Ramsey Methods in Analysis (Advanced Courses in Mathematics - CRM Barcelona)

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Manuel Castellet Spiros ...

Author: Spiros A. Argyros | Stevo Todorcevic

30 downloads 721 Views 1MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica

Managing Editor: Manuel Castellet

Spiros A. Argyros Stevo Todorcevic

Ramsey Methods in Analysis

Birkhäuser Verlag Basel • Boston • Berlin

Authors: Spiros A. Argyros National Technical University Department of Mathematics Zografou Campus 157 80 Athens Greece [email protected]

Stevo Todorcevic Université Paris 7 – C.N.R.S. U.M.R. 7057 2, Place Jussieu – Case 7012 75251 Paris Cedex 05 France and Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3, Canada [email protected]

2000 Mathematical Subject Classification 46B20, 05D10, 03E75

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar.

ISBN 3-7643-7264-8 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Micha Lotrovsky, 4106 Therwil, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ºº Printed in Germany ISBN-10: 3-7643-7264-8 ISBN-13: 978-3-7643-7264-4 987654321

www.birkhauser.ch

Contents

Foreword

A

Saturated and Conditional Structures in Banach Spaces Spiros A. Argyros

vii

1

Introduction

3

I. Tsirelson and Mixed Tsirelson Spaces

7

II. Tree Complete Extensions of a Ground Norm II.1 Mixed Tsirelson Extension of a Ground Norm . . . . . . . . . . . . II.2 R.I.S. Sequences and the Basic Inequality . . . . . . . . . . . . . .

21 21 26

III. Hereditarily Indecomposable Extensions with a Schauder Basis III.1 The HI Property in X[G, σ] . . . . . . . . . . . . . . . . . . . . . . III.2 The HI Property in X[G, σ]∗ . . . . . . . . . . . . . . . . . . . . . .

39 39 43

IV. The Space of the Operators for HI Banach Spaces IV.1 Some General Properties of HI Spaces . . . . . . . . . . . . . . . . IV.2 The Space of Operators L(X[G, σ]), L(X[G, σ]∗ ) . . . . . . . . . . .

47 47 52

V. Examples of Hereditarily Indecomposable Extensions V.1 A Quasi-reﬂexive HI Space . . . . . . . . . . . . . . . . . . . . . . V.2 The Spaces p , 1 < p < ∞, are Quotients of HI Spaces . . . . . . . V.3 A Non Separable HI Space . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 62

VI. The Space Xω1

71

VII. Finite Representability of JT0 and the Diagonal Space D(Xγ )

81

VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

87

Appendix A. Transﬁnite Schauder Basic Sequences

99

Appendix B. The Proof of the Finite Representability of JT0

105

Bibliography

117

vi

B

Contents

High-Dimensional Ramsey Theory and Banach Space Geometry Stevo Todorcevic

121

Introduction

123

I. Finite-Dimensional Ramsey Theory 127 I.1 Finite-Dimensional Ramsey Theorem . . . . . . . . . . . . . . . . . 127 I.2 Spreading Models of Banach Spaces . . . . . . . . . . . . . . . . . 130 I.3 Finite Representability of Banach Spaces . . . . . . . . . . . . . . . 135 II. Ramsey Theory of Finite and Inﬁnite Sequences II.1 The Theory of Well-Quasi-Ordered Sets . . . . . . . . . . . . . . II.2 Nash–Williams’ Theory of Fronts and Barriers . . . . . . . . . . II.3 Uniform Fronts and Barriers . . . . . . . . . . . . . . . . . . . . . II.4 Canonical Equivalence Relations on Uniform Fronts and Barriers II.5 Unconditional Subsequences of Weakly Null Sequences . . . . . . II.6 Topological Ramsey Theory . . . . . . . . . . . . . . . . . . . . . II.7 The Theory of Better-Quasi-Orderings . . . . . . . . . . . . . . . II.8 Ellentuck’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . II.9 Summability in Banach Spaces . . . . . . . . . . . . . . . . . . . II.10 Summability in Topological Abelian Groups . . . . . . . . . . . .

. . . . . . . . . .

143 143 147 153 165 169 177 180 185 188 192

III. Ramsey Theory of Finite and Inﬁnite Block Sequences III.1 Hindman’s Theorem . . . . . . . . . . . . . . . . . III.2 Canonical Equivalence Relations on FIN . . . . . . III.3 Fronts and Barriers on FIN[<∞] . . . . . . . . . . . III.4 Milliken’s Theorem . . . . . . . . . . . . . . . . . . III.5 An Approximate Ramsey Theorem . . . . . . . . .

. . . . .

197 197 200 201 205 209

IV. Approximate and Strategic Ramsey Theory of Banach Spaces IV.1 Gowers’ Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.2 Approximate and Strategic Ramsey Sets . . . . . . . . . . . . . . . IV.3 Combinatorial Forcing on Block Sequences in Banach Spaces . . . IV.4 Coding into Approximate and Strategic Ramsey Sets . . . . . . . . IV.5 Topological Ramsey Theory of Block Sequences in Banach Spaces . IV.6 An Application to Rough Classiﬁcation of Banach Spaces . . . . . IV.7 An Analytic Set whose Complement is not Approximately Ramsey

217 217 220 224 229 233 240 243

Bibliography

247

Index

253

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Foreword This book contains two sets of notes prepared for the Advanced Course on Ramsey Methods in Analysis given at the Centre de Recerca Matem`atica in January 2004, as part of its year-long research programme on Set Theory and its Applications. The common goal of the two sets of notes is to help young mathematicians enter a very active area of research lying on the borderline between analysis and combinatorics. The solution of the distortion problem for the Hilbert space, the unconditional basic sequence problem for Banach spaces, and the Banach homogeneous space problem are samples of the most important recent advances in this area, and our two sets of notes will give some account of this. But our main goal was to try to expose the general principles and methods that lie hidden behind and are most likely useful for further developments. The goal of the ﬁrst set of notes is to describe a general method of building norms with desired properties, a method that is clearly relevant when testing any sort of intuition about the inﬁnite-dimensional geometry of Banach spaces. The goal of the second set of notes is to expose Ramsey-theoretic methods relevant for describing the rough structure present in this sort of geometry. We would like to thank the coordinator of the Advanced Course, Joan Bagaria, and the director of the CRM, Manuel Castellet, for giving us this challenging but rewarding opportunity.

Part A

Saturated and Conditional Structures in Banach Spaces Spiros A. Argyros

Introduction In 1991, W.T. Gowers and B. Maurey [33], independently, constructed examples of reﬂexive Banach spaces with no unconditional basis. It is remarkable that, working completely separately, they arrived at the same space. This result is a fundamental discovery with important consequences. Thus, today we have a theory, or at least the beginning of a theory, initializing from Gowers–Maurey paper. Let us recall the basic ingredients of this theory. As it was noticed by W. Johnson the Gowers– Maurey example is a Hereditarily Indecomposable (HI) space. This means that no inﬁnite dimensional closed subspace is the topological direct sum of two further inﬁnite dimensional closed subspaces of it. This is equivalent to the following remarkable geometric property: For any two inﬁnite dimensional subspaces the distance between their unit spheres is zero (sometimes this property is called “the angle zero property”). This is a new concept, deﬁning a new class of Banach spaces, which has good stability properties; for example it is closed in taking subspaces of its members. Also HI spaces have some contradictory properties, resulting from the tightness of their structure. Thus no HI space is isomorphic to any proper subspace, answering in negative the long standing hyperplane problem [31], [33]. On the other side every two closed inﬁnite dimensional subspaces have further subspaces which are almost isometric. The new concept stands in the opposite of the unconditional basic sequence and W.T. Gowers’ famous dichotomy ([30], [32]) has provided the ﬁrst suﬃcient classiﬁcation of Banach spaces. Namely, every Banach space either is unconditionally saturated or contains an HI space. Unexpectedly this classiﬁcation provides a positive solution of the homogenous problem that yields a new characterization of Hilbert spaces. Another important discovery of Gowers and Maurey concerns the Banach spaces with few operators. As they have shown [33] every bounded linear operator on a complex HI space is of the form λI + S with S strictly singular. This has as consequence that every Fredholm operator on a HI (real or complex) space is of index zero, which yields the aforementioned property that such a space is not isomorphic to any proper subspace. The structure of L(X) for real HI spaces is studied in [25]. The problem of the existence of a Banach space with very few operators (i.e. every T ∈ L(X) is of the form λI + K with K a compact operator) remains open and if such a space existed it should be connected to the methods

4

Introduction

of HI constructions. Recently it has been shown that there exist spaces with few operators not containing any HI space ([14], [15]). The third important contribution of Gowers–Maurey discovery concerns the generic character of their method. This is a powerful method of constructing Banach spaces using saturated norms and delicate codings. This method is not related exclusively to HI spaces. As Gowers has shown [31] one can also obtain new spaces with an unconditional basis resulting from variants of Gowers–Maurey method. The recent paper of J. Lopez-Abad, S. Todorcevic and the present author [12], [13] extends this method to reﬂexive Banach spaces with transﬁnite Schauder basis where new phenomena concerning the operator spaces occur. I must admit that constructing such spaces requires several steps and although each one is not so difﬁcult putting everything together makes the understanding not easy. In particular dealing with these spaces, one faces three main parts. The ﬁrst is how we deﬁne such spaces. The deﬁnition of saturated norms uses induction and additional one has also to involve the special functionals resulting from the coding. The second concerns the form of vectors of which we compute the norms. In order to show the non-existence of unconditional basic sequences we have to compute the norms of certain vectors. To locate these vectors we have to follow several steps. Finally to compute the norms requires some new techniques which have been developed for this purpose. The goal of the present notes is, on the one hand, to develop the method of strictly singular HI extensions of a ground norm and, on the other hand, to apply this method in speciﬁc constructions of HI Banach spaces. In the later part we are mainly interested in two non separable constructions. The ﬁrst concerns a non separable HI space, such an example appeared in [16], and the second a non separable reﬂexive space containing no unconditional basic sequence [12], [13]. The method of strictly singular extensions is related to the following problem. Given Y a Banach space with a Schauder basis (yn )n and not containing 1 , we are interested in ﬁnding a HI space X with a Schauder basis (en )n such that the correspondence en → yn is extended to a bounded linear operator T : X → Y . This problem has been answered aﬃrmatively in [16] with the use of a transﬁnite hierarchy of saturation methods. In the same paper the strictly singular extensions were introduced and studied although not named there. In these notes we discuss a part missing from [16], namely the extensions with low complexity saturation methods. This does not answer the above stated problem in its complete generality, however it permits some of the main constructions to be carried out and moreover, extending with the methods (Anj , m1j )j , we deal exclusively with averages, as in [33], making our approach easier. The interest for the HI extensions arises from the fact that in some cases speciﬁc features of the space Y are preserved in the HI extension of it. For example if Y is c0 (N) and the basis (yn )n is the summing basis of c0 (N), then independently of the saturation method we use, the resulting space is a quasi-reﬂexive HI space. A remarkable consequence of the extension method is a dichotomy contained in

