2} .

Theorem 2.10. If all bounded subsets of 𝔛 are midpoint Bocce dentable, then 𝔛 has the complete continuity property. Proof. Let all bounded subsets of 𝔛 be midpoint Bocce dentable. Fix a bounded linear operator T from L_{1} into 𝔛. We shall show that the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion. Then an appeal to Fact 1.1.7 shows that 𝔛 has the complete continuity property.

MARIA GIRARDI 10

To this end, fix ϵ > 0 and B in Σ^{+}. Let F denote the vector measure from Σ into 𝔛 given by F (E) = T (χ_{E}). Since the subset \{\frac{F (E)}{μ(E)} : E ⊂ B and E ∈ Σ^{+}\} of 𝔛 is bounded, it is midpoint Bocce dentable. Accordingly, there is a finite collection F of subsets of B each in Σ^{+} such that for each x^{∗} in the unit ball of 𝔛^{∗} there is a set A in F such that if

\frac{F (A)}{μ(A)} = \frac{1}{2} \frac{F (E_{1})}{μ(E_{1})} + \frac{1}{2} \frac{F (E_{2})}{μ(E_{2})}

for some subsets E_{i} of B with E_{i} ∈ Σ^{+}, then

\frac{1}{2} |\frac{x^{∗}F (E_{1})}{μ(E_{1})} − \frac{x^{∗}F (A)}{μ(A)}| + \frac{1}{2} |\frac{x^{∗}F (E_{2})}{μ(E_{2})} − \frac{x^{∗}F (A)}{μ(A)}| < ϵ .  (1)

Fix x^{∗} in the unit ball of 𝔛^{∗} and find the associated A in F. By definition, the set T^{∗}(B(𝔛^{∗})) will satisfy the Bocce criterion provided that Bocce-osc (T^{∗}x^{∗})|_{A} ≤ ϵ.

If T^{∗}x^{∗} ∈ L_{1} is constant a.e. on A, then the Bocce-osc (T^{∗}x^{∗})|_{A} is zero and we are finished. So assume T^{∗}x^{∗} is not constant a.e. on A.

For a finite positive measurable partition π of A, denote

f_{π} = ∑_{E∈π} \frac{F (E)}{μ(E)}χ_{E}

and

E_{π}^{+} = ⋃\{ E ∈ π : \frac{x^{∗}F (E)}{μ(E)} ≥ \frac{x^{∗}F (A)}{μ(A)} \}

and

E_{π}^{−} = ⋃\{ E ∈ π : \frac{x^{∗}F (E)}{μ(E)} < \frac{x^{∗}F (A)}{μ(A)} \} .

Note that for E in Σ

x^{∗}F (E) = ∫_{E}(x^{∗}T^{∗}) dμ .

DUNFORD-PETTIS OPERATORS 11

Compute

∫A|x^{∗}f_{π} − \frac{x^{∗}F (A)}{μ(A)}| dμ

= ∑_{E∈π} ∫_{E}|\frac{x^{∗}F (E)}{μ(E)} − \frac{x^{∗}F (A)}{μ(A)}| dμ

= μ(A) ∑_{E∈π} \frac{μ(E)}{μ(A)} |\frac{x^{∗}F (E)}{μ(E)} − \frac{x^{∗}F (A)}{μ(A)}|  (2)

= μ(A)[ \frac{μ(E_{π}^{+})}{μ(A)} |\frac{x^{∗}F (E_{π}^{+})}{μ(E_{π}^{+})} − \frac{x^{∗}F (A)}{μ(A)}| + \frac{μ(E_{π}^{−})}{μ(A)} |\frac{x^{∗}F (E_{π}^{−})}{μ(E_{π}^{−})} − \frac{x^{∗}F (A)}{μ(A)}| ] .

Since the L_{1}-function T^{∗}x^{∗} is bounded, for now we may view T^{∗}x^{∗} as an element in L_{∞}. Lemma 2.9 allow us to apply property (1) to equation (2). For applying Lemma 2.9 to A with f ≡ T^{∗}x^{∗} produces an increasing sequence {π_{n}} of positive measurable partitions of A satisfying

∨ σ(π_{n}) = Σ ∩ A  and  μ(E_{π_{n}}^{+}) = \frac{μ(A)}{2} = μ(E_{π_{n}}^{−}) .

