Applications of Model Theory to Functional Analysis

Applications of

Model Theory

to

Functional

Analysis

Applications of

Model Theory

to

Functional

Analysis

José Iovino

Department of Mathematics

The University of Texas at San Antonio

San Antonio, Texas

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 2002, 2014 by José Iovino

All rights reserved.

Bibliographical Note

Applications of Model Theory to Functional Analysis, first published by Dover Publications, Inc., in 2014, is a slightly altered republication of the author’s minicourse textbook Ultraproductos en Análisis at the II Escuela de Matemáticas de America Latina y el Caribe and the XV Escuela Venezolana de Matemáticas, September 8–14, 2002, Mérida, Venezuela. Included in this edition are a new Preface, notes, and an updated bibliography.

International Standard Book Number

eISBN-13: 978-0-486-79861-5

Manufactured in the United States by Courier Corporation

78084801   2014

www.doverpublications.com

Preface to the Dover Edition

These notes evolved from a series of lectures given at Carnegie Mellon University during the late 1990’s. The notes were later adapted to be used as textbook for a minicourse on applications of ultraproducts in analysis, given at the Universidad de los Andes campus in Mérida, Venezuela, in 2002. The audience in Pittsburgh was composed of logicians and analysts; the course in Mérida was taken by students from graduate programs in Latin America. The goal of the lectures was to show how basic ideas that evolved independently in two different fields of mathematics — in this case, model theory and Banach space theory — melded to yield beautiful results; the showcases here are two theorems of Banach space theory, both of which bear the name of Jean-Louis Krivine, namely, Krivine’s Theorem on the finite representability of ℓp in all Banach lattices, and the Krivine-Maurey result that every stable Banach space contains some ℓp. At the time of preparing these notes there was no other elementary exposition in the literature that showed how the proofs of these theorems are related to fundamental ideas from logic, nor how the two proofs are related to each other. The same is true today, despite the explosion of activity in the areas of interaction between logic and functional analysis.

The original text has not been altered; only the historical remarks at the end have been edited slightly in order to bring them up to date. In the original edition, for the logical language we used Henson’s formalism of approximations of formulas; today, the preference is to use real-valued logic (see the historical remarks). Nevertheless, as pointed out in Section 1.5, for the restricted class of model-theoretic types used here (that is, quantifier-free types), the equivalence between both approaches is immediate.

The first edition was dedicated to my wife Martha and our daughter Abigail, who at the time was a baby. Since then, our son Luca was born. The monograph is now dedicated to him as well.

To Martha, Abigail, and Luca

Contents

Chapter 0.Introduction

Chapter 1.Preliminaries: Banach Space Models

1.Banach Space Structures and Banach Space Ultrapowers

2.Syntax: Positive Bounded Formulas

3.Semantics: Interpretations

4.Approximations of Formulas

5.Approximate Satisfaction

6.Beginning Model Theory

7.(1 + ε)-Isomorphism and (1 + ε)-Equivalence of Structures

8.Finite Representability

9.Types

10.Quantifier-Free Types

11.Saturated and Homogeneous Structures

12.General Normed Space Structures

13.The Monster Model

Chapter 2.Semidefinability of Types

Chapter 3.Maurey Strong Types and Convolutions

Chapter 4.Fundamental Sequences

Chapter 5.Quantifier-Free Types Over Banach Spaces

Chapter 6.Digression: Ramsey’s Theorem for Analysis

Chapter 7.Spreading Models

Chapter 8.ℓp- and c0-Types

Chapter 9.Extensions of Operators by Ultrapowers

Chapter 10.Where Does the Number p Come From?

Chapter 11.Block Representability of ℓp in Types

Chapter 12.Krivine’s Theorem

Chapter 13.Stable Banach Spaces

Chapter 14.Block Representability of ℓp in Types Over Stable Spaces

Chapter 15.ℓp-Subspaces of Stable Banach Spaces

Historical Remarks

Bibliography

Index of Notation

Index

Applications of

Model Theory

to

Functional

Analysis

CHAPTER 0

Introduction

If one were to compose a list of the most important results in of the last thirty years in Banach space theory, the following would have to be included:

1.Tsirelson’s example of a Banach space not containing ℓp or c0 [Tsi74],

2.Krivine’s Theorem [Kri76],

3.The Krivine-Maurey theorem that every stable space contains some ℓp almost isometrically [KM81],

4.The Bourgain-Rosenthal-Schechtman proof that there are uncountably many complemented subspaces of Lp [BRS81],

5.Gowers’ dichotomy [Gow96, Gow02, Gow03].

Apart from their importance, these results have in common the fact that they were proved by using concepts and techniques that originated in mathematical logic. Tsirelson’s construction was inspired by set-theoretic forcing. Krivine’s theorem was proved using classical model-theoretic tools such as types and indiscernible sequences. The Krivine-Maurey theorem was based on the notion of model-theoretic stability. The main tool of the Bourgain-Rosenthal-Schechtman paper is an ordinal-valued rank function of the type commonly used in model theory. Gowers’ dichotomy was proved using Gowers’ celebrated Ramsey theorem [Gow96, Gow02, Gow03], which resulted as a refinement of the methods used by Galvin-Prikry [GP73] and Ellentuck [Ell74] in proving partition theorems that emerged from problems about the existence of models of set theory with particular properties. (For a detailed historical account of this, see [Lar12].)

Rosenthal’s ℓ1 theorem [Ros74] is regarded as one of the most elegant theorems of Banach space theory. It was observed by Farahat [Far74] that Rosenthal’s paper contains an independent proof of the classical theorem (proved by Nash-Williams in the 1960’s [NW68], but independently also by Cohen and Ehrenfeucht, among others) that every open subset of (endowed with the product topology) is Ramsey. This observation unveiled infinite Ramsey theory as an important tool in Banach space theory and triggered a host of applications that peaked with Gowers’ Ramsey theorem [Gow96, Gow02, Gow03].

For a detailed exposition of how combinatorial methods from set theory have influenced Banach space theory, we refer the reader to Todorčević’s book on Ramsey spaces [Tod10].

In these notes, we will focus on a particular set of concepts where Banach space theory has made contact with logic, namely, concepts that originated in model theory. Among these are:

1.Ultraproducts,

2.Indiscernible sequences (called 1-subsymmetric sequences in Banach space theory),

3.Ordinal ranks (called ordinal indices in analysis),

4.Ehrenfeucht-Mostowski models (called spreading models in Banach space theory),

5.Spaces of types,

6.Stability.

Some of these concepts were introduced in analysis by direct adaptation of constructions from model theory (e.g., Banach space ultrapowers and indiscernible sequences, introduced in Krivine’s thesis [Kri67] and in the proof of Krivine’s Theorem [Kri76], respectively); others were inspired by analogies (e.g., Banach space stability, introduced by Krivine and Maurey in [KM81], motivated by the fact that in a stable theory every indiscernible sequence is totally indiscernible); and yet others were discovered independently by analysts (e.g., spreading models — and their construction using Ramsey’s Theorem — which were introduced by Brunel and Sucheston in the study of ergodic properties of Banach spaces; see [BS74]).

As we will see as well, certain basic notions from Banach space theory can be seen quite naturally from a model-theoretic perspective. An example is that of finite representability: a Banach space X is finitely representable in a Banach space Y (a