DUNFORD-PETTIS OPERATORS ON L_{1} AND THE COMPLETE CONTINUITY PROPERTY

BY

MARIA KATHRYN GIRARDI

B.S., Santa Clara University, 1984

ADVISOR

Dr. J.J. Uhl, Jr.

THESIS

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

in the Graduate College of the University of Illinois at Urbana-Champaign, 1990

Urbana, Illinois

ABSTRACT

The interplay between the behavior of bounded linear operators from L_{1} into a Banach space 𝔛 and the internal geometry of 𝔛 has long been evident. The Radon-Nikodým property (RNP) and strong regularity arose as operator theoretic properties but were later realized as geometric properties.

Another operator theoretic property, the complete continuity property (CCP), is a weakening of both the RNP and strong regularity. A Banach space 𝔛 has the CCP if all bounded linear operators from L_{1} into 𝔛 are Dunford-Pettis (i.e. take weakly convergent sequences to norm convergent sequences). There are motivating partial results suggesting that the CCP also can be realized as a geometric property. This thesis provides such a realization.

Our first step is to derive an oscillation characterization of Dunford-Pettis operators. Using this oscillation characterization, we obtain a geometric description of the CCP; namely, we show that 𝔛 has the CCP if and only if all bounded subsets of 𝔛 are Bocce dentable, or equivalently, all bounded subsets of 𝔛 are weak-norm-one dentable. This geometric description leads to yet another; 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. We also localize these results. We motivate these characterizations by the corresponding (known) characterizations of the RNP and of strong regularity.

ACKNOWLEDGEMENTS

The tree of wisdom and knowledge grows unboundedly in a nourishing environment. I wish to express my sincere gratitude to my thesis advisor, Professor J.J. Uhl, Jr., for providing the essentials for this growth through his generous help, guidance, encouragement, and patience during my life as a graduate student. More than a convivial advisor, he is above all else a true friend and exuberant teacher.

I would like to thank the many professors who have encouraged me with their stimulating discussions and kind words. I am also grateful to my family for always believing in me.

To my fellow chocolate and cappuccino connoisseurs, thank you for the comforting chats. To my racquetball partners, thank you for the many fine games in which we shared laughter and vented frustration. To my fellow baseball fans, maybe next year.

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. OSCILLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 3. DENTABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4. BUSHES AND TREES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5. LOCALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

CHAPTER 1 INTRODUCTION

The interplay between the behavior of bounded linear operators from L_{1} into a Banach space 𝔛 and the internal geometry of 𝔛 has long been evident. The Radon-Nikodým property (RNP) was originally defined in terms of the behavior of operators from L_{1} into 𝔛. Rieffel, Maynard, Huff, Davis, and Phelps provided the first concrete realization of the RNP as a geometric property of a Banach space when they showed that 𝔛 has the RNP if and only if all bounded subsets of 𝔛 are dentable. From this geometric characterization follows another; namely, 𝔛 has the RNP if and only if no bounded δ-bushes grow in 𝔛, and in the case that 𝔛 is a dual space (Stegall), if and only if no bounded δ-trees grow in 𝔛.

Similarly, the concept of strong regularity, a weakening of the RNP, arose as an operator theoretic property [B1] but was later realized as a geometric property [GGMS].

The complete continuity property (CCP), another operator theoretic property, has been around for many years. Dunford and Pettis showed that a Banach space with the RNP has the CCP; in fact, a strongly regular space also has the CCP.

There are motivating partial results suggesting that the CCP also can be realized as a geometric property. Bourgain showed that 𝔛 has the CCP if no bounded δtrees grow in 𝔛. As for the case in which 𝔛 is a dual space, work of Pelczynski, Hagler, and Bourgain gives that 𝔛 has the CCP if and only if 𝔛 is strongly regular.

In this thesis, we shall complete the cycle by showing that the complete continuity property is indeed an internal geometric property of a Banach space.

MARIA GIRARDI 2

Our first step is to derive an oscillation characterization of Dunford-Pettis operators (Chapter 2). Using this oscillation characterization, we obtain a geometric description of the CCP; namely, we show that 𝔛 has the CCP if and only if all bounded subsets of 𝔛 are Bocce dentable (Chapter 3). This geometric description leads to yet another; 𝔛 has the CCP if and only if no bounded separated δ-trees grow in 𝔛 (Chapter 4). We motivate these characterizations by the corresponding (known) characterizations of the Radon-Nikodým property and of strong regularity.

Throughout this thesis, 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, L_{1} to L_{1}(Ω, Σ, μ), and L_{1}-compact to compact in the L_{1}-norm topology. A subset K of L_{1} is uniformly bounded if sup_{f∈K} ∥ f ∥_{L_{∞}} is finite. All notation and terminology, not otherwise explained, are as in [DU]. For clarity, known results are presented as facts while new results are presented as theorems, corollaries, lemmas, and observations.

As for basic definitions, the RNP, strong regularity, and the CCP each have several equivalent formulations. We shall define these properties by examining the behavior of bounded linear operators from L_{1} into 𝔛.

Recall that a bounded linear operator T : L_{1} → 𝔛 is \underline{(Bochner) representable} if there is a g in L_{∞}(μ, 𝔛) such that T f = ∫_{Ω} fg dμ for each f in L_{1}(μ). A Banach space 𝔛 has the \underline{Radon-Nikodým property} if all bounded linear operators from L_{1} into 𝔛 are Bochner representable.

A bounded linear operator T : L_{1} → 𝔛 is \underline{strongly regular} if for each bounded subset D of L_{1} and ϵ > 0 there are finitely many slices S_{1}, S_{2}, …, S_{n} of D satisfying

diam (\frac{T S_{1} + … + T S_{n}}{n} ) < ϵ ,

THESIS 3

where a \underline{slice} of a subset D of Banach space 𝒴 is any non-empty set of the form {y ∈ D: y^{∗}(y) > α} where y^{∗} ∈ 𝒴^{∗} and α ∈ ℝ are fixed. A Banach space 𝔛 is \underline{strongly regular} if all bounded linear operators from L_{1} into 𝔛 are strongly regular.

An operator is \underline{Dunford-Pettis} if it maps weakly convergent sequences to norm convergent sequences. A Banach space 𝔛 has the \underline{complete continuity property} (CCP) if all bounded linear operators from L_{1} into 𝔛 are Dunford-Pettis.