For π = π_{n}, condition (2) becomes

∫A|x^{∗}f_{π_{n}} − \frac{x^{∗}F (A)}{μ(A)}| dμ
= μ(A)[ \frac{1}{2} |\frac{x^{∗}F (E_{π_{n}}^{+})}{μ(E_{π_{n}}^{+})} − \frac{x^{∗}F (A)}{μ(A)}| + \frac{1}{2} |\frac{x^{∗}F (E_{π_{n}}^{−})}{μ(E_{π_{n}}^{−})} − \frac{x^{∗}F (A)}{μ(A)}| ] .  (3)

Since F (A) / μ(A) has the form

\frac{F (A)}{μ(A)} = \frac{μ(E_{π_{n}}^{+})}{μ(A)} \frac{F (E_{π_{n}}^{+})}{μ(E_{π_{n}}^{+})} + \frac{μ(E_{π_{n}}^{−})}{μ(A)} \frac{F (E_{π_{n}}^{−})}{μ(E_{π_{n}}^{−})}

= \frac{1}{2} \frac{F (E_{π_{n}}^{+})}{μ(E_{π_{n}}^{+})} + \frac{1}{2} \frac{F (E_{π_{n}}^{−})}{μ(E_{π_{n}}^{−})} ,

applying property (1) to equation (3) yields that for each π_{n}

∫_{A} |x^{∗}f_{π_{n}} − \frac{x^{∗}F (A)}{μ(A)}| dμ < μ(A) ϵ .

Since ∨ σ(π_{n}) = Σ ∩ A and

(x^{∗}f_{π_{n}})|_{A} = ∑_{E∈π_{n}} \frac{x^{∗}F (E)}{μ(E)} χ_{E} = ∑_{E∈π_{n}} \frac{∫_{E}(T^{∗}x^{∗}) dμ}{μ(E)} χ_{E} = E_{π_{n}}(T^{∗}x^{∗})|_{A} ,

we have that (x^{∗}f_{π_{n}})|_{A} converges to (T^{∗}x^{∗})|_{A} in L_{1}-norm . Hence,

Bocce-osc (T^{∗}x^{∗})|_{A} ≡ \frac{∫_{A} |(T^{∗}x^{∗}) − ^{\frac{∫_{A}(T^{∗}x^{∗}) dμ}{μ(A)}} | dμ}{μ(A)} ≤ ϵ.

Thus T^{∗}(B(𝔛^{∗})) satisfies the Bocce criterion, and so as needed, 𝔛 has the complete continuity property.

We now verify Lemma 2.9. Proof of Lemma 2.9. Fix A in Σ^{+} and f in L_{∞}(μ). Without loss of generality, f is not constant a.e. on A and ∫_{A} f dμ = 0. Find P and N in Σ satisfying

A = P ∪ N  μ(P) = \frac{μ(A)}{2} = μ(N)  P ∩ N = ∅

and

∫_{P} f dμ ≡ 2M > 0 ∫_{N} f dμ ≡ −2M < 0 .

Approximate f by a simple function f̃(·) = ∑ α_{i} χ_{A_{i}}(·) satisfying (1) ∥ f − f̃ ∥_{L_{∞}}< M,

(2) ∪A_{i} = A and the A_{i} are disjoint,

(3) A_{i} ⊂ P if i ≤ m and A_{i} ⊂ N if i > m for some positive

integer m. Note that

P = ⋃_{i≤m} A_{i}  and  N = ⋃_{i>m} A_{i} .

To find the sequence {π_{n}}, we shall first find an increasing sequence {π_{n}^{P} } of partitions of P and an increasing sequence {π_{n}^{N}} of partitions of N. Then π_{n} will be the union of π_{n}^{P} and π_{n}^{N} . To this end, for each A_{i} obtain an increasing sequence of partitions of A_{i} :

A_{i} ≡ E_{1}^{i0}
↙ ↘
E_{1}^{i1} E_{2}^{i1}
↙ ↘ ↙ ↘
E_{1}^{i2} E_{2}^{i2} E_{3}^{i2} E_{4}^{i2}
⋯

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

E_{2k−1}^{i n+1} ∪ E_{2k}^{i n+1} = E_{k}^{in}  E_{2k−1}^{i n+1} ∩ E_{2k}^{i n+1} = ∅  μ(E_{k}^{in}) = \frac{μ(A_{i})}{2^{n}} .