Introduction

5

[16], not discussed here, that every separable Banach space either contains 1 or it is a quotient of a HI space. To some extend the HI extensions, thinking of the speciﬁc Y as a model of Banach space theory, attempt to do a work similar to the forcing method in set theory. Namely, they create a new model with a desired property (HI) and in some cases phenomena of absoluteness also occur, preserving properties of the initial space. The HI extensions are presented in Chapters II–IV. Chapter I is devoted to an introduction to Tsirelson type and Mixed Tsirelson spaces. The content of this chapter is not directly related to the HI extensions, thus the familiar reader could skip it. However we recommend to study the proof of Theorem I.4, where important ingredients, like the tree analysis and their usage, appear in a simple setting. Chapter V contains examples of HI extensions. Thus it is shown the existence of a quasi-reﬂexive HI space, that every p , 1 < p < +∞, is a quotient of a HI space and a non separable HI space Xns which is the dual as well as the second dual of a separable HI space. Chapter VI concerns the construction of Xω1 , a non separable reﬂexive space with no unconditional basic sequence, or according to Gowers’ dichotomy, HI saturated (i.e. every inﬁnite dimensional subspace contains a further HI subspace). The dimension of Xω1 is ω1 and it is unknown if there exists a space with the same properties and higher dimension. The main obstacle for non separable reﬂexive constructions with no unconditional basic sequence, concerns the deﬁnition of the conditional structure realized by the special functionals. The key ingredient for this is the coding function σ used by Gowers and Maurey and having its roots in the classical Maurey–Rosenthal construction [48]. The basic characteristic of σ, in the separable constructions, is that it is injective (one-toone), a property not extendable to the non separable setting. Thus the really new ingredient in the deﬁnition of Xω1 is the coding function σρ deﬁned with the use of Todorcevic’s ρ function [60],[61] . The ρ function is acting on the doubletons of ω1 , taking values in N and, although not injective, permits a deﬁnition of the special functionals sharing similar properties with the corresponding in the separable case. Chapter VII is devoted to the study of the space of diagonal non strictly singular operators of Xω1 . Let us mention that Xω1 , being a non separable reﬂexive space, admits many non trivial projections [40]. Therefore there exists a large portion of operators which are diﬀerent from the identity and non strictly singular. The space D(Xγ ) has a very precise representation, as a Banach space which is related to a long James like space. Chapter VIII is a continuation of Chapter VII in the study of the spaces of operators for subspaces of Xω1 . We have included two appendices. The ﬁrst concerns transﬁnite basic sequences and the second is devoted to a uniﬁed approach of the basic inequality and the ﬁnite block representability of the James like space JT0 . The present notes are based on a series of lectures delivered by the author in CRM Barcelona. The principal aim was to introduce the audience to the theory of HI Banach spaces. Thus there are directions of the theory not discussed here. One of them is the method developed by W.T. Gowers and B. Maurey [34] for constructing Banach spaces with selected a priori algebra of non strictly singu-

6

Introduction

lar operators. As consequence they provide a separable Banach space Xs with a Schauder basis (xn )n such that the space Xs does not contain an unconditional basic sequence and also the shift operator is an isometry. A second direction concerns the use of interpolation methods in HI constructions, as appeared in [10, 3, 27]. Recently has been studied the problem of the existence of Banach spaces with few operators (i.e. every T ∈ L(X) is of the form T = λI + S with S strictly singular) and not containing a HI subspace. The existence of few operators yields that the space X is indecomposable. Related to this type of questions are the papers [14], [15], [17]. Finally we recommend the two survey papers by W.T. Gowers [32] and B. Maurey [46] for results related to HI Banach spaces. In particular Maurey’s survey includes a beautiful presentation of some of the aforementioned methods and results not included in these notes. We extend our warm thanks to A. Arvanitakis, P. Dodos, A. Manoussakis, and A. Tolias for their valuable help during the preparation of the notes.

Chapter I

Tsirelson and Mixed Tsirelson Spaces The Gowers–Maurey example and all the subsequent constructions have their roots in B.S. Tsirelson’s fundamental discovery of a reﬂexive space with an unconditional basis not containing any p with 1 < p < ∞. Tsirelson’s space appeared in 1972. Almost twenty years later (1991) Th. Schlumprecht presented his space as an example of an arbitrarily distortable space. This is the unconditional frame for the Gowers–Maurey construction. The two norms share common features and our aim is to explain their generic character, their relation with the classical p norms and their diﬀerences.

Glossary For κ an ordinal we denote by c00 (κ) the vector space of all x : κ → R such that the set supp x = {α < κ : x(α) = 0} is ﬁnite. For x ∈ c00 (κ) we denote by ran x the minimal interval of κ containing supp x. We also denote by (eα )α<κ the natural basis of c00 (κ). For the special case of c00 (ω) we shall use the notation c00 . For E1 , E2 ⊂ κ we denote by E1 < E2 the property that sup E1 < min E2 . Further for x1 , x2 elements of c00 (κ) we denote by x1 < x2 the property that supp x1 < supp x2 . In most cases we shall use c00 (κ) as a vector space on which we shall deﬁne certain norms. These norms will be induced by an appropriate subset W of c00 (κ) and will be deﬁned as " # xW = sup |f (x)| = | f (α)x(α)| : f ∈ W . α<κ

Thus in some cases we shall denote the elements of c00 (κ) as f, g, etc. while its Hamel basis as (e∗α )α<κ meaning that we concern ourselves with the functionals of

8

Chapter I. Tsirelson and Mixed Tsirelson Spaces

the norming set. Finally for x ∈ c00 (κ) and E ⊂ κ we denote by Ex the restriction of x on E or equivalently the function xχE . We refer the reader to [24] and [41] for the fundamental background of the theory of Banach spaces. Tsirelson space Tsirelson’s norm is the ﬁrst norm deﬁned by induction. It satisﬁes a ﬁxed point property and it is implicitly deﬁned as follows. For x ∈ c00 , n # " 1 sup Ei xT . xT = max x0 , 2 n≤E1 <E2 <···<En i=1

(I.1)

The space T is the completion of c00 endowed with · T . The above deﬁnition is due to T. Figiel and W. Johnson [28]. Tsirelson’s original deﬁnition [63] was concerning the norm of the dual space. Notice that the norm of T ∗ does not admit any explicit or implicit description. Thus Tsirelson actually deﬁned the unit ball of T ∗ . Remarks I.1. (a) The existence of a norm satisfying the above implicit formula is established by induction. Namely we inductively deﬁne an increasing sequence of norms ( · n )n≥0 such that x0 = x∞ and n # " 1 sup xn+1 = max x0 , Ei xn . 2 n≤E1 <E2 <···<En i=1

Then xT = limn xn . (b) An important feature of Tsirelson’s norm is that it provides a method to saturate the structure of a Banach space with a property (P). For example the space T has the property that every normalized weakly null sequence contains no Cesaro summable subsequence, which actually shows that no p , 1 < p < ∞ is embedded in T . The existence of a weakly null normalized basic sequence with no Cesaro summable subsequence was a fundamental discovery of J. Schreier in the early 1930s. (c) Observe that in the deﬁnition of (I.1) appears the number 1/2 and the constrain n ≤ E1 < E2 < · · · < En . The number 1/2 could be substituted by any 0 < θ < 1 and the space retains all its properties. If θ = 1, then the space becomes isomorphic to 1 . The same will happen if we allow arbitrary families E1 < E2 < · · · < En . Tsirelson type norms In the sequel we shall denote by M a compact family (in the topology of pointwise convergence) of ﬁnite subsets of N which includes all singletons.

Chapter I. Tsirelson and Mixed Tsirelson Spaces

9

Deﬁnition I.2. A ﬁnite family E1 < E2 < · · · < En of subsets of N is said to be M-admissible if there exists M = {mi }ni=1 in M such that m1 ≤ E1 < m2 ≤ E2 < · · · < mn ≤ En . Next for M a compact family of subsets of N and 0 < θ < 1 we deﬁne the · (M,θ) as follows. For x ∈ c00 , n # " Ei x(M,θ) x(M,θ) = max x0 , θ sup i=1

where the above sup is taken over all M-admissible families E1 < E2 < · · · < En . Finally we denote by T (M, θ) the completion of (c00 , · (M,θ) ). It is easy to see that Tsirelson’s space T is the space T (S, 1/2), where S = {F ⊆ N : #F ≤ min F } is the Schreier family used by J. Schreier in the deﬁnition of his space [56]. The structure of T (M, θ) depends on the complexity of the underlying compact family M. In particular the structure of T (M, θ) is strongly related to the Cantor–Bendixson index of the family M denoted by i(M). We shall present later some results explaining the interference between i(M) and T (M, θ). Now we present an alternative description of the norm of T (M, θ) closer to the spirit of Tsirelson’s original deﬁnition. Let us denote by W (M, θ) the minimal subset of c00 containing ±e∗n , n ∈ N, and which is closed under the (M, θ)-operation; i.e. for every f1 < f2 < · · · < fn in W (M, θ) with supp f1 , supp f2 ,. . . , supp fn M-admissible, then θ(f1 + · · · + fn ) ∈ W (M, θ) (we shall call such f1 < f2 < · · · < fn an M-admissible family of functionals). The norm induced by W (M, θ) (i.e. for x ∈ c00 , xW (M,θ) = sup{f (x) : f ∈ W (M, θ)}) is exactly the norm x(M,θ) deﬁned above. Deﬁnition I.3. Let f ∈ W (M, θ). For a ﬁnite tree T , a family (ft )t∈T is said to be a tree analysis of f if the following are satisﬁed: (i) T has a unique root denoted by 0 and f0 = f . (ii) For every t ∈ T maximal (or terminal) ft = εt e∗k where εt = ±1 and k ∈ N. (iii) For every t ∈ T which is not maximal we have that {fs }s∈St is M-admissible and ft = θ s∈St fs (here St denotes the immediate successors of t). The tree analysis is a key ingredient which will follow us throughout these notes. Later we shall see some variants of it. Most of the estimations will be based on the tree analysis. It is not diﬃcult to see that every f ∈ W (M, θ) admits a tree analysis. Indeed we could consider the set W (M, θ) that contains ±e∗k , k ∈ N, and every f ∈ W (M, θ) satisﬁes f ∈ W (M, θ) and f admits a tree analysis. Then we show that W (M, θ) is closed under the (M, θ)-operation and from the minimality of W (M, θ) we obtain that W (M, θ) = W (M, θ). Among all possible compact families M there are two hierarchies that we are mainly concerned about. The ﬁrst is the low complexity hierarchy {An }n with An = {F ⊆ N : #F ≤ n} and the second is the family {Sξ }ξ<ω1 of Schreier

10

Chapter I. Tsirelson and Mixed Tsirelson Spaces

" # families. Each Sξ is deﬁned recursively as follows. We set S0 = {n} : n ∈ N ∪{∅}. For ξ = ζ + 1 we set Sξ

= {F ⊆ N : there exists k ≤ F1 < F2 < · · · < Fk k

with each Fi ∈ Sζ and F =

Fi } ∪ {∅}.

i=1

For ξ limit we choose ξn $ ξ and we set Sξ = {F : ∃n ≤ F with F ∈ Sξn } ∪ {∅}. It is quite possible, and partially proved, that the spaces T (M, θ) where M ∈ {An }n or M ∈ {Sξ }ξ<ω1 , describe the structure of T (M , θ) where M is an arbitrary family. The following theorem is quite remarkable showing that the classical p spaces admit an equivalent Tsirelson type norm. Theorem I.4. For every 0 < θ < 1 and every n ∈ N the space T (An , θ) is isomorphic to c0 or to some p with 1 < p < ∞. In particular: (i) If

1 n

≥ θ, then T (An , θ) is isomorphic to c0 .