It is well known that if a bounded linear operator T from L_{1} into 𝔛 is representable then T is strongly regular; and if T is strongly regular then T is Dunford-Pettis. Thus, if 𝔛 has the Radon-Nikodým property then 𝔛 is strongly regular, and if 𝔛 is strongly regular then 𝔛 has the CCP. The dual of the James tree space [cf. J] is strongly regular yet fails the RNP. Talagrand [cf. T1] has constructed a space that has the CCP yet is not strongly regular.

The following fact provides several equivalent formulations of the CCP.

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

(1) T is a Dunford-Pettis operator.

(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}-compact.

(7) E_{π}T^{∗} converges to T^{∗} in the operator topology of L(𝔛^{∗}, L_{1}).

(8) T E_{π} converges to T in the operator topology of L(L_{∞}, 𝔛).

MARIA GIRARDI 4

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. 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 1.2. 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}. 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.

It is easy to verify that a bounded linear operator from L_{1} into 𝔛 is representable by f ∈ L_{∞}(μ, 𝔛) if and only if the corresponding martingale converges in L_{1}(𝔛)- norm to f. 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 1.3. 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.

CHAPTER 2 OSCILLATION

Fix a bounded linear operator T from L_{1} into 𝔛 and consider the uniformly bounded subset T^{∗}(B(𝔛^{∗})) = { T^{∗}(x^{∗}) : ∥ x^{∗} ∥ ≤ 1} of L_{1}. In this chapter, we shall explore the oscillation behavior of elements in T^{∗}(B(𝔛^{∗})) in order to attain information about T, and thus also geometric information about 𝔛.

We can determine if T is representable by examining T^{∗}(B(𝔛^{∗})) [cf. GGMS]. Fact 2.1. The following statements are equivalent.

(1) T is representable.

(2) T^{∗}(B(𝔛^{∗})) is equi-measurable, i.e.

for each ϵ > 0 there is an A in Σ^{+} with μ(A) > 1 − ϵ such that, as a subset of L_{∞}, {(T^{∗}x^{∗})|_{A} : ∥ x^{∗} ∥ ≤ 1} is relatively L_{∞}-norm-compact.

(3) For each ϵ > 0 and B in Σ^{+} there is a subset A of B with positive measure satisfying

diam \{\frac{T (χ_{E})}{μ(E)} : E ⊂ A and E ∈ Σ^{+}\} < ϵ .

Recall that for a bounded function f in L_{1} and A in Σ^{+}, the \underline{oscillation of f} \underline{restricted to A} is given by

osc f|_{A} = ess sup_{ω∈A} f(ω) − ess inf_{ω∈A} f(ω) .

Condition (3) may be viewed as an oscillation condition on T^{∗}(B(𝔛^{∗})) since

diam \{\frac{T (χ_{E})}{μ(E)} : E ⊂ A and E ∈ Σ^{+}\} = sup_{x^{∗}∈B(𝔛^{∗})} osc (T^{∗}x^{∗})|_{A} .

THESIS 7

We can also determine whether T is strongly regular by examining the oscillation of elements in T^{∗}(B(𝔛^{∗})). Towards this, recall the following well-known fact [cf. GGMS]. Fact 2.2. If K is a uniformly bounded subset of L_{1}, then the following statements are equivalent.

(1) K is a \underline{set of small oscillation}, i.e. for each ϵ > 0 there is a finite positive measurable partition P of Ω such that for each f in K

∑_{A∈P} μ(A) osc f|_{A} < ϵ .

(2) K has the \underline{Bourgain property}, i.e. for each ϵ > 0 and B in Σ^{+} there is a finite collection F of subsets of B each with positive measure such that for each f in K there exists an A in F satisfying

osc f|_{A} < ϵ .

Applying this fact to the uniformly bounded subset T^{∗}(B(𝔛^{∗})) provides the following characterization of strongly regular operators [cf. GGMS]. Fact 2.3. The following statements are equivalent.

(1) T is a strongly regular operator.

(2) T^{∗}(B(𝔛^{∗})) is a set of small oscillation.

(3) T^{∗}(B(𝔛^{∗})) has the Bourgain property .

Clearly, the oscillation condition [Fact 2.1.3] on T^{∗}(B(𝔛^{∗})) that characterizes representable operators implies the oscillation condition [Fact 2.3.3] on T^{∗}(B(𝔛^{∗})) that characterizes strongly regular operators. The natural question to explore next

MARIA GIRARDI 8

is what oscillation condition on T^{∗}(B(𝔛^{∗})) characterizes Dunford-Pettis operators. Clearly, the condition must be some weakening of the strongly regular conditions. Towards this weakening, we now present a form of a L_{1}-oscillation that is a weakening of the usual L_{∞}-oscillation. Definition 2.4. 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)} , where 0/0 is 0 .

Since Bocce-osc f|_{A} ≤ osc f|_{A} for any bounded f in L_{1} and A in Σ^{+}, the following definitions are weakenings of the corresponding conditions in Fact 2.2. Definition 2.5. A subset K of L_{1} is a \underline{set of small Bocce oscillation} if for each ϵ > 0 there is a finite positive measurable partition P of Ω such that for each f in K

∑_{A∈P} μ(A) Bocce-osc f|_{A} < ϵ .

Definition 2.6. 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 .

We now state the result corresponding to Fact 2.2. Theorem 2.7. If K is a uniformly bounded subset of L_{1}, then the following statements are equivalent.

(1) K is relatively L_{1}-compact.

(2) K is a set of small Bocce oscillation.

(3) K satisfies the Bocce criterion.

THESIS 9

Before passing to the proof of Theorem 2.7, we note that a direct application of this theorem and Fact 1.1 to the uniformly bounded subset T^{∗}(B(𝔛^{∗})) of L_{1} yields the following characterization of Dunford-Pettis operators, this chapter’s main result.

Corollary 2.8. If T is a bounded linear operator from L_{1} into 𝔛, then the following statements are equivalent.

(1) T is a Dunford-Pettis operator.

(2) T^{∗}(B(𝔛^{∗})) is a set of small Bocce oscillation.

(3) T^{∗}(B(𝔛^{∗})) satisfies the Bocce criterion.