For each positive integer n, let π_{n}^{P} be the partition of P given by

π_{n}^{P} = {P_{k}^{n} : k = 1, …, 2^{n}}  where  P_{k}^{n} = ⋃_{i≤m} E_{k}^{in} ,

π_{n}^{N} be the partition of N given by

π_{n}^{N} = {N_{k}^{n} : k = 1, …, 2^{n}}  where  N_{k}^{n} = ⋃_{i>m} E_{k}^{in} ,

and π_{n} be the partition of A given by

π_{n} = π_{n}^{P} ∪ π_{n}^{N} .

The sequence {π_{n}} has the desired properties. Since

μ(P_{k}^{n}) = ∑_{i≤m} \frac{μ(A_{i})}{2^{n}} = \frac{μ(P)}{2^{n}} = \frac{μ(A)}{2^{n+1}}

and

μ(N_{k}^{n}) = ∑_{i<m} \frac{μ(A_{i})}{2^{n}} = \frac{μ(N)}{2^{n}} = \frac{μ(A)}{2^{n+1}} ,

any element in π_{n} has measure μ(A) / 2^{n+1}. Thus ∨ σ(π_{n}) = Σ ∩ A .

As for the other properties, since f̃ takes the value α_{i} on E_{k}^{in} ⊂ A_{i} we have

∫_{P_{k}^{n}} f̃dμ = ∑_{i≤m} ∫_{E_{k}^{in}} f̃dμ

= ∑_{i≤m} α_{i} μ(E_{k}^{in})

= \frac{1}{2^{n}} ∑_{i≤m} α_{i} μ(A_{i})

= \frac{1}{2^{n}} ∫_{P} f̃dμ

> 0

and likewise

∫_{N_{k}^{n}} f̃dμ = \frac{1}{2^{n}} ∫_{N} f̃dμ < 0 .

We chose f̃ close enough to f so that the above inequalities still hold when we replace f̃ by f ,

∫_{P_{k}^{n}} f dμ ≥ ∫_{P_{k}^{n}}(f̃− M) dμ

= \frac{1}{2^{n}} ∫_{P} f̃dμ − Mμ(P_{k}^{n})

≥ \frac{1}{2^{n}} ∫_{P} (f − M) dμ − \frac{Mμ(A)}{2^{n+1}}

= \frac{1}{2^{n}} ∫_{P} f dμ − \frac{Mμ(A)}{2^{n+1}} − \frac{Mμ(A)}{2^{n+1}}

> \frac{M}{2^{n}} − \frac{Mμ(A)}{2^{n}} = \frac{M [1 − μ(A)]}{2^{n}}

≥ 0

and likewise

∫_{N_{k}^{n}} f dμ < \frac{M [μ(A) − 1]}{2^{n}} ≤ 0 .

DUNFORD-PETTIS OPERATORS 15

Thus the other properties of the lemma are satisfied since for each n,

μ ( ⋃ {E : E ∈ π_{n}, ∫_{E} f dμ ≥ 0 }) = μ ( ⋃ {E : E ∈ π_{n}^{P} })

= μ(P)

= \frac{μ(A)}{2}

and so

μ ( ⋃ \{E : E ∈ π_{n}, ∫_{E} f dμ < 0 \}) = \frac{μ(A)}{2} .

Note that the partitions {π_{n}} are nested by construction.

MARIA GIRARDI 16

3 : Bushes and Trees

In this section, we examine which Banach spaces allow certain types of bushes and trees to grow in them. First let us review some known implications.

A Banach space 𝔛 fails the RNP precisely when a bounded δ-bush grows in 𝔛. Thus if a bounded δ-tree grows in 𝔛 then 𝔛 fails the RNP. The converse is false; the Bourgain-Rosenthal space [BR] fails the RNP yet has no bounded δ-trees. However, if 𝔛 is a dual space then the converse does hold.