(ii) If

1 n

< θ, then T (An , θ) ∼ = p , where θ =

Proof. (i) Let

1 n

1 n1/q

and

1 p

+

1 q

= 1.

≥ θ. It is enough to show that for all f ∈ W it holds f(

an en ) ≤ max |an | for all coeﬃcients (an )n .

n

n

Let f ∈ W and (ft )t∈T be a tree analysis of f . An easy inductive argument gives us that ft ( ak ek ) ≤ max |ak | , k

k

for all t ∈ T , and this inequality yields that the space T (An , θ) is isomorphic to c0 . (ii) Let n1 < θ. Let p, q be as in the assumption. We prove that the basis (en )n of T (An , θ) is equivalent to the standard basis of p . The proof goes through the following four steps. Step 1. For every x ∈ c00 ,

x ≤ xp .

Proof. It is enough to show that for every f ∈ W (M, θ) it holds |f (x)| ≤ xp .

(I.2)

Let (ft )t∈T be a tree analysis of f . It is trivial that for a terminal node t ∈ T , ft satisﬁes (I.2).

Chapter I. Tsirelson and Mixed Tsirelson Spaces

11

1 and (fs )s∈St is An -admissible. Let t ∈ T , ft = θ s∈St fs where θ = n1/q Assume that for every s ∈ St we have that for all y ∈ c00 , |fs (y)| ≤ yp . Let x ∈ c00 . Then, setting xi = (supp(fi ))(x) we have

|ft (x)| ≤θ

|fs (x)| =

s∈St

≤

1 1

nq

1

xi p ≤

s∈St

1

nq

|fs (xi )|

s∈St

d $ #St % q1 1 ( xi pp ) p ≤ xp , n i=i

where in the previous inequality we use the inductive assumption and H¨ older’s inequality. 1 m 1 p ≤ Step 2. For all m ∈ N, 1 m i=1 ei . np

1 Proof. Suppose ﬁrst that m = ns for some s ∈ N. The functional f= ns/q clearly belongs to W (M, θ). So s

n

s

ei ≥ f

n % $ ei =

i=1

i=1

$ ns

i=1 ei

%

1 s n = ns(1−1/q) = ns/p = m1/p . ns/q

Now let m ∈ N and ﬁnd s such that ns ≤ m ≤ ns+1 . Then

m

s

ei ≥

i=1

n

ei = ns/p =

i=1

1 n1/p

n(s+1)/p ≥

1 n1/p

m1/p .

∞ Step 3. For every normalized block sequence (xk )∞ k=1 of the basis (en )n=1 we have

a k xk ≤

2 ak ek θ

for all coeﬃcients (ak ) . Proof. It is enough to show that for every f ∈ W one gets 2 ak ek . f( a k xk ) ≤ θ For the proof we shall need the following deﬁnition. Deﬁnition I.5. Let f ∈ W (M, θ), (ft )t∈T be a tree analysis of f and let (xk )k be a ﬁnite block sequence. a) For every k ∈ N we consider the set of nodes Tk = {t ∈ T

:

(i) ran ft ∩ ran xk = ∅ for all s t if s ∈ Su (ii) ran fs ∩ ran xk = ran fu ∩ ran xk (iii) for all s ∈ St ran fs ∩ ran xk ran ft ∩ ran xk }

12

Chapter I. Tsirelson and Mixed Tsirelson Spaces

b) For every t ∈ T we set Dt =

st {k

: s ∈ Tk }.

It is easy to see that for every k ∈ N, #Tk ≤ 1. Also a t ∈ T could belong to more than one Tk . The proof of Step 3 will be an immediate consequence of the following two lemmas. Lemma I.6. Let f ∈ W (M, θ) with a tree analysis (ft )t∈T and (xk )k be a ﬁnite normalized block sequence. Assume that for all t ∈ T it holds that # ({s ∈ St : Ds = ∅} ∪ {k : Tk = {t}}) ≤ n . Then there exists g ∈ W (M, θ) such that for all k it holds that 1 g(ek ) . θ Proof. For every t ∈ T we set K = Dt \ s∈St Ds = {k : Tk = {t}}. We also set E = {s ∈ St : Ds = ∅}. It follows from the hypothesis that #(K ∪ E) ≤ n. We shall prove by induction that for every t ∈ T there exists gt ∈ W (M, θ) such that f (xk ) ≤

supp gt ⊂ Dt and ft (xk ) ≤ Indeed let t ∈ T and ft = θ ft (

s∈St

a k xk ) = θ

fs . We have that

fs (

s∈St

k

1 gt (ek ) for every k ∈ Dt . θ

We set gt = θ(

a k xk ) + ( fs )(ak xk )

s∈E

gs +

.

k∈K s∈St

k∈Ds

e∗k ) .

k∈K

e∗k , gs

∈ W (M, θ) for every k ∈ K, s ∈ E and #(K ∪ E) ≤ n, it follows that Since gt ∈ W (M, θ). From the fact that for every k s∈St

fs (xk ) ≤

1 1 xk = e∗k (ek ), θ θ

we get, ft (xk ) ≤ 1θ gt (ek ) for every k ∈ K. If k ∈ Ds for some s ∈ E, from the inductive hypothesis we get ft (xk ) = θfs (xk ) ≤ gs (ek ) =

1 gt (ek ) . θ

In order to complete the proof of Step 3, we show that there exists a partition of xk such that the assumptions of the previous lemma are satisﬁed.

Chapter I. Tsirelson and Mixed Tsirelson Spaces

13

Lemma I.7. Let f ∈ W with a tree analysis (ft )t∈T and (xk )k be a ﬁnite block sequence. Then there exists a partition of xk , xk = xk + xk such that (ft )t∈T , (xk )k and (ft )t∈T , (xk )k satisfy the assumptions of the previous lemma. Proof. Let f , (xk )k=1 be as above. Let (ft )t∈T be a ﬁxed analysis of f . For k = 1, . . . , we set sk to be the minimum of n ∈ N ∪ {0} such that there exist two t1 , t2 ∈ T with (i) |t1 | = |t2 | = n, (ii) ran ftj ∩ ran xk = ∅ for j = 1, 2 if such an n exists. Otherwise we set sk to be the maximum of all n ∈ N such that there exists t ∈ T with (i) |t| = n, (ii) if supp(xk ) is a singleton ran ft ∩ ran xk = ∅. For every k, let {ft : t ∈ T , |t| = sk and ran ft ∩ ran xk = ∅} = {f1 < · · · < fd }. We deﬁne xk = xk |supp f1 and xk = xk | d supp fi . i=2

Then the pairs (ft )t∈T , (xk )k and (ft )t∈T , (xk )k satisﬁes the conclusion. Indeed, for t ∈ T let K = {k : Tk = {t}} and E = {s ∈ St : Ds = ∅}. For k ∈ K there exists tk ∈ St with ran ftk ∩ ran xk = ∅ and max supp ftk = max supp xk . It follows that tk ∈ E. Therefore we can deﬁne a one-to-one map G : K → St \ E, hence #K + #E ≤ #St = n. The proof for the pair (ft )t∈T , (xk )k is similar. The previous two lemmas completes the proof of Step 3.

Step 4. For all and all rational non-negative (rj )j=1 ,

1/p

rj ej ≥

j=1

Proof. Write rj = sj i=sj−1 +1 ei ,

kj k

, kj , k ∈ N. Set s0 = 0 , sj = k1 + · · · + kj and uj = 1/p

j = 1, . . . , . By Step 1, uj ≤ kj

by unconditionality.

1

1/p

rj ej =

j=1

1 1/p ( rj ) . 2n j=1

k

1/p

. So

j=1

1

1/p

kj ej ≥

k

1/p

j=1

uj ej

14

Chapter I. Tsirelson and Mixed Tsirelson Spaces By Step 3, 1

k =

1/p

uj ej ≥

j=1

θ 1 uj uj 2 k1/p j=1 uj

sj s θ 1 θ 1 e = ei . i 2 k1/p j=1 i=s +1 2 k1/p i=1 j−1

By Step 2,

s

i=1 ei

≥

1/p 1 s n1/p

; so using that θ =

s 1/p θ 1 1 s 1 $ e ≥ = i 1/p 1/p 2 k 2n k 2n i=1

j=1

kj %1/p

k

1 n1/q

we get

1 $ %1/p rj . 2n j=1

=

Step 4 and the unconditionality of (en )n∈N imply that ak ek ≥ 1 |ak |p )1/p for all coeﬃcients (ak ) . This fact combined with Step 1 completes 2n ( the proof of the Theorem. Remark. The result of the Theorem can also be deduced by Steps 1, 2 and 3 using a well known theorem of Zippin [65]. This is a signiﬁcant result permitting a uniﬁed approach of Tsirelson’s space and the classical p spaces. Hence the saturated norms enable us to extend the class of the classical sequence spaces. Next we state the following result without proof. Theorem I.8. Let M be a compact family of ﬁnite subsets of N containing the subsets of its elements and all singletons. Let also 0 < θ < 1. Then the following hold: (i) If i(M) < ω and

1 i(M)

≥ θ, then T (M, θ) ∼ = c0 .

(ii) If i(M) < ω and

1 i(M)

< θ, then T (M, θ) ∼ = p , for some 1 < p < ∞.

(iii) If i(M) ≥ ω, then T (M, θ) is reﬂexive with an unconditional basis not containing any p , 1 < p < ∞. (iv) If i(M) = ω, then there exists a normalized sequence (xn )n in T (M, θ) and a subsequence (enk )k in Tsirelson’s space T (S, θ) which are equivalent. " # Here i(M) = min α < ω1 : M(α) ⊂ {∅} and M(α) is the usual α-Cantor– Bendixson derivative of the countable compact space {χF : F ∈ M} ∪ {∅}. It is open whether or not for M as in the theorem with i(M) > ω there exists some ξ < ω1 such that for the spaces T (M, θ) and T (Sξ , θ) the corresponding conclusion (iv) holds. For results related to this problem we refer the reader to [42]. A proof of the above theorem can be found in [20].