The remainder of this chapter is devoted to the proof of Theorem 2.7. Because of its length and complexity and also for the sake of the clarity of the exposition, we present the implications in this theorem as three separate theorems.

Theorem 2.10 below shows the equivalence of (1) and (2) in Theorem 2.7.

Remark 2.9. For a finite measurable partition π of Ω, let E_{π}(f) denote the conditional expectation of f relative to the sigma field generated by π. Thus

E_{π}(f) = ∑_{B∈π} \frac{∫_{B} f dμ}{μ(B)}χ_{B} ,

observing the convention that 0/0 is 0.

Theorem 2.10. If K is a bounded subset of L_{1}, then the following statements are equivalent.

(1) K is relatively L_{1}-compact.

(2) For each ϵ > 0 there is a finite measurable partition π of Ω such that ∥ f − E_{π}(f) ∥_{L_{1}}< ϵ for each f in K.

(3) K is a set of small Bocce oscillation.

MARIA GIRARDI 10

Proof. For bounded subsets of L_{1}, the equivalence of (1) and (2) is well-known and easy to check. The equivalence of (2) and (3) follows directly from

∥ f − E_{π}(f) ∥_{L_{1}} = ∫_{Ω} | f − ∑_{B∈π} \frac{∫_{B} f dμ}{μ(B)}χ_{B} | dμ

= ∑_{B∈π} ∫_{B} | f − \frac{∫_{B} f dμ}{μ(B)} | dμ

= ∑_{B∈π} μ(B) Bocce-osc f|_{B} ,

and the definition of a set of small Bocce oscillation. □

Viewed as a function from Σ into the real numbers, Bocce-osc f|_{(·)} is not increasing (see Example 2.14); however, μ(·) Bocce-osc f|_{(·)} is increasing (see Remark 2.15). With this in mind, we now proceed with Theorem 2.11 which shows that (2) implies (3) in Theorem 2.7. Theorem 2.11. If a subset of L_{1} is a set of small Bocce oscillation, then it satisfies the Bocce criterion. Proof. Let the subset K of L_{1} be a set of small Bocce oscillation. Fix ϵ > 0 and B in Σ^{+}. Since K is a set of small Bocce oscillation, there is a positive measurable partition π = {A_{1}, A_{2}, …, A_{n}} of Ω such that for each f in K

∑_{i=1}^{n} μ(A_{i}) (Bocce-osc f|_{A_{i}}) < ϵ μ(B) .

Set

F = {A_{i} ∩ B : A_{i} ∈ π and μ(A_{i} ∩ B) > 0}.

Fix f in K.

Since the function μ(·) Bocce-osc f|_{(·)} is increasing, if Bocce-osc f|_{A_{i}∩B} ≥ ϵ for each set A_{i} ∩ B in F then we would have that

ϵ · μ(B) = ∑_{i=1}^{n} ϵ · μ(A_{i} ∩ B)

THESIS 11

≤ ∑_{i=1}^{n} μ(A_{i} ∩ B) Bocce-osc f|_{A_{i}∩B}

≤ ∑_{i=1}^{n} μ(A_{i}) Bocce-osc f|_{A_{i}}

< ϵ · μ(B) .

This cannot be, so there is a set A_{i} ∩ B in F such that the Bocce-osc f|_{A_{i}∩B} < ϵ. Thus the set K satisfies the Bocce criterion. □

Theorem 2.12 shows that (3) implies (2) in Theorem 2.7. We shall present the author’s original proof of this implication. For a new short proof, we refer the reader to [G3]. Theorem 2.12. If a uniformly bounded subset of L_{1} satisfies the Bocce criterion, then it is relatively L_{1}-compact. We need the following lemma which we will prove after the proof of Theorem 2.12. Lemma 2.13. Let the uniformly bounded subset K of L_{1} satisfy the Bocce criterion. If ϵ > 0 and {f_{n}} is a sequence in K, then there are disjoint sets B_{1}, …, B_{p} in Σ^{+} and a subsequence {g_{n}} of {f_{n}} satisfying

∫_{Ω} | g_{n} − ∑_{j=1}^{p} \frac{∫_{B_{j}} g_{n} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ 2 ϵ .  (∗∗)

Proof of Theorem 2.12. Let the uniformly bounded subset K of L_{1} satisfy the Bocce criterion. Choose a sequence {f_{n}} in K and a sequence {ϵ_{k}} of positive real numbers decreasing to zero.

It suffices to find a sequence {π_{k}} of finite measurable partitions of Ω and a nested sequence {{f_{n}^{k}}_{n}}_{k} of subsequences of {f_{n}} such that for all positive integers n and k

∥ f_{n}^{k} − E_{k}(f_{n}^{k}) ∥_{L_{1}} ≤ 2 ϵ_{k} ,  (∗)

MARIA GIRARDI 12

where E_{k}(f) denotes the conditional expectation of f relative to the σ-field generated by π_{k} (see Remark 2.9). For then the set {f_{n}^{n}} is relatively L_{1}-compact since it can be uniformly approximated in the L_{1}-norm within 2ϵ_{k} by the relatively compact set

{E_{k}(f_{n}^{n})}_{n≥k} ∪ {f_{n}^{n}}_{n<k} ;

hence, there is a L_{1}-convergent subsequence of {f_{n}}.

Towards (∗), repeated applications of Lemma 2.13 yield for each positive integer k:

(1) disjoint subsets B_{1}^{k}, …, B_{n_{k}}^{k} of Ω each with positive measure, and

(2) a subsequence {f_{n}^{k}}_{n} of {f_{n}^{k−1}}_{n} (where we write {f_{n}^{0}} for {f_{n}} ) satisfying for each positive integer n

∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ ≤ ϵ_{k} .

Set

B_{0}^{k} = Ω \ ⋃_{j=1}^{n_{k}} B_{j}^{k}

and let π_{k} be the partition generated by B_{0}^{k}, B_{1}^{k}, …, B_{n_{k}}^{k}. Still observing the convention that 0/0 is 0, two appeals to the above inequality yield for each positive integer n and k

∥ f_{n}^{k} − E_{k}(f_{n}^{k}) ∥_{L_{1}} = ∫_{Ω} | f_{n}^{k} − ∑_{j=0}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ ∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ + ∫_{Ω} | \frac{∫_{B_{0}^{k}} f_{n}^{k} dμ}{μ(B_{0}^{k})}χ_{B_{0}^{k}} | dμ

≤ ϵ_{k} + ∫_{B_{0}^{k}} | f_{n}^{k} | dμ

= ϵ_{k} + ∫_{B_{0}^{k}} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ ϵ_{k} + ∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ 2 ϵ_{k} .