Bourgain [B2] showed that if 𝔛 fails the CCP then a bounded δ-tree grows in 𝔛. The converse is false; the dual of the James Tree space has a bounded δ-tree and the CCP. It is well-known that if a bounded δ-Rademacher tree grows in 𝔛 then 𝔛 fails the CCP. Riddle and Uhl [RU] showed that the converse holds in a dual space. This section’s main theorem, Theorem 3.1 below, makes precise exactly which types of bushes and trees grow in a Banach space failing the CCP.

Theorem 3.1. The following statements are equivalent.

(1) 𝔛 fails the CCP.

(2) A bounded separated δ-tree grows in 𝔛.

(3) A bounded separated δ-bush grows in 𝔛.

(4) A bounded δ-Rademacher tree grows in 𝔛.

The remainder of this section is devoted to proving Theorem 3.1. That (1) implies (2) will follow from Theorem 3.2 below. All the other implications are straightforward and will be verified shortly. As usual, we start with some definitions.

Perhaps it is easiest to define a bush via martingales. If {π_{n}}_{n≥0} is an increasing sequence of finite positive interval partitions of [0, 1) with ∨ σ(π_{n}) = Σ and π_{0} =

While typing this paper, I learned that H.P. Rosenthal has also recently obtained the result that if 𝔛 fails the CCP then a bounded δ-Rademacher tree grows in 𝔛.

DUNFORD-PETTIS OPERATORS 17

{Ω} and if {f_{n}, σ(π_{n})}_{n≥0} is an 𝔛-valued martingale, then each f_{n} has the form

f_{n} = ∑_{E∈π_{n}} x_{E}^{n}χ_{E}

and the system

{x_{E}^{n} : n = 0, 1, 2, … and E ∈ π_{n}}

is a bush in 𝔛. Moreover, every bush is realized this way. A bush is a \underline{δ-bush} if the corresponding martingale satisfies for each positive integer n

∥ f_{n}(t) − f_{n−1}(t) ∥ > δ .  (i)

A bush is a \underline{separated δ-bush} if there exists a sequence \{x_{n}^{∗}\}_{n≥1} in S(𝔛^{∗}) such that the corresponding martingale satisfies for each positive integer n

| x_{n}^{∗} (f_{n}(t) − f_{n−1}(t)) | > δ .  (ii)

In this case we say that the bush is separated by {x_{n}^{∗}}. Clearly a separated δ-bush is also a δ-bush. Observation that (3) implies (1) in Theorem 3.1. If a bounded separated δ-bush grows in a subset D of 𝔛, then condition (ii) guarantees that the corresponding D-valued martingale {f_{n}, σ(π_{n})} is not Pettis-Cauchy since

∥ f_{n} − f_{n−1} ∥_{P ettis} ≥ ∫_{Ω} | x_{n}^{∗} (f_{n}(t) − f_{n−1}(t)) | dμ > δ .

So if a bounded separated δ-bush grows in 𝔛 then 𝔛 fails the CCP (Fact 1.5).

If each π_{n} is the n^{th} dyadic partition then we call the bush a (dyadic) tree. Let us rephrase the above definitions for this case, without the help of martingales. A \underline{tree} in 𝔛 is a system of the form \{x_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}\} satisfying for n = 1, 2, … and k = 1, …, 2^{n−1}

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

MARIA GIRARDI 18

Condition (iii) guarantees that {f_{n}} is indeed a martingale. It is often helpful to think of a tree diagrammatically:

x_{1}^{0}
x_{1}^{1} x_{2}^{1}
x_{1}^{2} x_{2}^{2} x_{3}^{2} x_{4}^{2}
x_{1}^{3} x_{2}^{3} x_{3}^{3} x_{4}^{3} x_{5}^{3} x_{6}^{3} x_{7}^{3} x_{8}^{3}
⋯

It is easy to see that (iii) is equivalent to

x_{2k−1}^{n} − x_{2k}^{n} = 2 ( x_{2k−1}^{n} − x_{k}^{n−1} ) = 2 ( x_{k}^{n−1} − x_{2k}^{n}) .  (iii’)

