Kent State University On the Borel Complexity of some classes of Banach spaces Bruno de Mendon¸caBraga Kent, Ohio July 2013 To anyone that, somehow, enjoy it. i Acknowledgements: First of all, I would like to thank my adviser Joe Diestel. I thank him not only for all the beers we had together and the amazing stories he shared but also (and mainly) for all the mathematical help he gave me. Without Joe I would still not even know about the existence of this amazing branch of mathematics, i.e., descriptive set theory and its applications to the geometry of Banach spaces. His advices and guru-like guidance were essential for this dissertation. I am really happy to say that Kent State University provided me with a great study environment and I thank all the professors I had while a student here. Specially, I would like to thank Prof. Richard Aron for being one of the friendliest people I have ever met, for being always available to help me not only with math but also with random advices, and also for being one of the only people I could practice my (beautiful) Portuguese with. I would also like to express my sincere gratitude to Kent State University and its depart- ment of mathematics for the financial support they provided me. I also thank all members of my committee for spending their time reading this dissertation. I cannot forget to thank all my monkey friends and family. Although faraway their sup- port and mere existence were of great importance to me. My vacations back home worked to recharge my body with both love and caipirinhas. I specially thank Pedro Igor and Caio Guimares for all the personal assistance they gave me. Also, the friends I made here in Kent were essential for my sanity and I sincerely thank them for that. I was able to find some really amazing people in this tiny university town. I thank my officemate, Willian Franca, for all the mathematical brainstorms we had to- ii gether and for not carrying about an (annoying) officemate listening to loud music all day long. At last, I thank Masha, the best girlfriend in the world, for all the support and help. iii \A guy goes to a psychiatrist and says: `Doc, my brother's crazy; he thinks he's a chicken.' And the doctor says, `Well, why don't you turn him in?' The guy says, `I would, but I need the eggs.' " Alvy Singer quoting Groucho Marx. iv Contents 1 Background. 7 1.1 Banach spaces. .8 1.1.1 Basis and Sequences. .8 1.1.2 Operators. .9 1.1.3 Some Classes of Banach Spaces. 10 1.1.4 Local Theory. 12 1.2 Descriptive set theory. 13 1.2.1 Standard Borel spaces. 13 1.2.2 Trees. 16 2 Complementability of some ideals in L(X). 19 2.1 Unconditionally Converging Operators. 19 2.2 Weakly Compact Operators. 23 3 Geometry of Banach spaces. 28 3.1 Banach-Saks Property. 28 3.2 Alternating Banach-Saks Property. 31 3.3 Weak Banach-Saks property. 35 4 Complementability of some ideals in L(X), Part II. 40 1 CONTENTS 4.1 Banach-Saks Operators. 40 5 Geometry of Banach spaces, Part II. 43 5.1 Schur Property. 43 5.2 Dunford-Pettis Property. 45 5.3 B-convex Banach Spaces. 46 5.4 Daugavet Property. 47 5.5 Complete Continuous Property. 49 5.6 Radon-Nikodym property. 53 5.7 Analytic Radon-Nikodym property. 58 5.8 The Equivalence Relation of Isomorphisms. 59 6 Local Structure of Banach Spaces. 61 6.1 Finite Representability. 61 6.2 Super Reflexibility. 63 6.3 Local Unconditional Structure. 65 7 On the Borel complexity of CP . 68 7.1 Pure classes not cointaining some `p...................... 69 7.2 Spaces containing a minimal subspace. 73 8 Non-Universality Results. 79 8.1 Non-Universality Results. 79 9 Hausdorff Distance and Openings Between Banach Spaces. 83 9.1 The Hausdorff Distance on F(X)........................ 85 9.2 Openings Between Banach Spaces. 87 2 Preface. \Happiness is a Banach space." (Dewitt-Morette and Choquet Bruhat) Our goal for this dissertation is to study the Borel complexity of certain classes of Ba- nach spaces, hence, these notes lie in the intersection of descriptive set theory and the theory of Banach spaces. With this dissertation we intend to introduce and familiarize the reader with general techniques to compute the Borel complexity of classes of Banach spaces. Therefore, we do not intend to be concise and we take our time making the text as complete as possible for the understanding of the general ideas behind the proofs. With this in mind, we start this dissertation with chapter 1, which is a brief introduction to all the background the reader may need regarding descriptive set theory and the theory of Banach spaces. In chapter 2, we study