Thus the proof of Theorem 2.12 is finished as soon as we verify Lemma 2.13. Proof of Lemma 2.13. Let the uniformly bounded subset K of L_{1} satisfy the Bocce criterion. Fix ϵ > 0 and a sequence {f_{n}} in K. The proof is an exhaustion-type argument.

Let E_{1} denote the collection of subsets B of Ω such that there are disjoint subsets B_{1}, …, B_{p} of B each with positive measure and a subsequence {f_{n_{k}}} of {f_{n}} satisfying

∫_{B} | f_{n_{k}} − ∑_{j=1}^{p} \frac{∫_{B_{j}} f_{n_{k}} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ μ(B) .

Since K satisfies the Bocce criterion, there is an A in Σ^{+} and a subsequence {f_{n_{k}}} of {f_{n}} satisfying

∫_{A} | f_{n_{k}} − \frac{∫_{A} f_{n_{k}} dμ}{μ(A)} | dμ ≤ ϵ μ(A) .

Thus the collection E_{1} is not empty. If there is a set B in E_{1} with corresponding subsets B_{1}, …, B_{p} and a subsequence {g_{n}} of {f_{n}} satisfying

∫_{Ω} | g_{n} − ∑_{j=1}^{p} \frac{∫_{B_{j}} g_{n} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ ,  (1)

then we are finished since (1) implies (∗∗).

Otherwise, let j_{1} be the smallest positive integer for which there is a C_{1} in E_{1} with \frac{1}{j_{1}} ≤ μ(C_{1}). Accordingly, there is a finite sequence \{C_{j}^{1}\} of disjoint subsets of C_{1} each with positive measure and a subsequence {f_{n}^{1}} of {f_{n}} satisfying

∫_{C_{1}} | f_{n}^{1} − ∑_{j} \frac{∫_{C_{j}^{1}} f_{n}^{1} dμ}{μ(C_{j}^{1})}χ_{C_{j}^{1}} | dμ ≤ ϵ μ(C_{1}) .  (2)

MARIA GIRARDI 14

Note that since condition (1) was not satisfied, μ(C_{1}) < μ(Ω) = 1 .

Let E_{2} denote the collection of subsets B of Ω \ C_{1} such that there are disjoint subsets B_{1}, …, B_{p} of B each with positive measure and a subsequence {f_{n_{k}}} of {f_{n}^{1}} satisfying

∫_{B} | f_{n_{k}} − ∑_{j=1}^{p} \frac{∫_{B_{j}} f_{n_{k}} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ μ(B) .

Since K satisfies the Bocce criterion and μ(Ω\C_{1}) > 0, we see that E_{2} is not empty. If there is a set B in E_{2} with corresponding finite sequence {C_{j}^{2}} of subsets of B and a subsequence {g_{n}} of {f_{n}^{1}} satisfying

∫_{Ω\C_{1}} | g_{n} − ∑_{j} \frac{∫_{C_{j}^{2}} g_{n} dμ}{μ(C_{j}^{2})}χ_{C_{j}^{2}} | dμ ≤ ϵ μ(Ω \ C_{1}) ,  (3)

then we are finished. For in this case, (∗∗) holds since inequalities (2) and (3) insure that

∫_{Ω} | g_{n} − ∑_{\begin{matrix}k=1,2 \\ j \end{matrix}} \frac{∫_{C_{j}^{k}} g_{n} dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ ≤ ϵ μ(C_{1}) + ϵ μ(Ω \ C_{1}) = ϵ .

Otherwise, let j_{2} be the smallest positive integer for which there is a C_{2} in E_{2} with \frac{1}{j_{2}} ≤ μ(C_{2}). Accordingly, there is a finite sequence \{C_{j}^{2}\} of disjoint subsets of C_{2} each with positive measure and a subsequence {f_{n}^{2}} of {f_{n}^{1}} satisfying

∫_{C_{2}} | f_{n}^{2} − ∑_{j} \frac{∫_{C_{j}^{2}} f_{n}^{2} dμ}{μ(C_{j}^{2})}χ_{C_{j}^{2}} | dμ ≤ ϵ μ(C_{2}) .

Note that since condition (3) was not satisfied, μ(C_{2}) < μ(Ω \ C_{1}) .

Continue in this way. If the process stops in a finite number of steps then we are finished. If the process does not stop, then diagonalize the resultant sequence { {f_{n}^{k}}_{n=1}^{∞} }_{k=1}^{∞} of sequences to obtain the sequence {f_{n}^{n}}_{n=1}^{∞} and set C_{∞} = ⋃_{k=1}^{∞} C_{k}.

THESIS 15

Note that μ(Ω \ C_{∞}) = 0. For if μ(Ω \ C_{∞}) > 0 then, since K satisfies the Bocce criterion, there is a subset B of Ω \ C_{∞} with positive measure and a subsequence {h_{n}} of {f_{n}^{n}} satisfying

∫_{B} | h_{n} − \frac{∫_{B} h_{n} dμ}{μ(B)} | dμ < ϵ μ(B) .

Since for each positive integer m

∑_{k=1}^{m} \frac{1}{j_{k}} ≤ μ( ⋃_{k=1}^{m} C_{k}) ≤ μ(Ω)

and j_{m} > 1, we can choose an integer m > 1 such that

\frac{1}{j_{m} − 1} ≤ μ(B) .

But this implies that B is in E_{m} since

B ⊂ Ω \ C_{∞} ⊂ Ω \ ⋃_{k=1}^{m−1} C_{k}

and {h_{n}}_{n=m−1}^{∞} is a subsequence of {f_{n}^{n}}_{n=m−1}^{∞}, which in turn is a subsequence of {f_{n}^{m−1}}_{n=1}^{∞}. This contradicts the choice of j_{m}. Thus μ(Ω \ C_{∞}) = 0.