A tree \{x_{k}^{n}\} is a \underline{δ-tree} if for n = 1, 2, … and k = 1, …, 2^{n−1}

∥ x_{2k−1}^{n} − x_{k}^{n−1} ∥ ≡ ∥ x_{2k}^{n} − x_{k}^{n−1} ∥ > δ .  (iv)

An appeal to (iii’) shows that (iv) is equivalent to

∥ x_{2k−1}^{n} − x_{2k}^{n} ∥ > 2 δ .  (iv’)

A tree \{x_{k}^{n}\} is a \underline{separated δ-tree} if there exists a sequence \{x_{n}^{∗}\}_{n≥1} in S(𝔛^{∗}) such that for n = 1, 2, … and k = 1, …, 2^{n−1}

| x_{n}^{∗}(x_{2k−1}^{n} − x_{k}^{n−1}) | ≡ | x_{n}^{∗}(x_{2k}^{n} − x_{k}^{n−1}) | > δ .  (v)

Another appeal to (iii’) shows that (v) is equivalent to

| x_{n}^{∗}(x_{2k−1}^{n} − x_{2k}^{n}) | > 2 δ .  (v’)

Furthermore, by switching x_{2k−1}^{n} and x_{2k}^{n} when necessary, we may assume that (v’) is equivalent to

x_{n}^{∗}(x_{2k−1}^{n} − x_{2k}^{n}) > 2 δ .  (v”)

Since a separated δ-tree is also a separated δ-bush, (2) implies (3) in Theorem 3.1.

A tree \{x_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}\} is called a \underline{δ-Rademacher tree} [RU] if for each positive integer n

∥ ∑_{k=1}^{2^{n−1}} (x_{2k−1}^{n} − x_{2k}^{n}) ∥ > 2^{n}δ .

Perhaps a short word on the connection between Rademacher trees and the Rademacher functions {r_{n}} is in order. In light of our discussion in section 1, there is a one-to-one correspondence between all bounded trees in 𝔛 and all bounded linear operators from L_{1} into 𝔛. If {x_{k}^{n}} is a bounded tree in 𝔛 with associated operator T, then it is easy to verify that {x_{k}^{n}} is a δ-Rademacher tree precisely when ∥ T (r_{n}) ∥ > δ for all positive integers n.

Fact that (4) implies (1) in Theorem 3.1 [RU]. Let {f_{n}} be the (dyadic) martingale associated with a δ-Rademacher tree {x_{k}^{n}}. If x^{∗} is in 𝔛^{∗} and I_{k}^{n} is the dyadic interval [ (k − 1)/2^{n} , k/2^{n} ) then

∫_{Ω} | x^{∗} (f_{n} − f_{n−1}) | dμ = ∑_{k=1}^{2^{n−1}} ∫_{I_{k}^{n−1}} | x^{∗} (f_{n} − f_{n−1}) | dμ

= ∑_{k=1}^{2^{n−1}}[ ∫_{I_{2k−1}^{n}} | x^{∗} (x_{2k−1}^{n} − x_{k}^{n−1}) | dμ + ∫_{I_{2k}^{n}} | x^{∗} (x_{2k}^{n} − x_{k}^{n−1}) | dμ ]

= \frac{1}{2^{n}} ∑_{k=1}^{2^{n−1}}[ | x^{∗} (x_{2k−1}^{n} − x_{k}^{n−1}) | + | x^{∗} (x_{2k}^{n} − x_{k}^{n−1}) | ]

= \frac{1}{2^{n}} ∑_{k=1}^{2^{n−1}} | x^{∗} (x_{2k−1}^{n} − x_{2k}^{n}) | by (iii’)

≥ \frac{1}{2^{n}} | x^{∗} ( ∑_{k=1}^{2^{n−1}} (x_{2k−1}^{n} − x_{2k}^{n}) ) | .