Chapter I. Tsirelson and Mixed Tsirelson Spaces

15

Mixed Tsirelson norms Schlumprecht space S is the completion of c00 endowed with the following norm. For x ∈ c00 , " xS = max x0 , sup sup n

# 1 Ei xS , log2 (n + 1) i=1 n

where the inside sup is taken over all choices E1 < E2 < · · · < En of subsets of N. The motivation for the construction of such a space was to provide an example of an arbitrarily distortable Banach space. The space S appeared as an ad hoc construction. However the next deﬁnition makes the relation of S with the Tsirelson type spaces more transparent. Let (Mn )n , (θn )n be two sequences with each Mn a compact family of ﬁnite subsets of N, 0 < θn < 1 and limn θn = 0. The mixed Tsirelson space T [(Mn , θn )n ] is the completion of c00 endowed with the norm k # " Ei x∗ , x∗ = max x0 , sup sup θn n

i=1

where the inside sup is taken over all choices E1 < E2 < · · · < Ek of Mn admissible families. Remark I.9. (a) In the above notation the Schlumprecht space S is the mixed 1 Tsirelson space T [(An , log (n+1) )n ], where An = {F ⊆ N : #F ≤ n}. 2

(b) The mixed Tsirelson space T [(Mk , θk )nk=1 ] deﬁned by a ﬁnite family (Mk , θk )nk=1 is isomorphic to T (Mk0 , θk0 ) for some 1 ≤ k0 ≤ n. Thus in order to obtain really new spaces we need to use inﬁnite sequences (Mn , θn )n . As in the case of Tsirelson spaces there is an alternative deﬁnition of the norm of mixed Tsirelson spaces resulting from a norming set W [(Mn , θn )n ]. This set is deﬁned as the minimal subset of c00 satisfying the following properties. (i) W [(Mn , θn )n ] contains all ±e∗k , k ∈ N. (ii) It is closed under the operations (Mn , θn )n . It follows that for an f ∈ W [(Mn , θn )n ] the tree analysis (ft )t∈T is also deﬁned, taking into account the necessary modiﬁcations. We shall discuss this topic more extensively in the next chapter. The main property of mixed Tsirelson spaces is that for appropriate choices (Mn , θn )n , they provide countable many equivalent norms ( · n )n such that for each (xk )k block sequence in T [(Mn , θn )n ] and every n ∈ N there exists a vector yn ∈ (xk )k such that yn = yn n and for m = n yn m ≤ εm where εm → 0. In other words, the sequence (·n )n has the property that in every block subspace none of them dominates the rest but also it is not dominated by the rest.

16

Chapter I. Tsirelson and Mixed Tsirelson Spaces

Theorem I.10. Let X = T [(Mn , θn )n ]. If for some n ∈ N it holds that i(Mn ) ≥ ω or i(Mn ) = r < ω and θn > 1r , then X is reﬂexive. Moreover if the ﬁrst alternative holds, then X does not contain isomorphically any of the spaces p , 1 ≤ p < ∞, c0 . Proof. The proof of the reﬂexivity is similar to the original proof of Tsirelson [T]. According to a classical result of R.C. James [35], it is enough to show that the basis of X is boundedly complete and shrinking. (i) The basis (en )n of T [(Mn , θn )n ] is boundedly complete. On the contrary, assume n that there exist ε > 0 and a block sequence (xk )k of (en )n such that supn k=1 xk ≤ 1 and xk ≥ ε for all k = 1, 2, . . .. Since the basis is unconditional it is enough to ﬁnd a ﬁnite subset A of N such that k∈A xk > 1. According to the assumption there exist n, r ∈ N such that i(Mn ) ≥ r (r−1) and θn > 1r . Therefore there exists n0 ∈ Mn . It follows that if (yk )k is a block sequence of (en )n there exist a sequence (nt )t∈N of positive integers and a subsequence (ykt )t∈N of (yk )k such that nt ≤ supp ykt+1 < nt+1 and for every l = 0, 1, . . . it holds {n0 , nlr+1 , . . . , nlr+(r−1) } ∈ Mn . It follows that the sequence (ylr+1 , . . . , y(l+1)r ) is Mn -admissible. Let s ∈ N be such that (θn r)s > 1ε . From the previous observation, we can choose a subsequence (xt )t of (xk )k such that for every l = 0, 1, . . . the rsequence (xlr+1 , . . . , x(l+1)r ) is Mn -admissible. For l = 0, 1, . . . , we set x(1,l) = i=1 xlr+i . It follows that r x(1,l) ≥ θn xlr+i ≥ θn rε . i=1

We repeat the same procedure for the sequence (x(1,l) )l∈N and we get a block sequence (x(2,l) )l∈N of (xn )n such that x(2,l) ≥ (θn r)2 ε. Repeating this procedure s times we get a block sequence (x(s,l) )l , where xs,l is sum of terms of the sequence (xn )n , such that x(s,l) ≥ (θn r)s ε > 1, a contradiction. (ii) The basis {en }∞ n=1 is a shrinking. Let θ = maxk θk < 1. For f ∈ X ∗ and m ∈ N, denote by Qm (f ) the restriction of f to the space generated by {ek }k≥m . It suﬃces to prove the following: For every f ∈ BX ∗ there is m ∈ N such that Qm (f ) ∈ θBX ∗ . Recall that BX ∗ = co(W ), where the closure is in the topology of pointwise convergence. We shall ﬁrst prove the following: Claim. For every f ∈ W there is m such that Qm (f ) ∈ θ co(W ). To prove this, let f ∈ W and let {f n }∞ n=1 be a sequence in W converging pointwise to f . If f n = e∗kn for an inﬁnite number of n, we have nothing to prove. So suppose that for every n there are kn ∈ N, a set {mn1 , . . . , mndn } ∈ Mkn and vectors fin ∈ W , i = 1, . . . , dn such that mn1 ≤ supp f1n < mn2 ≤ supp f2n < · · · < mndn ≤ supp fdnn and f n = θkn (f1n + · · · + fdnn ). If there is a subsequence of {θkn } converging to 0,

Chapter I. Tsirelson and Mixed Tsirelson Spaces

17

then f = 0. So we may suppose that there is a k such that kn = k for all n, i.e. θkn = θk and {mn1 , . . . , mndn } ∈ Mk . Since Mk is compact, if we substitute {f n } with a subsequence we get that there is a set {m1 , . . . , md } ∈ Mk such that the sequence of indicator functions of the sets {mn1 , . . . , mndn } converges to the indicator function of {m1 , . . . , md }. So, for large n, mni = mi , i = 1, . . . , d, and mnd+1 → ∞ as n → ∞. Passing to a further subsequence of (f n )∞ n=1 , we get that there exist fi ∈ W , i = 1, . . . , d, with supp fi ⊂ [mi , mi+1 ), i = 1, . . . , d − 1, and supp fd ⊂ [md , ∞) such that fjn → fj pointwise for j = 1, . . . , d. We conclude that f = θk (f1 + · · · + fd ), so Qmd (f ) = θk fd ∈ θ co(W ). The proof of the claim is complete. In particular we get that W is a weakly compact subset of c0 . Consider now an f ∈ BX ∗ = co(W ). In the set BX ∗ = co(W ) the pointwise topology coincides with the restriction of the w∗ topology of ∞ onto co(W ). According to Choquet’s theorem there exists a measure µ1 ∈ M1 (W ) such that Gdµ = G(f0 ) W ∗

for every linear w -continuous function G. Consider the sets Am = {f ∈ W , Qm (f ) ∈ θ 2 co(W )}, m ∈ N . The sequence (Am )m is an increasing sequence of closed sets converging to W , and hence there exists m0 such that |µ|(Am0 ) > 1 − θ(1 − θ). We claim that 1 θ Qm0 (f0 ) ∈ co(W ). Indeed if not, then by the Hahn–Banach theorem there exists a linear w∗ -continuous function G such that sup{G(f ) : f ∈ co(W )} = 1 <

1 G(Qm0 (f0 )) . θ

On the other hand, 1 1 G(Qm0 (f0 )) = (G ◦ Qm0 )dµ θ θ W 1 1 = (G ◦ Qm0 )dµ + (G ◦ Qm0 )dµ θ A m0 θ W \Am0 ≤

1 1 2 θ |µ|(Am0 ) + |µ|(W \ Am0 ) = θ + 1 − θ = 1 , θ θ

a contradiction. Hence Qm0 (f0 ) ∈ θco(W ). This completes the proof that the basis is shrinking and hence it follows that X is reﬂexive. For the remaining part, since X is reﬂexive it follows that 1 and c0 do not embed in X. Assume that for some p, 1 < p < ∞,p , embeds in X. By standard arguments, there exists a block sequence (xn )n∈N of (en )n equivalent

18

Chapter I. Tsirelson and Mixed Tsirelson Spaces

to the standard basis of p . Hence there exists C > 0 such that n an xn ≤ 1 1 C( n |an |p ) p for all coeﬃcients (an )n . Choose m ∈ N such that m1− p > θCn . Since i(Mn ) ≥ ω, there exist n1 , . . . , nm ∈ N such that the sequence (xn1 , . . . , xnm ) is Mn -admissible. It follows θn m ≤

m

1

xi ≤ Cm p ,

i=1

a contradiction.

Notes and Remarks. Tsirelson space is an important discovery in Banach space theory. It is the “ﬁrst truly non-classical space” according to E. Odell and Th. Schlumprecht [53]. It is the ﬁrst space with its norm inductively deﬁned, and, more important, it introduces a fundamental method for saturating the structure of a Banach space with a property (P). To illustrate this we recall that J. Schreier [56] provided the ﬁrst example of a weakly null sequence (xn )n with no norm Cesaro summable subsequence. Tsirelson space retains this property for all seminormalized weakly null sequences and it remains reﬂexive. Let us mention that Tsirelson’s initial construction used the forcing method and he actually deﬁned the dual of what we call Tsirelson space. The implicit form is due to T. Figiel and W.B. Johnson [28]. We recommend the interested reader Tsirelson’s web page where it is explained how he discovered his space. A tentative study followed Tsirelson’s discovery. Thus T. Fiegel and W.B. Johnson [28], introduced the p-convexiﬁcation of T and W.B. Johnson [39], deﬁned the modiﬁed version of T spaces with remarkable properties. We refer the reader to [21] for a comprehensive presentation of the results concerning Tsirelson space. Tsirelson-type spaces of the form T (M, θ) were introduced in an unpublished paper [5], where also the proof of Theorem I.4 was presented. The later result was initially proved in [18] with a diﬀerent proof. It is worth noticing that Lemmas I.6, I.7 are the simplest approach of the basic inequality, which will be discussed in the following chapters. The ﬁrst mixed Tsirelson space is Schlumprecht’s space [55]. This space was the decisive ingredient for Gowers–Maurey example [33]. The concept of Mixed Tsirelson space was introduced in [6]. The modiﬁed versions of mixed Tsirelson spaces are discussed in [8] and [9]. It is interesting that while Tsirelson space is isomorphic to its modiﬁed version, genuine mixed Tsirelson spaces are totally incomparable to their modiﬁed version. A. Manoussakis in [43] introduced and studied a class of mixed Tsirelson deﬁned as p-spaces. These are spaces of the form T [(Anj , θj )j ] satisfying the following property: For j ∈ N we denote by Tj = T [(Ani , θi )ji=1 ]. Then Tj is isomorphic to pj with 1 < pj ≤ ∞, [20]. A p-space is a space of the above such that the sequence (pj )j strictly decreases to p. Under this consideration Schlumprecht space S is an 1-space.

Chapter I. Tsirelson and Mixed Tsirelson Spaces

19

Theorem I.10 is proved in [6] and Theorem I.8 which is stated with no proof can be found in [20]. We refer the reader to survey papers [11] and [51] for related results to the content of the present section. The hierarchy of Schreier families (Sξ )ξ<ω was introduced in [1]. We refer the reader to S. Todorcevic’s part of this book for a systematic presentation of the compact families of ﬁnite subsets of N and their remarkable Ramsey properties.