Since K is uniformly bounded, there is a real number M such that ∥ f ∥_{∞}≤ M for each f in K. Pick an integer m so that

M μ(Ω \ ⋃_{k=1}^{m} C_{k}) ≤ ϵ .

Since {f_{n}^{n}}_{n=m}^{∞} is a subsequence of {f_{n}^{m}}_{n=1}^{∞}, for each f in {f_{n}^{n}}_{n=m}^{∞} we have

∫_{Ω} | f− ∑_{\begin{matrix}1≤k≤m \\ j \end{matrix}} \frac{∫_{C_{j}^{k}} f dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ

= ∑_{k=1}^{m} ∫_{C_{k}} | f − ∑_{j} \frac{∫_{C_{j}^{k}} f dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ + ∫_{Ω\⋃_{k=1}^{m} C_{k}} |f| dμ

≤ ∑_{k=1}^{m} ϵ μ(C_{k}) + Mμ(Ω \ ⋃_{k=1}^{m} C_{k})

≤ ϵ + ϵ = 2 ϵ .

MARIA GIRARDI 16

So the disjoint subsets {C_{j}^{k} : j ≥ 1 and k = 1, …, m} along with the subsequence {f_{n}^{n}}_{n=m}^{∞} of {f_{n}} satisfy the conditions of the lemma. □

This completes the proof of Theorem 2.7.

The proof that if K has the Bourgain property then K is a set of small oscillation uses the fact that the function osc f|_{(·)} from Σ into the real numbers is increasing [cf. GGMS]. The proof of Theorem 2.12 is complicated by the fact that the function Bocce-osc f|_{(·)} does not enjoy this property, as our next example illustrates.

Example 2.14. Let A = [0, 1/4] and B = [0, 1]. Define the function f from [0,1] into the real numbers by f(t) = χ_{C}(t) where C = [1/8, 1]. It is straightforward to verify that Bocce-osc f|_{A} = 1/2 but Bocce-osc f|_{B} = 7/32.

Remark 2.15. In the proof of Theorem 2.11, we used the fact that the function

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

is an increasing function from Σ into the reals numbers. To see this, let A be a subset of B with A and B in Σ^{+}. Let

m_{A} = \frac{∫_{A} f dμ}{μ(A)} and m_{B} = \frac{∫_{B} f dμ}{μ(B)} .

Note that

∫_{A} | f − m_{A} | dμ
≤ ∫_{A} | f − m_{B} | dμ + ∫_{A} | m_{B} − m_{A} | dμ
= ∫_{B} | f − m_{B} | dμ − ∫_{B\A} | f − m_{B} | dμ + ∫_{A} | m_{B} − m_{A} | dμ .

Thus

∫_{A} | f − m_{A} | dμ ≤ ∫_{B} | f − m_{B} | dμ

since

∫_{A} | m_{B} − m_{A} | dμ = μ(A) | m_{B} − m_{A} |

= | \frac{μ(A)}{μ(B)} ∫_{B} f dμ − ∫_{A} f dμ |

= | \frac{μ(A)}{μ(B)} ∫_{B} f dμ − (∫_{B} f dμ − ∫_{B\A} f dμ) |

= | ∫_{B\A} f dμ − \frac{μ(B) − μ(A)}{μ(B)} ∫_{B} f dμ |

= | ∫_{B\A}(f − m_{B}) dμ | .

Remark 2.16. Fix f ∈ L_{1}. Viewed as a function from Σ into the real numbers, Bocce-osc f|_{(·)} is continuous, i.e. if a sequence {A_{n}} in Σ converges to A ∈ Σ in the sense that μ(A △ A_{n}) tends to 0, then Bocce-osc f|_{A_{n}} tends to Bocce-osc f|_{A}. We can verify this using methods similar to the methods in Remark 2.15; or we can note that this is an easy consequence of the Lebesgue Dominated Convergence Theorem.

Remark 2.17. If a subset K of L_{1} satisfies the Bocce criterion, then the translate of K by a L_{1}-function f also satisfies the Bocce criterion.

If K is uniformly bounded and f is bounded, then this is an easy consequence of Theorem 2.7. For the general case, let the subset K of L_{1} satisfy the Bocce criterion and f ∈ L_{1}. We shall to show that the set K + f ≡ { g + f : g ∈ K } satisfies the Bocce criterion. Towards this end, fix ϵ > 0 and B ∈ Σ^{+}. Find B_{0} ⊂ B with B_{0} ∈ Σ^{+} such that f is bounded on B_{0}.

Approximate fχ_{B_{0}} in L_{∞}-norm within \frac{ϵ}{4} by a simple function f̃. Find C ⊂ B_{0} with C ∈ Σ^{+} such that f̃ is constant on C. Since K satisfies the Bocce criterion, we can find a finite collection F of subsets corresponding to \frac{ϵ}{2} and C.

Fix g + f ∈ K + f. Find A ∈ F such that Bocce-osc g|_{A} < \frac{ϵ}{2} . Note that since

MARIA GIRARDI 18

f̃ is constant on A, Bocce-osc g|_{A} = Bocce-osc (g + f̃)|_{A} . Compute,

Bocce-osc (g + f)|_{A} ≤ Bocce-osc (g + f̃)|_{A} + Bocce-osc (f̃− f)|_{A}

≤ Bocce-osc g|_{A} + 2 ∥ (f̃− f) χ_{A} ∥_{L_{∞}}

< ϵ .

Thus K + f satisfies the Bocce criterion.

CHAPTER 3 DENTABILITY

In this chapter, 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 3.1. The following statements are equivalent.

(1) 𝔛 has the RNP.

(2) Every bounded subset D of 𝔛 is dentable.

Definition 3.2 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 3.3 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 .

(4) Every bounded subset D of 𝔛 has slices of arbitrarily small diameter.

A weakening of condition (4) provides a dentability characterization of strongly regular spaces [cf. GGMS]. This characterization is often used as the definition. Fact 3.4. 𝔛 is strongly regular if and only if for each bounded subset D of 𝔛 and ϵ > 0 there are finitely many slices S_{1}, …, S_{n} of D such that

diam \{\frac{S_{1} + … + S_{n}}{n} \} < ϵ .

The natural question to explore next is what dentability condition characterizes the CCP. Towards this, the next definition is a weakening of Definition 3.2.