From this we see that {f_{n}} is not Cauchy in the Pettis norm since

∥ f_{n} − f_{n−1} ∥_{P ettis} = sup_{x^{∗}∈B(𝔛^{∗})} ∫_{Ω} | x^{∗} (f_{n} − f_{n−1}) | dμ

≥ sup_{x^{∗}∈B(𝔛^{∗})} \frac{1}{2^{n}} | x^{∗} ( ∑_{k=1}^{2^{n−1}} (x_{2k−1}^{n} − x_{2k}^{n}) ) |

= \frac{1}{2^{n}} ∥ ∑_{k=1}^{2^{n−1}} (x_{2k−1}^{n} − x_{2k}^{n}) ∥

> \frac{1}{2^{n}} 2^{n} δ = δ .

Thus if a bounded δ-Rademacher tree grows in a subset D of 𝔛, then there is a bounded D-valued martingale that in not Pettis-Cauchy and so 𝔛 fails the CCP (Fact 1.5). Observation that (2) implies (4) in Theorem 3.1. A separated δ-tree can easily be reshuffled so that it is a δ-Rademacher tree. For if {x_{k}^{n}} is a separated δ-tree then we may assume, by switching x_{2k−1}^{n} and x_{2k}^{n} when necessary, that there is a sequence {x_{n}^{∗}} in S(𝔛^{∗}) satisfying

x_{n}^{∗}(x_{2k−1}^{n} − x_{2k}^{n}) > 2 δ .

With this modification {x_{k}^{n}} is a δ-Rademacher tree since

∥ ∑_{k=1}^{2^{n−1}}(x_{2k−1}^{n} − x_{2k}^{n}) ∥ ≥ | ∑_{k=1}^{2^{n−1}} x_{n}^{∗} (x_{2k−1}^{n} − x_{2k}^{n}) |

= ∑_{k=1}^{2^{n−1}} x_{n}^{∗} (x_{2k−1}^{n} − x_{2k}^{n})

> ∑_{k=1}^{2^{n−1}} 2 δ = 2^{n} δ .

It should be noted that a bounded δ̃-Rademacher tree need neither be a δ-tree nor a separated δ-tree. For example, consider the c_{0}-valued dyadic martingale {f_{n}} given by

f_{n} = (s_{0}, …, s_{n}, 0, 0, …) ,

where the function s_{n} from [0, 1] into [−1, 1] is defined by

s_{n} = \{ \begin{matrix}(−1)^{k} 2^{−n} & if ω ∈ I_{k}^{n} with k ≤ 2 ; \\ (−1)^{k} & if ω ∈ I_{k}^{n} with k > 2 . \end{matrix}

DUNFORD-PETTIS OPERATORS 21

The tree associated with \{f_{n}\} is a \frac{1}{4}-Rademacher tree but is neither a δ-tree nor a separated δ-tree for any positive δ. Thus, since a δ-tree grows in a space failing the CCP, the notion of a separated δ-tree is more desirable than that of a δ-Rademacher tree for characterizing the CCP.

To complete the proof of Theorem 3.1, we need only to show that (1) implies (2). Towards this end, let 𝔛 fail the CCP. An appeal to Theorem 2.7 gives that there is a bounded non-midpoint-Bocce-dentable subset of 𝔛. In such a set, we can construct a separated δ-tree. This construction is made precise in the following theorem. Theorem 3.2. A separated δ-tree grows in a non-midpoint-Bocce-dentable set. Proof. Let D be a subset of 𝔛 that is not midpoint Bocce dentable. Accordingly, there is a δ > 0 satisfying:

for each finite subset F of D there is a norm one linear functional x_{F}^{∗} such that

each x in F has the form x = \frac{x_{1} + x_{2}}{2} with |x_{F}^{∗} (x_{1} − x_{2})| > δ for a suitable choice of x_{1} and x_{2} in D .

(∗)

We shall use the property (∗) to construct a tree {x_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}} in D that is separated by a sequence {x_{n}^{∗}}_{n≥1} of norm one linear functionals.

Towards this construction, pick an arbitrary x_{1}^{0} in D. Apply (∗) with F = {x_{1}^{0}} and find x_{F}^{∗} = x_{1}^{∗}. Property (∗) provides x_{1}^{1} and x_{2}^{1} in D satisfying

x_{1}^{0} = \frac{x_{1}^{1} + x_{2}^{1}}{2}  and  |x_{1}^{∗}(x_{1}^{1} − x_{2}^{1})| > δ .