Chapter II

Tree Complete Extensions of a Ground Norm In this chapter we start the novel construction of HI extensions of a ground norm. Our approach shares some common metamathematical ideas with the extension of models in set theory. Namely, one may think of a ground norm as an initial model, and the HI extension of it, being a new Banach space, which is HI and at the same time preserves some of the properties of the initial space. In this part we shall discuss the mixed Tsirelson extensions which correspond to the unconditional extensions.

II.1

Mixed Tsirelson Extension of a Ground Norm

We ﬁx two sequences of integers (mj )j and (nj )j such that m1 = 2, mj+1 = m5j and n1 = 4, nj+1 = (5nj )sj where sj = log2 m3j+1 . These sequences will follow us throughout these notes. Deﬁnition II.1. Let κ be an ordinal and G ⊂ c00 (κ). The set G is said to be a ground set provided the following are fulﬁlled: (i) For α < κ, e∗α ∈ G, the set G is symmetric (i.e. g ∈ G iﬀ −g ∈ G) and closed in the restriction of its elements on intervals of κ (i.e. for E ⊂ κ interval and g ∈ G, Eg = g · χE ∈ G). (ii) For g ∈ G, g∞ ≤ 1 and g(α) ∈ Q for α < κ. A ground norm is the norm induced on c00 (κ) by a ground set G. Namely for x ∈ c00 (κ) xG = sup{g(x) : g ∈ G}. We shall denote by YG the completion of (c00 (κ), · G ).

22

Chapter II. Tree Complete Extensions of a Ground Norm

A property of YG is that the natural Hamel basis (eα )α<κ of c00 (κ) deﬁnes a normalized bimonotone transﬁnite Schauder basis of YG . This is a consequence of the fact that the norming set is closed in the interval projections and also the property that g∞ ≤ 1 for all g ∈ G. In the opposite, for every Banach space Z with a transﬁnite Schauder basis (zα )α<κ there exists a ground set G such that the natural correspondence eα → zα from YG to Z deﬁnes an isomorphism. The mixed Tsirelson extension of the ground norm · G is deﬁned on c00 (κ) by the next formula. For x ∈ c00 (κ) we set d " # 1 Ei ∗ , E1 < · · · < Ed , d ≤ nj } . x∗ = max xG , sup sup{ mj i=1 j

The space Tκ [G] is the completion of (c00 (κ), · ∗ ). Observe that the above deﬁned norm is greater than or equal to · G and its diﬀerence from the usual mixed Tsirelson norms is that we have substituted x0 by xG . Deﬁnition II.2. Let · G be a ground norm in c00 (κ). The mixed Tsirelson extension Tκ [G] is said to be a strictly singular extension of YG if the identity map I : Tκ [G] → YG is a strictly singular operator. We remind the reader that an operator T : X → Y is strictly singular if its restriction to any inﬁnite dimensional closed subspace of X is not an isomorphism. The following is a well known result from the theory of strictly singular operators [41]. Theorem II.3. Let T : X → Y be a strictly singular operator. Then for every inﬁnite dimensional subspace Z of X and every ε > 0 there exists an inﬁnite dimensional subspace W of Z such that T |W < ε. If moreover X has a transﬁnite Schauder basis and Z is a block subspace of X then W may be selected as a block subspace of Z. The norming sets Wκ [G], Wκ [G] As in the case of mixed Tsirelson spaces there is an alternative deﬁnition of the norm of the space Tκ [G] through employing a norming set of functionals as follows. Deﬁnition II.4. We shall denote by Wκ [G] the minimal subset of c00 (κ) satisfying the following conditions: (i) It contains the ground set G. (ii) It is closed in the (Anj , m1j ) operations. (iii) It is rationally convex. It is easy to check that the set Wκ [G] is symmetric and closed in the restriction of its elements on the intervals of κ.

II.1. Mixed Tsirelson Extension of a Ground Norm

23

We shall also denote by Wκ [G] the minimal subset of c00 (κ) satisfying the above (i) and (ii) (i.e. we do not require the set Wκ [G] to be closed in rational convex combinations). The tree analysis of f ∈ Wκ [G], f ∈ Wκ [G] and their relation In the sequel for f ∈ Wκ [G] (f ∈ Wκ [G]) resulting from the operation (Anj , m1j ) for some j ∈ N we shall denote by w(f ) the weight of f which is equal to mj . The weight w(f ) is not necessarily uniquely determined. Also for f ∈ Wκ [G] we say that f is of type 0 if f ∈ G, f is of type I if w(f ) exists and f is of type II if it is a rational convex combination. It is easy to show that every f ∈ Wκ [G] is of one of the above deﬁned types. For f ∈ Wκ [G] there are only two possibilities, namely type 0 and type I. As it happens with the weight of f the type of f is not necessarily unique. Deﬁnition II.5. Let f ∈ Wκ [G]. A family (ft )t∈T with T a rooted ﬁnite tree, is a tree analysis of f if the following hold: (i) The functional f0 equals to f where 0 denotes the root of T . (ii) Each ft belongs to Wκ [G] and if t ≺ s then ran(fs ) ⊂ ran(ft ). (iii) If t is a maximal element of T then ft ∈ G. (iv) For t ∈ T which is not maximal, denoting by St the set of 1immediate successors of t, either ft = m1j fs as a result of an (Anj , mj ) operation of s∈St the functionals (fs )s∈St , or ft = rs fs as a rational convex combination s∈St

of the functionals (fs )s∈St and each fs is not a convex combination of its immediate successors. Proposition II.6. Every f ∈ Wκ [G] admits a tree analysis. The proof is easy. We simply consider all f ∈ Wκ [G] admitting a tree analysis and then we show that this set satisﬁes (i), (ii) and (iii) of Deﬁnition II.5. The result follows from the minimality of the set Wκ [G]. Remark II.7. For f ∈ Wκ [G] the tree analysis (ft )t∈T is deﬁned in the same manner. The only diﬀerence concerns condition (iv) in the above deﬁnition where the second alternative does not occur. The conclusion of the above proposition remains valid for f ∈ Wκ [G]. The next proposition connects the sets Wκ [G], Wκ [G].

24

Chapter II. Tree Complete Extensions of a Ground Norm

Proposition II.8. Let G be a ground subset of c00 (κ). (i) If f ∈ Wκ [G] is of type I with w(f ) = mj then there exist (fi )ni=1 in Wκ [G] with w(fi ) = mj such that f is a rational convex combination of (fi )ni=1 . Moreover, if f admits a tree analysis (ft )t∈T such that for every t ∈ T with ft of type I, w(ft ) = mj0 then each fi admits a tree analysis with the same property. (ii) If f ∈ Wκ [G] is of type II then f is a rational convex combination of a family (fi )ni=1 of elements of Wκ [G]. (iii) Wκ [G] = convQ (Wκ [G]). (iv) The sets Wκ [G], Wκ [G] induce the same norm on c00 (κ) which is the mixed Tsirelson extension deﬁned above. The proof is easy and we leave it to the reader. The auxiliary space T [(A5nj , m1j )j ] and the norm of the averages of its basis We consider the mixed Tsirelson space T [(A5nj , m1j )j ] and we denote by W0 , W0 the norming sets corresponding to the norm of this space where W0 is closed in the (A5nj , m1j ) operations and the rational convex combinations while W is closed only in the corresponding operations. We shall prove the following lemma. Lemma II.9. Let j0 ∈ N and f ∈ W0 . Then for every k1 < k2 < . . . < knj0 we have that nj0 2 , if w(f ) = mi , i < j0 1 |f ( ekl )| ≤ m1i ·mj0 (II.1) n j0 if w(f ) = mi , i ≥ j0 . mi , l=1 The same estimates hold for f ∈ W0 with w(f ) = mi . If we additionally assume that the functional f admits a tree (ft )t∈T such that w(ft ) = mj0 for all t ∈ T , then we have that nj0 1 ekl )| ≤ |f ( n j0

l=1

In particular |f ( n1j

0

nj0

l=1

ekl )| ≤

1 m2j

2 mi ·m2j 0 1 mi ,

,

if w(f ) = mi , i < j0

(II.2)

if w(f ) = mi , i > j0 .

. The same estimates remain valid for f ∈ W0

0

with w(f ) = mi admitting a tree (ft )t∈T such that w(ft ) = mj0 for all t ∈ T . Proof. We ﬁrst prove the following claim. Claim. Let h ∈ W0 . Then (i) #{k : |h(ek )| >

1 mj0

} ≤ (5nj0 −1 )log2 (mj0 )−1 .

II.1. Mixed Tsirelson Extension of a Ground Norm

25

(ii) If the functional h admits a tree (ha )a∈A with w(ha ) = mj0 for each a ∈ A then #{k : |h(ek )| > m12 } ≤ (5nj0 −1 )2 log2 (mj0 )−1 . j0

Proof of the claim. We shall prove only part (i) of the claim, as the proof of (ii) is similar. Let (ha )a∈A be a tree of h and let n be its height (i.e. the maximal length of its branches). We may assume that |h(ek )| > m1j for all k ∈ supp h. 0 Let h = (h0 , h1 , . . . , hn ) be a branch (then hn = ±e∗p ) and let k ∈ supp hn . Then n−1 & 1 1 1 mj < |h(ek )| = w(hl ) ≤ 2n , hence n < log2 (mj0 ). 0

l=0

On the other hand, since |h(ek )| >

1 mj0

for all k ∈ supp h, each hα with

α non maximal node is a result of an (A5nj , m1j ) operation for j ≤ j0 − 1. An inductive argument yields that for i ≤ n the cardinality of the set {ht : |t| = i} is less or equal to (5nj0 −1 )i . The fact that n < log2 (mj0 ) yields that #(supp(h)) ≤ (5nj0 −1 )log2 (mj0 )−1 . The proof of the claim is complete. We pass to the proof the lemma. The case w(f ) = mi , i ≥ j0 is straightford ward. Let f ∈ W0 with w(f ) = mi , i < j. Then f = m1i fl where f1 < · · · < fd belong to W0 and d ≤ ni . For p = 1, . . . , d we set Hp = {k : |fp (ek )| >

l=1

}. Part (i) of the claim d yields that #(Hp ) < (5nj0 −1 )log2 (mj0 ) . Thus, setting H = Hp , we get that 1 mj0

p=1

#(H) ≤ d(5nj0 −1 )log2 (mj0 )−1 ≤ (5nj0 −1 )log2 (mj0 ) . Therefore nj0 nj0 nj0 d d

( 1 '

1 1 1

( ekl )| ≤ fp )|H ( ekl ) + ( fp )|(N\H) ( ekl ) |f ( n j0 mi p=1 n j0 n j0 p=1 l=1

l=1

l=1

1 1 2 1 (#(H) + )< . ≤ mi n j0 m j0 m i m j0 The result for f ∈ W0 follows from Proposition II.8(i) and the above estimates. The second part is proved similarly by using part (ii) of the claim. Notes and Remarks. The space Tκ [G] is the frame for the HI extensions which will be presented in the next chapter. As we will see the norming set DG of the HI extensions will be a subset of the norming set W [G]. This actually means that the HI space X[G, σ] is interpolated between T [G] and YG . The results of the next section yield that if Tκ [G] is a strictly singular extension of YG , then it is reﬂexive and unconditionally saturated. Moreover Tκ [G] does not contain any p , 1 ≤ p < ∞. The transﬁnite basis of Tκ [G] is boundedly complete (for the deﬁnition we refer to Appendix A) hence Tκ [G] is the dual of the space generated by the biorthogonals of the basis (eα )α<κ .