Definition 3.5. 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 3.3 that are useful in showing the converse of the above implication of [PU].

Definition 3.6. A subset D of 𝔛 is \underline{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 = ∑_{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 3.7. 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 chapter’s main result, shows that these dentability conditions provide an internal geometric characterization of the CCP.

THESIS 21

Theorem 3.8. 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 chapter is devoted to the proof of Theorem 3.8. 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). For completeness sake, Fact 3.14 presents the proof of [PU] that (1) implies (2). Using Corollary 2.8, Theorem 3.11 shows that (3) implies (1). That (1) implies (4) follows from Theorem 3.9 and the martingale characterization (Fact 1.3) of the CCP. Theorem 3.9. 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 a non-Bocce-dentable subset of 𝔛. Accordingly, there is 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

MARIA GIRARDI 22

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).

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 .

THESIS 23

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 3.11. Lemma 3.10. 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 3.11. 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 Corollary 2.8 shows that 𝔛 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 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.

THESIS 25

If the L_{1}-function T^{∗}x^{∗} 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 .

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 3.10 allow us to apply property (1) to equation (2). For applying Lemma 3.10 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}}^{−}) .

MARIA GIRARDI 26

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 3.10. Proof of Lemma 3.10. 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 .

THESIS 27

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 .

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. □

[PU, Theorem II.7] shows that (1) implies (2) in Theorem 3.8 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. Since quasi-martingales enter into their proof, let

MARIA GIRARDI 30

us pause to recall a few definitions and facts. Fix the notation as in Definition 1.2. Let {ϵ_{n}}_{n≥0} be a summable sequence of non-negative numbers.

Definition 3.12. A sequence \{f_{n}\}_{n≥0} in L_{1}(𝔛) is a 𝔛-valued \underline{quasi-martingale} with respect to {F_{n}} and corresponding to {ϵ_{n}} if for each n we have that f_{n} is F_{n}- measurable and ∥ E_{n}(f_{n+1}) − f_{n} ∥_{L_{1}} ≤ ϵ_{n} .

A self-contained presentation of quasi-martingales along with the decomposition theorem below may be found in [KR].

Fact 3.13 quasi-martingale decomposition theorem. Let K be a closed bounded convex subset of 𝔛. If {f_{n}, F_{n}} is a K-valued quasi-martingale corresponding to {ϵ_{n}}, then there is a K-valued martingale {g_{n}, F_{n}} satisfying

∥ f_{n} − g_{n} ∥_{L_{1}} ≤ ∑_{j=n}^{∞} ϵ_{j} .

Note that a quasi-martingale is Pettis-Cauchy if and only if the corresponding martingale from the decomposition theorem is Pettis-Cauchy.

Fact 3.14 [PU, Theorem II.7]. Let {ϵ_{n}}_{n≥0} be a summable sequence of positive numbers. If a subset D of 𝔛 is not weak-norm-one dentable, then there is an increasing sequence {π_{n}} of partition of [0, 1) and a D-valued quasi-martingale {f_{n}, σ(π_{n})} corresponding to {ϵ_{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. Consequently, if there is a bounded nonweak-norm-one dentable subset D of 𝔛, then there is a (\overline{co} D)-valued martingale that is not Pettis-Cauchy.

Proof. Fix a summable sequence {ϵ_{n}}_{n≥0} of positive numbers. Let D be subset of

THESIS 31

𝔛 that is not weak-norm-one dentable. Accordingly, there is an ϵ > 0 satisfying:

for each finite subset F of D there is an x_{F}^{∗} in S(𝔛^{∗}) such that if x is in F then x ∈ \overline{co} (z ∈ D : | x_{F}^{∗} (z − x) | ≥ ϵ ) .

(∗)

We shall use property (∗) to construct an increasing sequence {π_{n}}_{n≥0} of finite partitions of [0, 1), a quasi-martingale {f_{n}, σ(π_{n})}_{n≥0} corresponding to {ϵ_{n}}, 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) | x_{n+1}^{∗}(f_{n+1} − f_{n}) | ≥ ϵ a.e. ,

(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 into intervals of length at most

1/2^{n} and f_{n} = ∑_{E∈π_{n}} x_{E}χ_{E} with x_{E} ∈ D have been defined. We now construct f_{n+1}, π_{n+1} and x_{n+1}^{∗} satisfying conditions (1), (2) and (3).

To this end, 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 for a suitable choice of x_{1}, …, x_{m} in D and positive real numbers α_{1}, …, α_{m} whose sum is one,

∥ x_{E} − ∑_{i=1}^{m} α_{i}x_{i} ∥ ≤ ϵ_{n}  and for each i  | x_{n+1}^{∗}(x_{i} − x_{E}) | ≥ ϵ . (∗∗)

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

MARIA GIRARDI 32

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}, π_{n+1} and x_{n+1}^{∗} satisfy conditions (1), (2) and (3).

Towards insuring that {f_{n}} is indeed a quasi-martingale corresponding to {ϵ_{n}}, fix an E = [a, b) in π_{n}. Using the above notation, we have that for almost all t in E, f_{n}(t) = x_{E} and

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} .

An appeal to condition (∗∗) yields the necessary inequality,

∥ E_{n}(f_{n+1}) − f_{n} ∥_{L_{1}}≤ ϵ_{n} .

This completes the necessary constructions. □ Remark 3.15. For a convex set D, Bocce dentability and midpoint Bocce dentability coincide since, for a fixed x^{∗} ∈ S(𝔛^{∗}), if x ∈ D satisfies the condition

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, then x ∈ D also satisfies the condition

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

THESIS 33

then ∑_{i=1}^{n} α_{i} | x^{∗}(x − z_{i}) | < 2ϵ.

Remark 3.16. It is easy to verify that B(L_{1}) is not weak-norm-one dentable, Bocce dentable, nor midpoint Bocce dentability but that

B^{+}(L_{1}) ≡ {f ∈ L_{1} : ∥ f ∥_{L_{1}}≤ 1 and f ≥ 0 a.e. }

is weak-norm-one dentable, Bocce dentable, and midpoint Bocce dentability.