Next apply (∗) with F = {x_{1}^{1}, x_{2}^{1}} and find x_{F}^{∗} = x_{2}^{∗}. For k = 1 and 2, property (∗) provides x_{2k−1}^{2} and x_{2k}^{2} in D satisfying

x_{k}^{1} = \frac{x_{2k−1}^{2} + x_{2k}^{2}}{2}  and  |x_{2}^{∗}(x_{2k−1}^{2} − x_{2k}^{2})| > δ .

MARIA GIRARDI 22

Instead of giving a formal inductive proof we shall be satisfied by finding x_{1}^{3}, x_{2}^{3}, …, x_{8}^{3} in D along with x_{3}^{∗} . Apply (∗) with F = {x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}} and find x_{F}^{∗} = x_{3}^{∗}. For k = 1, 2, 3 and 4, property (∗) provides x_{2k−1}^{3} and x_{2k}^{3} in D satisfying

x_{k}^{2} = \frac{1}{2}(x_{2k−1}^{3} + x_{2k}^{3})  and  |x_{3}^{∗}(x_{2k−1}^{3} − x_{2k}^{3})| > δ .

It is now clear that a separated δ-tree grows in such a set D. Remark 3.3. Theorem 2.7 presents several dentability characterizations of the CCP. Our proof that (1) implies (2) in Theorem 3.1 uses part of one of these characterizations; namely, if 𝔛 fails the CCP then there is a bounded non-midpoint-Bocce-dentable subset of 𝔛. If 𝔛 fails the CCP, then there is also a bounded non-weak-norm-one-dentable subset of 𝔛 (Theorem 2.7). In the closed convex hull of such a set we can construct a martingale that is not Pettis-Cauchy [PU, Theorem II.7]; furthermore, the bush associated with this martingale is a separated δ-bush. However, it is unclear whether this martingale is a dyadic martingale thus the separated δ-bush may not be a tree. If 𝔛 fails the CCP, then there is also a bounded non-Bocce-dentable subset of 𝔛 (Theorem 2.7). In such a set we can construct a martingale that is not Pettis-Cauchy (Theorem 2.8), but it is unclear whether the bush associated with this martingale is a separated δ-bush. Remark 3.4. The δ-tree that Bourgain [B2] constructed in a space failing the CCP is neither a separated δ-tree nor a δ-Rademacher tree since the operator associated with his tree is Dunford-Pettis.

DUNFORD-PETTIS OPERATORS 23

4 : Localization

We now localize the results thus far. We define the CCP for bounded subsets of 𝔛 by examining the behavior of certain bounded linear operators from L_{1} into 𝔛. Before determining precisely which operators let us set some notation and consider an example.

Let F (L_{1}) denote the positive face of the unit ball of L_{1}, i.e.

F (L_{1}) = {f ∈ L_{1} : f ≥ 0 a.e. and ∥ f ∥= 1} ,

and Δ denote the subset of F (L_{1}) given by

Δ = \{\frac{χ_{E}}{μ(E)} : E ∈ Σ^{+}\} .

Note that the L_{1}-norm closed convex hull of Δ is F (L_{1}).

Some care is needed in localizing the CCP. The example below (due to Stegall) illustrates the trouble one can encounter in localizing the RNP. Example 4.1. We would like to define the RNP for sets in such a way that if a subset D has the RNP then the \overline{co} D also has the RNP. For now, let us agree that a subset D of 𝔛 has the RNP if all bounded linear operators from L_{1} into 𝔛 with T (Δ) ⊂ D are representable. Let 𝔛 be any separable Banach space without the RNP (e.g. L_{1}). Renorm 𝔛 to be a strictly convex Banach space. Let D be the unit sphere of 𝔛 and T : L_{1} → 𝔛 satisfy T (Δ) ⊂ D. Since 𝔛 is strictly convex, it is easy to verify that T (Δ) is a singleton in 𝔛. Thus T is representable and so D has the RNP. If this is to imply that \overline{co} D also has the RNP, then the unit ball of 𝔛 would have the RNP. But if the unit ball of 𝔛 has the RNP then 𝔛 has the RNP; but, 𝔛 fails the RNP. The same problem arises if we replace T (Δ) ⊂ D by either T (F (L_{1})) ⊂ D or T (B(L_{1})) ⊂ D.