two problems related to special classes of operators on separable Banach spaces being complemented in the space of its bounded operators or not. Specifi- cally, we will show that both the set of Banach spaces with its unconditionally converging operators complemented in its bounded operators, and the set of Banach spaces with its weakly compact operators complemented in its bounded operators, are non Borel. The first one is actually complete coanalytic. In both of these problems, we will be using re- 3 CONTENTS sults of [BBG] concerning the complementability of those ideals in its space of all bounded operators and the fact that the space itself contains c0. Next, in chapter 3, we study the Borel complexity of other classes of Banach spaces, namely, Banach spaces with the so called Banach-Saks property, alternating Banach-Saks property, and weak Banach-Saks property. We show that the first two of them are com- plete coanalytic sets in the class of separable Banach spaces, and that the third one is at 1 least non Borel (it is also shown that the weak Banach-Saks property is at most Σ2). In order to show some of those results we use the geometric sequential characterizations of Banach spaces with the Banach-Saks property and the alternating Banach-Saks property given by B. Beauzamy (see [Be]). The stability under `2-sums of the Banach-Saks prop- erty shown by J. R. Partington ([P]) will also be of great importance in our proofs. In chapter 4 we revisit the main concern of chapter 2 by studying another problem re- lated to the complementability of an ideal of L(X). Precisely, it is shown that the set of Banach spaces whose set of Banach-Saks operators is complemented in its bounded operators is non Borel. For this, a result by J. Diestel and C. J. Seifert ([DiSe]) that says that weakly compact operators T : C(K) ! X, where K is a compact Hausdorff space, are Banach-Saks operators, will be essential. In chapter 5 we study the Borel complexity of more geometric classes of separable Ba- nach spaces. In order to show that the class of Banach spaces with the Schur property is non Borel we will rely on the stability of this property shown by B. Tanbay ([T]), and, when dealing with the Dunford-Pettis property, the same will be shown using one of its characterizations (see [FLR], and [Fa]) and Tanbay's result. It is also shown that the 1 Schur property is at least Σ2. After that, we show that both the set of B-convex Banach 4 CONTENTS spaces and the set of Banach spaces with the Daugavet property are Borel sets. For the latter, our proof uses the geometric characterization of the Daugavet property given in [KaSSiW], and [KaW]. Following, we show that the set of separable Banach spaces with the complete continu- ous property (CCP), and the set of Banach spaces with the Radon-Nikodym property (RNP), are complete coanalytic. For this we use characterizations of these properties in terms of the existence of a special kind of bush on the space ([G], [J3]). Also, we show that the analytic Radon-Nikodym property is non Borel. Chapter 6 is dedicated to the local structure of separable Banach spaces. We show that the set of Banach spaces Y that are finitely representable in a given space X, the set of super reflexive spaces and spaces with local unconditional structure are all Borel. It is also shown that if a property P is Borel then property super-P is at least coanalytic. Chapter 7 deals with the Borel complexity of CP = fY 2 SBj9Z 2 P; Z ,! Y g, for some specific classes of Banach spaces P ⊂ SB. Specifically, we show that CP is non Borel if P is the class of minimal spaces, tight spaces, HI spaces, Banach-Saks spaces, alternating Banach-Saks spaces, Schur spaces, B-convex spaces, and infinite dimentional super reflexive Banach spaces. In chapter 8 we give several applications of the theorems obtained along these notes to non-universality like results. In all the results proven in these notes we will be applying techniques related to descriptive set theory and its applications to the geometry of Banach spaces that can be found in [D], and [S]. 5 CONTENTS At last, in chapter 9, we turn ourselves to a completely different kind of problem, there- fore, this chapter can be read independently from the rest of this dissertation. In this chapter we study the Hausdorff metric in SB2 and several openings (or gaps) between Banach spaces. Our goal in this chapter is to notice that those functions are Borel in re- lation to the Effros-Borel structure and to study the Borel complexity of other distances that are commonly defined on SB.