Remark 3.17. For a bounded subset D and ϵ > 0, a point x ∈ D is called an \underline{ϵ-strong extreme point} of D if there is a δ > 0 such that if x_{1}, x_{2} belong to D and there is a point u on the line segment joining x_{1} and x_{2} with ∥ u − x ∥< δ, then ∥ u − x_{1} ∥< ϵ or ∥ u − x_{2} ∥< ϵ. A closed bounded convex set has the \underline{Approximate Krein-Milman} \underline{property} (AKMP) if each of its nonempty subsets has an ϵ-strong extreme point for every ϵ > 0. It is clear from the definitions that if a closed bounded convex set has the AKMP, then it is midpoint Bocce dentable.

Remark 3.18. Analogous to the RNP case, in Theorem 3.8 we would like to reduce our test sets from bounded sets to closed bounded convex sets. However, the author is not certain whether this is possible. A close look at the proofs reveal that we only need that, for each bounded linear operator T from L_{1} into 𝔛, the sets of the form

Δ_{B} ≡ \{\frac{T (χ_{A})}{μ(A)} : A ⊂ B and A ∈ Σ^{+}\} where B ∈ Σ^{+}

to be midpoint Bocce dentable. But such sets need not be closed nor convex. If T is the identity operator on L_{1}, then Δ_{Ω} is closed but is not convex. If T : L_{1} → L_{2} is given by (T f)(·) ≡ ∫_{0}^{·} f dμ, then Δ_{Ω} is neither closed nor

convex.

MARIA GIRARDI 34

If T is representable by a simple function f and B ∈ Σ^{+}, then Δ_{B} is the convex hull of the values that f takes on B. Thus, if T is a bounded linear operator into a Banach space with the RNP and B ∈ Σ^{+}, then the \overline{Δ_{B}} is convex.

If T is a bounded linear operator into a Banach space with the CCP and B ∈ Σ^{+}, then we have already seen that Δ_{B} need not be closed nor convex; however, \overline{Δ_{B}} is convex. This follows from the fact that for each ϵ > 0 there is an operator T_{ϵ} (Fact 1.1.8) that is representable by a simple function such that

sup_{f∈B(L_{∞})} ∥ (T − T_{ϵ}) (f) ∥_{𝔛} < ϵ .

CHAPTER 4 BUSHES AND TREES

In this chapter, 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 chapter’s main theorem, Theorem 4.1 below, makes precise exactly which types of bushes and trees grow in a Banach space failing the CCP. Theorem 4.1. 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.

That (1) implies (2) will follow from Theorem 4.2 below. All the other implications are straightforward and will be verified shortly. As usual, we start with definitions.

While typing this thesis, 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.

MARIA GIRARDI 36

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} = {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 4.1. 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 .

Thus, if a bounded separated δ-bush grows in 𝔛 then 𝔛 fails the CCP (Fact 1.3). □

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)

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)

MARIA GIRARDI 38

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 4.1.

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 the Rademacher functions {r_{n}} and Rademacher trees is in order. In light of our discussion in Chapter 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 4.1 [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 1.3). □ Observation that (2) implies (4) in Theorem 4.1. 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 . □

MARIA GIRARDI 40

To complete the proof of Theorem 4.1, we need only to show that (1) implies (2). Towards this end, let 𝔛 fail the CCP. An appeal to Theorem 3.8 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 4.2. 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 x_{F}^{∗} ∈ S(𝔛^{∗}) 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, let x_{1}^{0} be any element of 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 .

Instead of giving a formal inductive proof we shall be satisfied by finding x_{3}^{∗} along with x_{1}^{3}, x_{2}^{3}, …, x_{8}^{3} in D . 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 .

THESIS 41

It is now clear that a separated δ-tree grows in such a set D. □ Remark 4.3. Theorem 3.8 presents several dentability characterizations of the CCP. Our proof that (1) implies (2) in Theorem 4.1 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 3.8). In the closed convex hull of such a set we can construct a martingale that is not Pettis-Cauchy [Fact 3.14]; furthermore, the bush associated with this martingale is a separated δ-bush. However, it is unclear whether this martingale is a dyadic 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 3.8). In such a set we can construct a martingale that is not Pettis-Cauchy (Theorem 3.9), but it is unclear whether the bush associated with this martingale is a separated δ-bush. Remark 4.4. The Rademacher functions {r_{n}} may be viewed as a test sequence for the CCP. For if 𝔛 has the CCP and T is a bounded linear operator from L_{1} into 𝔛, then {T (r_{n})} converges to 0 in norm. If 𝔛 fails the CCP, then there is a bounded linear operator T from L_{1} into 𝔛 such that inf_{n} ∥ T (r_{n}) ∥ > δ some δ > 0 . Remark 4.5. Recall that a separated δ-tree is also a δ-tree and, after a reshuffling, a δ-Rademacher tree. However, a bounded δ̃-Rademacher tree need neither be a δtree nor a separated δ-tree. For example, consider the c_{0}-valued dyadic martingale {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}

The tree associated with \{f_{n}\} is a bounded \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

MARIA GIRARDI 42

space failing the CCP, the notion of a separated δ-tree is preferred to that of a δ-Rademacher tree for characterizing the CCP. Remark 4.6. 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. Modifying Bourgain’s construction, we can produce a bounded δ-Rademacher tree; however, it is not clear that this tree is a δ-tree. We now present this construction. A direct proof that a bounded δ-Rademacher tree grows in a space failing the CCP.

Let 𝔛 be a Banach space failing the CCP. Fix a non-Dunford-Pettis operator T from L_{1} into 𝔛. By a lemma of Bourgain [B2], there exists A ∈ Σ^{+}, δ > 0, and a sequence {g_{r}} of simple functions in L_{1} such that

(i) ∥ g_{r} ∥_{L_{∞}}≤ 1 (ii) {g_{r}} is weakly null in L_{1} (iii) \overline{lim} ∥ T ( f g_{r} ) ∥ ≥ 3δ ∥ f ∥_{L_{1}} if f ∈ F(A)

where F(A) ≡ {h ∈ L_{1} : ∥ h ∥_{L_{1}}=∥ hχ_{A} ∥_{L_{1}} , h ≥ 0 a.e. } . We shall construct, by induction on n, a system

{f_{k}^{n} : n = 0, 1, … and k = 1, …, 2^{n}}

of L_{1}-functions satisfying for each admissible n and k (1) f_{k}^{n} is a simple function in F(A)

(2) ∥ f_{k}^{n} ∥_{L_{1}}= 1

(3) 2 f_{k}^{n} = f_{2k−1}^{n+1} + f_{2k}^{n+1}

(4) ∥ ∑_{k=1}^{2^{n}} (T f_{2k−1}^{n+1} − T f_{2k}^{n+1}) ∥ > δ2^{n+1} . It is easy to see that {T f_{k}^{n}} will be a bounded δ-Rademacher tree in 𝔛 (in fact the tree is in the image of the probability densities of L_{1} with support in A).