Because of such difficulties, we localize properties to nonconvex sets by considering their closed convex hull. We now make precise the localized definitions.

MARIA GIRARDI 24

Definition 4.2. If D is a closed bounded convex subset of 𝔛, then D has the \underline{complete} \underline{continuity property} if all bounded linear operators T from L_{1} into 𝔛 satisfying T (Δ) ⊂ D are Dunford-Pettis. If D is an arbitrary bounded subset of 𝔛, then D has the \underline{complete continuity property} if the \overline{co} D has the complete continuity property.

The RNP for subsets is defined similarly. We obtain equivalent formulations of the above definitions by replacing T (Δ) ⊂ D with T (F (L_{1})) ⊂ D. Because of the definitions we restrict our attention to closed bounded convex subsets of 𝔛.

We can derive a martingale characterization of the CCP for a closed bounded convex subset K of 𝔛. As in section 1, fix an increasing sequence {π_{n}}_{n≥0} of finite positive interval partitions of Ω such that ∨ σ(π_{n}) = Σ and π_{0} = {Ω}. Set F_{n} = σ(π_{n}). It is easy to see that a martingale {f_{n}, F_{n}} takes values in K precisely when the corresponding bounded linear operator T satisfies T (Δ) ⊂ K. In light of Fact 1.5, we now have the following fact.

Fact 4.3. If K is a closed bounded convex subset of 𝔛, then K has the CCP precisely when all K-valued martingales are Cauchy in the Pettis norm.

Theorem 2.7 localizes to provide the following characterization.

Theorem 4.4. Let K be a closed bounded convex subset of 𝔛. The following statements are equivalent.

(1) K has the CCP.

(2) All the subsets of K are weak-norm-one dentable.

(3) All the subsets of K are midpoint Bocce dentable.

(4) All the subsets of K are Bocce dentable.

Proof. It is clear from the definitions that (2) implies (3) and that (4) implies (3).

DUNFORD-PETTIS OPERATORS 25

Theorem 2.8 and Fact 4.3 show that (1) implies (4) while [PU, Theorem II.7] and Fact 4.3 show that (1) implies (2). So we only need to show that (3) implies (1). For this, slight modifications in the proof of Theorem 2.10 suffice.

Let all subsets of K be midpoint Bocce dentable. Fix a bounded linear operator T from L_{1} into 𝔛 satisfying T (Δ) ⊂ K. We shall show that the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion. Then an appeal to Fact 1.1.7 gives that K has the complete continuity property. To this end, fix ϵ > 0 and B in Σ^{+}. Let F denote the vector measure from Σ into 𝔛 given by F (E) = T (χ_{E}). Since T (Δ) ⊂ K, the set \{\frac{F (E)}{μ(E)} : E ⊂ B and E ∈ Σ^{+}\} is a subset of K and thus is midpoint Bocce dentable. The proof now proceeds as the proof of Theorem 2.10.

Towards a localized tree characterization, let K be a closed bounded convex subset of 𝔛. If K fails the CCP, then there is a subset of K that is not midpoint Bocce dentable (Theorem 4.4) and hence a separated δ-tree grows in K (Theorem 3.2). A separated δ-tree is a separated δ-bush and, with slight modifications, a δ-Rademacher tree. In light of our discussion in section 3, if a separated δ-bush or a δ-Rademacher tree grows in K, then the associated K-valued martingale is not Pettis-Cauchy and so K fails the CCP (Fact 4.3). Thus Theorem 3.1 localizes to provide the following characterization. Theorem 4.5. Let K be a closed bounded convex subset of 𝔛. The following statements are equivalent.

(1) K fails the CCP.

(2) A separated δ-tree grows in K.

(3) A separated δ-bush grows in K.

(4) A δ-Rademacher tree grows in K.

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University of Illinois, Mathematics Department, 1409 W. Green St., Urbana, IL 61801