26

Chapter II. Tree Complete Extensions of a Ground Norm

The auxiliary space T [(A5nj , m1j )j ] will be used in the estimates of the norm of certain block averages. For this reason we compute the norm of the averages of the basis in this space. The tool to reduce from the block averages to the averages of the basis of the auxiliary space, is the basic inequality stated and proved in the next section and also in Appendix B.

II.2

R.I.S. Sequences and the Basic Inequality

The tree complete extension DG of a ground set is a subset of Wk [G] which satisﬁes certain properties. The most important one is that DG is closed in the even operations (An2j , m12j ) while the behavior of DG with respect to the odd operations 1 ) is left open. We shall obtain some properties of the spaces Xk [DG ] (An2j+1 , m2j+1 in this section. In the next chapter we shall be more speciﬁc for the behavior of the DG with respect to the odd operations in order to obtain HI extensions of the ground norm. The main result of this part concerns the basic inequality and its applications. Deﬁnition II.10. Let G be a ground set. A subset DG of Wκ [G] is said to be a tree complete extension of G if the following are fulﬁlled. (i) The ground set G is a subset of DG . (ii) DG is symmetric, closed in the restrictions of its elements on intervals of κ and closed in the (An2j , m12j )j operations. (iii) Every f ∈ DG admits a tree analysis (ft )t∈T with ft ∈ DG for all t ∈ T . We shall denote by Xκ [DG ] the completion of (c00 (κ), · DG ) and for κ = ω we write X[DG ]. Remark II.11. Let’s observe that the diﬀerence between Wκ [G] and DG concerns 1 that in the later set the behavior of the odd operations (An2j+1 , m2j+1 )j is remaining undeﬁned. Actually we shall use the odd operations in order to impose the conditional structure on the space normed by the set DG . The complete deﬁnition of the set DG will be given in the next chapter. Deﬁnition II.12. The tree complete extension is said to be strictly singular if the identity operator I : Xκ [DG ] → YG is strictly singular. For f ∈ DG we say that is of type I (or II) if it is a result of an (Anj , m1j ) operation (or a rational convex combination) of a family (ft )t with ft ∈ DG . Deﬁnition II.13 (R.I.S.). A block sequence (xk )k in Xκ [DG ] is said to be a (C, ε) rapidly increasing sequence (R.I.S.), if xk ≤ C, and there exists a strictly increasing sequence (jk ) of positive integers such that (a) mj1 · # supp xk < ε. k+1

(b) For every k = 1, 2, . . . and every f ∈ DG with w(f ) = mi , i < jk we have C that |f (xk )| ≤ m . i

II.2. R.I.S. Sequences and the Basic Inequality

27

The next proposition is the fundamental tool for the computation of the norm for certain vectors in Xκ [DG ]. Proposition II.14 (The basic inequality). Let (xk )k be a (C, ε) R.I.S. in Xκ [DG ] such that for every g ∈ G we have that |g(xk )| > ε for at most one k. Let also (λk )k ∈ c00 be a sequence of scalars. Then for every f ∈ DG of type I we can ﬁnd g1 , such that either g1 = h1 or g1 = e∗t + h1 with t ∈ supp h1 where h1 ∈ W0 with w(h1 ) = w(f ) and g2 ∈ c00 (N) with g2 ∞ ≤ ε with g1 , g2 having nonnegative coordinates and such that |f ( λk xk )| ≤ C(g1 + g2 )( |λk |ek ). (II.3) If we additionally assume that there exists j0 ∈ N such that for every h ∈ DG with w(h) = mj0 and every interval E of the natural numbers we have that λk xk )| ≤ C(max |λk | + ε |λk |), (II.4) |h( k∈E

k∈E

k∈E

then, if w(f ) = mj0 , we may select h1 to have a tree analysis (ht )t∈T with w(ht ) = mj0 for all t ∈ T with ht of type I. Proof. The proof in the general case (where (II.4) is not assumed) and in the special case (where we assume (II.4)) is actually the same. We shall give the proof only in the special case. The proof in the general case arises by omitting any reference to distinguishing cases whether a functional has weight mj0 or not and treating the functionals with w(f ) = mj0 as for any other j. We ﬁx a tree analysis (ft )t∈T of f . Before passing to the proof we adopt some useful notation and state two lemmas. Deﬁnition II.15. For each k we deﬁne the set Tk as follows: ) t ∈ T such that ft is of type 0 or I and Tk = (i) ran ft ∩ ran xk = ∅ (ii) ∀ u < t, if fu is of type I, then w(fu ) = mj0 (iii) ∀ s ≤ t if s ∈ Su and fu is of type I, then ran fs ∩ ran xk = ran fu ∩ ran xk (iv) if w(ft ) = mj0 , then for all s ∈ St * ran fs ∩ ran xk ran ft ∩ ran xk The next lemma describes the properties of the set Tk . Lemma II.16. For every k we have the following: (i) If t ∈ T and ft is of type II then t ∈ Tk . (ii) If t ∈ Tk , then for every s < t: if fs is of type I, then w(fs ) = mj0 .

28

Chapter II. Tree Complete Extensions of a Ground Norm

(iii) If Tk is not a singleton, then its members are incomparable members of the tree T . Moreover if t1 , t2 are two diﬀerent elements of Tk and s is the (necessarily uniquely determined) maximal element of T satisfying s < t1 and s < t2 then fs is of type II. (iv) If t ∈ T is such that supp ft ∩ ran xk = ∅ and u ∈ Tk for all u < t, then there exists s ∈ Tk with t ≤ s. In particular if supp f ∩ ran xk = ∅, then Tk = ∅. {k : s ∈ Tk }. Deﬁnition II.17. For every t ∈ T we deﬁne Dt = s≥t

Lemma II.18. According to the notation above we have the following: (i) If supp f ∩ ran xk = ∅, then k∈ D0 (remember that 0 denotes the unique root of T and f = f0 ). Hence f ( λk xk ) = f ( λk xk ). k∈D0

(ii) If ft is of type I with w(ft ) = mj0 , then Dt is an interval of N. (iii) If ft is of type I with w(ft ) = mj0 , then {k} : k ∈ Dt \ Ds ∪ {Ds : s ∈ St } s∈St

is a family of successive subsets of N. Moreover for every k ∈ Dt \

Ds

s∈St

(i.e. for k such that t ∈ Tk ) such that supp ft ∩ran xk = ∅ there exists a s ∈ St such that either min supp xk ≤ max supp fs < max supp xk or min supp xk < min supp fs ≤ max supp xk . (iv) If ft is of type II, s ∈ St and k ∈ Dt \ Ds , then supp fs ∩ ran xk = ∅ and hence fs (xk ) = 0. Recall that we have ﬁxed a tree analysis (ft )t∈T for the given f . We shall construct two families (gt1 )t∈T and (gt2 )t∈T such that the following conditions are fulﬁlled. (i) For every t ∈ T such that ft is not of type II, gt1 = ht or gt1 = e∗kt + ht with kt ∈ supp ht , where ht ∈ W0 and gt2 ∈ c00 (N) with gt2 ∞ ≤ ε. (ii) For every t ∈ T , supp gt1 ⊂ Dt and supp gt2 ⊂ Dt and the functionals gt1 , gt2 have nonnegative coordinates. (iii) For t ∈ T with ft ∈ G and Dt = ∅ we have that gt1 = e∗p . rs fs (where rs ∈ Q+ for every s ∈ St and (iv) For ft of type II with f = s∈St rs = 1) we have gt1 = rs gs1 and gt2 = rs gs2 . s∈St

s∈St

s∈St

we have gt1 = e∗kt where kt ∈ Dt is such (v) For ft of type I with w(f ) = mj 0 2 that |λkt | = max |λk | and gt = εe∗k . k∈Dt

k∈Dt

II.2. R.I.S. Sequences and the Basic Inequality

29

(vi) For ft of type I with w(f ) = mj for j = j0 we have gt1 = ht or gt1 = e∗kt + ht where ht ∈ W0 with w(ht ) = mj and kt ∈ supp ht . (vii) For every t ∈ T the following inequality holds: |ft (

λk xk )| ≤ C(gt1 + gt2 )(

k∈Dt

|λk |ek ).

k∈Dt

When the construction of (gt1 )t∈T and (gt2 )t∈T has been accomplished, we set g1 = g01 and g2 = g02 (where 0 is the root of T and f = f0 ) and we observe that these are the desired functionals. To show that such (gt1 )t∈T and (gt2 )t∈T exist we use ﬁnite induction starting with those t ∈ T which are maximal and in the general inductive step we assume that gs1 , gs2 have been deﬁned for all s > t satisfying the inductive assumptions and we deﬁne gt1 and gt2 . 1st inductive step Let t ∈ T be maximal; then ft ∈ G. If Dt = ∅, we deﬁne gt1 = 0 and gt2 = 0. If Dt = ∅, we set Et = {k ∈ Dt : |ft (xk )| > ε} and Ft = Dt \ Et . If Et = ∅, then from our assumption in the statement of the proposition we have that Et = {kt }. We deﬁne gt1 =

k∈Et

e∗k . gt2 =

εe∗k ,

k∈Ft

and gt2 ∞ ≤ ε. Inequality (vii) is easily checked. General inductive step Let t ∈ T and suppose that gu1 and gu2 have been deﬁned for every u > t satisfying the inductive assumptions. If Dt = ∅, we set gt1 = 0 and gt2 = 0. In the remainder of the proof we assume that Dt = ∅. We consider the following three cases: 1st case: The functional ft is of +type II. Let ft = rs fs where rs ∈ Q are such that rs = 1. In this case, we have s∈S s∈St t that Dt = Ds . We deﬁne s∈St

gt1 =

s∈St

rs gs1 and gt2 =

rs gs2 .

s∈St

Inequality (vii) is easily veriﬁed. 2nd case: The functional ft is of type I with w(f ) = mj0 . In this case Dt is an interval of the natural numbers (Lemma II.18(ii)). Let kt ∈ Dt

30

Chapter II. Tree Complete Extensions of a Ground Norm

be such that |λkt | = max |λk |. We deﬁne k∈Dt

gt1 = e∗kt and gt2 =

εe∗k .

k∈Dt

Inequality (vii) is easily established. 3rd case: The functional ft is of type I with w(f ) = mj for j = j0 . Then ft = m1j fs and the family {fs : s ∈ St } is a family of successive s∈St

functionals with #(St ) ≤ nj . We set Et

{k : t ∈ Tk and supp ft ∩ ran xk = ∅} (= {k ∈ Dt \ Ds : supp ft ∩ ran xk = ∅}). =

s∈St

We consider the following partition of Et . Et2 = {k ∈ Et : mjk+1 ≤ mj } and Et1 = Et \ Et2 . We deﬁne gt2 =

k∈Et2

εe∗k +

gs2 .

s∈St

Observe that gt2 ∞ ≤ ε. Let Et1 = {k1 < k2 < · · · < kl }. From the deﬁnition of Et1 we get that mj < mjk2 < · · · < mjkl . We set kt = k1 and gt1 = e∗kt + ht where ht =

l 1 ∗ ( eki + gs1 ). mj i=2 s∈St

(The term e∗kt does not appear if Et1 = ∅.) It is easy to verify that inequality (vii) holds. By the second part of Lemma II.18(iii), for every k ∈ Et there exists an element of the set N = {min supp fs , max supp fs : s ∈ St } belonging to ran xk . Hence #(Et1 ) ≤ #(Et ) ≤ 2nj . We next show that ht ∈ W0 with w(ft ) = mj . We ﬁrst examine the case that for every s ∈ St the functional fs is not of type II. Then for every s ∈ St one of the following holds: (i) fs ∈ G. In this case gs1 = e∗ks ∈ W0 (by the ﬁrst inductive step). (ii) fs is of type I with w(fs ) = mj0 . In this case gs1 = e∗ks ∈ W0 . (iii) fs is of type I with w(fs ) = mj for j = j0 . In this case gs1 = e∗ks + hs (or gs1 = hs ) where hs ∈ W0 and ks ∈ supp hs . We set Es1 = {n ∈ N : n < ks }, Es2 = {n ∈ N : n > ks } and h1s = Es1 hs , h2s = Es2 hs . The functionals h1s , e∗ts , h2s are successive and belong to W0 .