THESIS 43

Set f_{1}^{0} = \frac{χ_{A}}{μ(A)} . Clearly, f_{1}^{0} satisfies conditions (1) and (2).

Fix n ≥ 0. Assume that we have constructed a system {f_{k}^{n} : k = 1, …, 2^{n}} that satisfies conditions (1) and (2). We shall now construct {f_{k}^{n+1} : k = 1, …, 2^{n+1}} such that the system {f_{k}^{m} : m = n, n + 1 and k = 1, …, 2^{m}} satisfies conditions (1) through (4).

Set f_{n} = 2^{−n} ∑_{k=1}^{2^{n}} f_{k}^{n} . Note that f_{n} ∈ F(A) and ∥ f_{n} ∥_{L_{1}}= 1. Applying Bourgain’s lemma, we may assume (just pass to a subsequence) that for each positive integer r, ∥ T ( f_{n} g_{r} ) ∥ ≥ 3δ .

Let A_{n} be the (finite) algebra generated by {f_{k}^{n} : k = 1, …, 2^{n}} and E_{n}(·) be the conditional expectation with respect to A_{n}. Set

g′_{r} = g_{r} − E_{n}(g_{r}) .

Since {g_{r}} tends weakly to 0, ∥ E_{n}(g_{r}) ∥_{L_{∞}} tends to 0. Thus, we may choose a positive integer R_{n} ≡ R such that

∥ g′_{R} ∥_{L_{∞}} ≤ 2  and  ∥ E_{n}(g_{R}) ∥_{L_{∞}} < \frac{δ}{∥ T ∥} .

Note that for k = 1, …, 2^{n}, since each f_{k}^{n} is A_{n}- measurable, ∫_{Ω} f_{k}^{n}g′_{R} dμ = 0 . Set, for k = 1, …, 2^{n},

f_{2k−1}^{n+1} = f_{k}^{n} + \frac{1}{2}f_{k}^{n}g′_{R} ≡ f_{k}^{n} (1 + \frac{1}{2}g′_{R})

and

f_{2k}^{n+1} = f_{k}^{n} − \frac{1}{2}f_{k}^{n}g′_{R} ≡ f_{k}^{n} (1 − \frac{1}{2}g′_{R}) .

Clearly, {f_{k}^{m} : m = n, n + 1 and k = 1, …, 2^{m}} satisfies conditions (1), (2), and

MARIA GIRARDI 44

(3). As for condition (4),

∥ ∑_{k=1}^{2^{n}} (T f_{2k−1}^{n+1} − T f_{2k}^{n+1}) ∥ = ∥ ∑_{k=1}^{2^{n}} T (f_{k}^{n}g′_{R}) ∥

= ∥ T (2^{n}f_{n}g′_{R}) ∥ ≥ ∥ T (2^{n}f_{n}g_{R}) ∥ − ∥ T (2^{n}f_{n}E_{n}(g_{R})) ∥ ≥ 3δ2^{n} − 2^{n} ∥ T ∥ ∥ f_{n} ∥_{L_{1}} ∥ E_{n}(g_{R}) ∥_{L_{∞}} > 3δ2^{n} − 2^{n} δ = δ 2^{n+1} .

Thus, condition (4) is also satisfied. □

CHAPTER 5 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 5.1. 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

MARIA GIRARDI 46

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.

Definition 5.2. If D is a closed bounded convex subset of 𝔛, then D has the \underline{complete 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 and strong regularity for subsets are defined similarly. We obtain equivalent formulations of the above definitions by replacing T (Δ) with T (F (L_{1})). Because of the definitions we restrict our attention to closed bounded convex subsets of 𝔛.

The oscillation results of Chapter 2 localize easily using Corollary 2.8.

Theorem 5.3. If K is a closed bounded convex subset of 𝔛, then the following statements are equivalent.

(1) K has the CCP.

(2) For each bounded linear operator T : L_{1} → 𝔛 with T (Δ) ⊂ K, the subset T^{∗}(B(𝔛^{∗})) of L_{1} is a set of small Bocce oscillation.

(3) For each bounded linear operator T : L_{1} → 𝔛 with T (Δ) ⊂ K, the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion.

We can localize the martingale characterization of the CCP. As in Chapter 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

THESIS 47

{f_{n}, F_{n}} takes values in K precisely when the corresponding operator T satisfies T (Δ) ⊂ K. In light of Fact 1.3, we now have the following fact.

Fact 5.4. 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 3.8 localizes to provide the following characterization.

Theorem 5.5. 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). Theorem 3.9 and Fact 5.4 show that (1) implies (4) while Fact 3.14 and Fact 5.4 show that (1) implies (2). So we only need to show that (3) implies (1). For this, slight modifications in the proof of Theorem 3.11 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 Theorem 5.3 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 3.11. □

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

MARIA GIRARDI 48

Bocce dentable (Theorem 5.5) and hence a separated δ-tree grows in K (Theorem 4.2). A separated δ-tree is a separated δ-bush and, with slight modifications, a δ-Rademacher tree. In light of our discussion in Chapter 4, 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 5.4). Thus Theorem 4.1 localizes to provide the following characterization. Theorem 5.6. 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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VITA

Maria Kathryn Girardi was born 4 March 1963 in Peoria, Illinois. She received a B.S. in Mathematics from Santa Clara University in June 1984. From 1984 through 1990, she studied mathematics at the University of Illinois where she received support in the form of University fellowships, research assistantships, and teaching assistantships. She is a member of Phi Kappa Phi, Pi Mu Epsilon, Alpha Sigma Nu, the American Mathematical Society, and the University of Illinois Mathematics Department Bocce Ball Association.