II.2. R.I.S. Sequences and the Basic Inequality

31

We set Tt1 Tt2 Tt3

= {s ∈ St : fs ∈ G} = {s ∈ St : fs of type I and w(fs ) = mj0 } = {s ∈ St : fs of type I and w(fs ) = mj0 }.

The family of successive (see Lemma II.18(iii)) functionals of W0 , {e∗ki : i = 2, . . . , l} ∪ {gs1 : s ∈ Tt1 } ∪ {gs1 : s ∈ Tt2 } ∪{h1s : s ∈ Tt3 } ∪ {e∗ks : s ∈ Tt3 } ∪ {h2s : s ∈ Tt3 } has cardinality ≤ 5nj , thus we get that ht ∈ W0 with w(ht ) = mj . For the case that for some s ∈ St the functional fs is of type II, the conclusion follows from the previous case and Lemma II.8. n

j0 Proposition II.19. Let (xk )k=1 be a (C, ε) R.I.S. with ε ≤

2 m2j

such that for every

0

g ∈ G, |g(xk )| > ε for at most one k. Then: 1) For every f ∈ D with w(f ) = mi ,

nj0 1 |f ( xk )| ≤ n j0 k=1

In particular n1j

0

nj0

xk ≤

k=1

2C mj0

3C mj0 mi , C C nj0 + mi

if i < j0 + Cε ,

if i ≥ j0 .

.

n

j0 are scalars with |bk | ≤ 1 such that 2) If (bk )k=1

|h(

bk xk )| ≤ C(max |bk | + ε k∈E

k∈E

|bk |)

(II.5)

k∈E

for every interval E of positive integers and every h ∈ D with w(h) = mj0 , then nj0 1 4C bk xk ≤ 2 . n j0 m j0 k=1

Proof. The proof is an application of the basic inequality (Proposition II.14) and Lemma II.9. Indeed, let f ∈ D with w(f ) = mi . Proposition II.14 yields the existence of an h1 ∈ W0 with w(h1 ) = mi , a t ∈ N and an h2 ∈ c00 (N) with h2 ∞ ≤ ε, such that nj0 nj0 % $ 1 1 ∗ |f ( xk )| ≤ C(et + h1 + h2 ) ek . n j0 n j0 k=1

k=1

32

Chapter II. Tree Complete Extensions of a Ground Norm If i ≥ j0 , we get that |f ( n1j

0

nj0

k=1

xk )| ≤ C( n1j + 0

If i < j0 , using Lemma II.9 we get that |f ( n1j

0

3C mi ·mj0

nj0

k=1

1 mi

+ ε) <

C nj0

xk )| ≤ C( n1j + 0

+

C mi

2 mi ·mj0

+ Cε. + ε) <

. n

j0 be scalars with |bk | ≤ 1 such that (II.5) is In order to prove 2) let (bk )k=1 satisﬁed. Then condition (II.4) of the basic inequality is satisﬁed for the linear nj0 combination n1j bk xk . Thus for every f ∈ D with w(f ) = mi , i = j0 there 0

k=1

exist t ∈ N, h1 ∈ W0 , h2 ∈ c00 (N) with h1 , h2 having nonnegative coordinates, h2 ∞ ≤ ε and h1 admitting a tree (ht )t∈T with w(ht ) = mj0 for every t ∈ T with ht of type I, such that nj0 nj0 nj0 % % $ 1 $ 1 1 bk xk )| ≤ C(e∗t + h1 + h2 ) |bk |ek ≤ C(e∗t + h1 + h2 ) ek n j0 n j0 n j0 k=1 k=1 k=1 (II.6) Using the second part of Lemma II.9 we deduce that

|f (

nj0 1 1 1 4C |f ( bk xk )| ≤ C( + 2 + ε) < 2 . n j0 n j0 m j0 m j0 k=1

For f ∈ D with w(f ) = mj0 , from condition (II.5) we get that |f ( n1j

0

C nj0

(1 +

2 m2j

n j0 ) <

0

4C m2j

.

nj0

k=1

bk xk )| ≤

0

Proposition II.20. Suppose that there exists a universal constant C > 0 such that for every (xk )k block sequence in Xκ [DG ] and every j ∈ N there exists a (C, m21 ) 2j+1

R.I.S. y1 , y2 , . . . , yn2j+1 in span{xk : k ∈ N} such that 1 (i) n2j+1

n2j+1 k=1

yk ≥

1 m2j+1 .

1 (ii) For every f ∈ DG with w(f ) = m2j+1 we have |f ( n2j+1

n2j+1

(−1)k+1 yk )| <

k=1

C . m22j+1

Then the space Xκ [DG ] contains no unconditional basic sequence. Proof. Proposition II.19 yields that + + +

1 n2j+1

n2j+1

k=1

+ + (−1)k+1 yk + <

4C . m22j+1

This and assumption (i) of the statement yield that for every block sequence (xk )k∈N is not unconditional.

II.2. R.I.S. Sequences and the Basic Inequality

33

The result follows from the fact that every subspace of Xκ [DG ] contains a further subspace isomorphic to a subspace generated by a block sequence (xk )k∈N (Proposition A.3). Deﬁnition II.21 (k1 -averages). Let k ∈ N. A ﬁnitely supported vector x ∈ Xκ [DG ] is said to be a C − k1 average if x > 1 and there exist x1 < . . . < xk with k xi ≤ C such that x = k1 xi . i=1

Lemma II.22. Let k ∈ N and ε > 0. Then every block subspace of Xκ [DG ] contains a vector x which is a 2 − k1 average. If Xκ [DG ] is a strictly singular extension of YG then we may select x satisfying additionally xG < ε. Proof. If Xκ [DG ] is a strictly singular extension of YG we may pass to a further subspace Z on which the restriction of the identity map I : Xκ [DG ] → YG has norm less than 2ε . We choose j, s ∈ N such that ks ≤ n2j and 2s > m2j . Such a choice is possible from the deﬁnition of the sequences (mj )j , (nj )j . Let (xi )i be a normalized block p k q xl for p = 0, 1, 2, . . . sequence in the block subspace Z. We set xp,q = l=kp (q−1)+1

and q = 1, 2, . . .. Assume that the conclusion of the lemma fails. Then an easy inductive argument yields that xp,q < ( k2 )p for all p, q ≥ 1. In particular xs,1 < ( k2 )s . Since ks ≤ n2j , using the fact that the set DG is closed in the (An2j , m12j ) ks s operations we get that xs,1 ≥ m12j xl = mk2j . Thus 2s < m2j , a contradicl=1

tion. Therefore there exists a 2 − k1 average z in Z, while our choice of Z yields that zG < ε. Lemma II.23. Let x be a C − k1 average. Then for every n ≤ k and every sequence n of intervals E1 < . . . < En , we have that El x ≤ C(1 + 2n k ). In particular if l=1

n

x is an C − 1 j average, then for every f ∈ D with w(f ) = mi , i < j, we have 2n 1 ) ≤ 3C |f (x)| ≤ m1i C(1 + nj−1 2 mi . j Proof. Let x =

1 k

k

xi be a C − k1 average. Let also E1 < E2 < · · · < En be a

i=1

sequence of intervals, where n ≤ k. For l = 1, . . . , n, let Il (Jl resp.) be the set of n all i such that supp xi is contained in (resp. intersects) El . Clearly #Il ≤ k, l=1 while for each l we have that El x ≤ k1 El xi ≤ k1 C(#Il + 2). Therefore n l=1

El x ≤

C k1 (

n l=1

i∈Jl

#Il + 2n) ≤ C(1 +

2n k ).

34

Chapter II. Tree Complete Extensions of a Ground Norm

Lemma II.24. Let (xk )k∈N be a block sequence in Xκ [DG ] such that each xk is a C − l1k average, where (lk )k∈N is a strictly increasing sequence of integers, and let ε > 0. Then there exists a subsequence of (xk )k∈N which is a ( 3C 2 , ε) R.I.S. Proof. For each k we set jk = max{j : nj ≤ lk }. There exists a subsequence of (xk )k∈N (we denote this subsequence again by (xk )k∈N ) such that (jk )k∈N is a strictly increasing sequence and mjk+1 > 1ε # supp(xk ) for all k. From Lemma II.23 we also get that for each k and every f ∈ DG with w(f ) = mi , i < jk , we 1 have that |f (xk )| ≤ 3C 2 mi . Therefore this subsequence is a ( 3C 2 , ε) R.I.S. Proposition II.25 (Existence of R.I.S.). If Xκ [DG ] is a strictly singular extension of YG , then for every ε > 0 and every block subspace Z of Xκ [DG ] there exists a (3, ε) R.I.S. (xk )k in Z with xk > 1 and xk G < ε.

Proof. It follows from Lemma II.22 and Lemma II.24.

Proposition II.26. Let G be a ground subset of c00 (ω). If X[DG ] is a strictly singular extension of YG , then the dual space X[DG ]∗ of X[DG ] is the norm closed linear span of the w∗ closure of G. w∗

X[DG ]∗ = span(G

). w∗

Proof. Assume the contrary. Then setting Z = span(G ) there exists ∗∗ x∗ ∈ Xκ [DG ]∗ \ Z with x∗ = 1 and x∗∗ ∈ XG such that Z ⊂ Kerx∗∗ , x∗∗ = 2 ∗∗ ∗ and x (x ) = 2. First we observe that the space X[DG ] contains no isomorphic copy of 1 . Indeed, if not, then there exists a normalized block sequence (zn )n∈N equivalent to the usual 1 basis. But from Propositions II.19 and II.25 for every n2j block sequence (un )n∈N and all j there exist (yi )i=1 normalized block sequence for n2j 1 6 of (un )n∈N such that n2j i=1 yi ≤ m2j . This leads to a contradiction. From Odell–Rosenthal’s theorem there exists a sequence (xk )k∈N in X[DG ] w∗

with xk ≤ 2 such that xk −→ x∗∗ . Since each e∗n belongs to Z we get that lim e∗n (xk ) = 0 for all n, thus, using a sliding hump argument, we may assume k

that (xk )k∈N is a block sequence. Since also x∗ (xk ) → x∗∗ (x∗ ) = 2 we may also assume that 1 < x∗ (xk ) for all k. Let’s observe that every convex combination of (xk )k∈N has norm greater than 1. w∗

Considering each xk as a continuous function xk : G → R we have that the sequence (xk )k∈N is uniformly bounded and tends pointwise to 0, hence it is w∗

w∗

a weakly null sequence in C(G ). Since YG is isometric to a subspace of C(G ) w we get that xk −→ 0 in YG , thus there exists a convex block sequence (yk )k∈N of (xk )k∈N with yk G → 0. We may thus assume that yk G < 2ε for all k, where ε = n14 . We may construct a block sequence (zk )k∈N of (yk )k∈N such that (zk )k∈N is a (3, ε) R.I.S. of 1 averages and each zk is an average of (yk )k∈N with zk < ε.

II.2. R.I.S. Sequences and the Basic Inequality Proposition II.19 yields that the vector z =

1 n4

35 n4

zk satisﬁes z ≤

k=1

2·3 m4

< 1.

On the other hand, the vector z, being a convex combination of (xk )k∈N , satisﬁes z < 1. This contradiction completes the proof of the proposition. Remark II.27. The content of the above proposition is that the strictly singular w∗ extension X[DG ] of the space YG is actually a reﬂexive extension. Namely if G is a subset of c00 (N) then a consequence of Proposition II.26 is that the space X[DG ] is reﬂexive. Furthermore, if X[DG ] is nonreﬂexive then the quotient space X[DG ]∗ /X[DG ]∗ is norm generated by the classes of the elements of the set G

w∗

.

We denote by G0 (κ) the ground set {±eα : α < κ}. Proposition II.28. For every ordinal κ and every DG0 (κ) tree complete extension of G0 (κ) the bimonotone transﬁnite basis (eα )α<κ of X[DG0 (κ) ] is boundedly complete and shrinking, hence the space X[DG0 (κ) ] is reﬂexive. We refer the reader to Appendix A for the deﬁnitions of boundedly complete and shrinking transﬁnite bases and the proof that this properties yield the reﬂexivity of the space. Proof. We prove it for the case κ = ω. The general case is reduced to this one. The fact that the norming set DG0 (κ) is closed in the (An2j , m12j ) operations implies that for every j ∈ N and every sequence of ﬁnite intervals E1 < E2 < n 2j · · · < En2j we have that for every x ∈ c00 , x ≥ m12j Ei x. Thus, since the i=1

n

sequence ( m2j )j increases to inﬁnity, it follows that the basis (en )n is boundedly 2j complete. We next show that the basis (en )n is shrinking. Assume the contrary. Then ∞ there exists a x∗ ∈ X[DG0 (κ) ]∗ \span{e∗n : n ∈ N}. Let x∗ = w∗ − bn en . We may n=1

choose an ε > 0 and a sequence of successive intervals (En )n with En x∗ > ε. We choose j ∈ N with m2j > 3ε . Pick (xn )n a sequence in X[DG0 (κ) ] such that supp xn ⊂ En , xn = 1ε and ∗ x (xn ) > 1. The action of x∗ yields that every convex combination of (xn )n has norm greater than 1. 3 On the other hand we may select a ( 2ε , m12j ) R.I.S. of 1 averages y1 , y2 , . . . , yn2j such that each yl is an average of (xn )n . Proposition II.19 yields that the n 2j vector y = n12j yl satisﬁes y < εm32j . But since y is a convex combination of l=1

(xn )n we also have that y > 1, a contradiction.

Remark II.29. It is easy to see that similar arguments yield that the basis (eα )α<κ of X[DG ] is boundedly complete for an arbitrary ground subset G of c00 (κ) and any tree complete extension DG .

36

Chapter II. Tree Complete Extensions of a Ground Norm

Proposition II.30. Let G be a ground subset of c00 (ω). If X[DG ] is a strictly singular extension of YG , then X[DG ] is reﬂexive saturated (or somewhat reﬂexive). Proof. Let Z be a block subspace of X[DG ]. From the fact that the identity operator I : X[DG ] → YG is strictly singular we may choose a normalized block sequence ∞ (zn )n∈N in Z, with zn G < 12 . We claim that the space Z = span{zn n ∈ N} n=1

is a reﬂexive subspace of Z. It is enough to show that the Schauder basis (zn )n∈N of Z is boundedly complete and shrinking. The ﬁrst follows from the fact that (zn )n∈N is a block sequence of the boundedly complete basis (en )n∈N of X[DG ]. To see that (zn )n∈N n→∞ ∗ is shrinking it is enough to show that f |span{zi i≥n} −→ 0 for every f ∈ XG . ∞ w∗ 1 zn G < 2 From Proposition II.26 it is enough to prove it for f ∈ G . Since n=1

the conclusion follows.

Deﬁnition II.31 (exact pair). A pair (x, φ) with x ∈ Xκ [DG ] and φ ∈ DG is said to be a (θ, C, j) exact pair (where θ ∈ {0, 1}, C ≥ 1, j ∈ N) if the following conditions are satisﬁed: (i) 1 ≤ x ≤ C, for every ψ ∈ DG of type I with w(ψ) = mi , i = j we have 3C that |ψ(x)| ≤ m if i < j, while |ψ(x)| ≤ mC2 if i > j. i j

(ii) φ is of type I with w(φ) = mj . (iii) φ(x) = θ and ran x = ran φ. Proposition II.32. If Xκ [DG ] is a strictly singular extension of YG , then for every block subspace Z of Xκ [DG ], every ε > 0 and j ∈ N there exists a (1, 6, 2j) exact pair (x, φ) with x ∈ Z and xG < ε. n

2j a (3, ε)-R.I.S. with ε ≤ 1/m32j . Proof. From Proposition II.25 there exists (xk )k=1 ∗ ∗ ∗ Choose xk ∈ DG with xk (xk ) = 1 and ran xk ⊂ ran xk . Then Proposition II.19 yields that n2j n2j m2j 1 ∗ ( xk , xk ) n2j m2j

k=1

k=1

is a (1, 6, 2j) exact pair.

Deﬁnition II.33. Let (X, · ) be a Banach space. X is said to be λ-distortable, λ > 1, if there exists an equivalent norm | · | on X such that, ,

|x| inf sup : x, y ∈ Y and x = y = 1 ≥ λ . Y ⊂X |y| X is said to be arbitrarily distortable if it is λ-distortable for every λ > 1. Theorem II.34. If Xω [DG ] is a strictly singular extension of YG , then Xω [DG ] is arbitrarily distortable.

II.2. R.I.S. Sequences and the Basic Inequality

37

Proof. Let λ > 1 and choose j0 ∈ N such that m2j0 /144 > λ. Deﬁne |x| =

1 x + sup{|φ(x)| : φ ∈ DG and w(φ) = m2j0 } . m2j0

Then | · | is an equivalent norm . Let Y be a subspace of Xω [DG ]. By standard arguments we may assume that Y is a block subspace. By Proposition II.32 there exists a (1, 6, 2j0 ) exact pair (x1 , φ1 ) with x1 ∈ Y and w(φ1 ) = m2j0 . It follows that |x1 | ≥ φ1 (x1 ) = 1 . (II.7) Also by Proposition II.32 there exists a (1, 6, 2j0 + 2) exact pair (x2 , φ2 ) with x2 ∈ Y . From the properties of the exact pairs, in particular from Deﬁnition II.31(i), it follows that 6 18 24 |x2 | ≤ + = . (II.8) m2j0 m2j0 m2j0 Setting x =

x1 x1

and y =

x2 x2 ,

it follows from (II.7) and (II.8) that

|x| ≥ m2j0 /144 > λ . |y| Since Y was arbitrary chosen we have that Xω [DG ] is arbitrarily distortable.

Remark II.35. For a transﬁnite ordinal κ and every λ > 1 it can be shown, using the previous method, that there exists an equivalent norm which λ-distorts every block subspace of Xκ [DG ]. Since the block subspaces do not describe all subspaces of Xκ [DG ], κ > ω, it remains unknown if the previous result holds in full generality. Notes and Remarks. As we have mentioned in the introduction, strictly singular extensions were introduced in [16], for higher complexity saturation methods. In the present form they are contained in [4]. The tree complete extensions and the basic inequality were also introduced and studied in [16]. With the tree complete extensions we attempt to isolate the results obtained from the unconditional part of the deﬁnition of the norming set DG . Thus, independently of the deﬁnition of the special functions, resulting from (An2j+1 , m12j )j operations which act on special sequences, one can have seminormalized 1 averages and from these the exact pairs. As we will see in the next chapter exact pairs are the key component in the n2j+1 -dependent sequences. There is a very interesting interference of the distortion of Banach spaces with the discovery of HI spaces, that we would like to present. R.C. James proved that the spaces c0 and 1 are not distortable [37]. In the late 60’s V. Milman [49] proved the following. If X is a Banach space with the property that for every equivalent norm on X and every subspace Y of X and > 0 there exists a subspace Z of Y such that the initial and the new norm are (1 + ) equivalent, then either some p or c0 is isomorphic to a subspace of X. When Tsirelson discovered his space it became clear that there are Banach spaces not satisfying Milman’s condition.

38

Chapter II. Tree Complete Extensions of a Ground Norm

In the early 1990s E. Odell constructed a norm on Tsirelson space which (2 − ) distorts the original norm. After this, Schlumprecht presented his space as the ﬁrst example of an arbitrarily distortable Banach space, and based on this, Gowers and Maurey proceeded to the construction of their space. Subsequently Odell and Schlumprecht [52], [53], showed that p , 1 < p < ∞ are arbitrarily distortable and B. Maurey extended this to Banach spaces not containing uniformly n1 [45]. Finally N. Tomczack [63], proved that every HI space is arbitrarily distortable. It remains open whether or not the notions of distortable and arbitrarily distortable are equivalent. In particular we do not know if Tsirelson space is arbitrarily distortable. Notice that the p-convexiﬁcation (1 < p < ∞) of Tsirelson space is arbitrarily distortable. This is a consequence of the aforementioned Maurey theorem. The key role of the Tsirelson space in the study of the above problem arises from a result of V. Milman and N. Tomczak [50] that asserts the following. If X is a Banach space with no arbitrarily distortable subspace, then X contains an asymptotic p (or c0 ) space. Results related to the distortion of asymptotic 1 spaces are contained in [9] and [44].

Chapter III

Hereditarily Indecomposable Extensions with a Schauder Basis III.1 The HI Property in X[G, σ] In this chapter we make the ﬁnal step in the deﬁnition of HI extensions for spaces with a Schauder basis. Thus we introduce a coding σ and then we deﬁne the n2j+1 1 special sequences. The tree complete norming set is closed for all (An2j+1 , m2j+1 ) operations acting on n2j+1 -special sequences. We also provide suﬃcient conditions for the HI property of the predual X[G, σ]∗ . Deﬁnition III.1. A Banach space X is said to be Hereditarily Indecomposable (HI) if for every inﬁnite dimensional closed subspace Y of X there is no nontrivial projection P : Y → Y . (A projection is said to be trivial if either the dimension or the codimension of its kernel is ﬁnite). For equivalent reformulations of the above deﬁnition we refer to Proposition IV.4. Throughout this section we shall work in c00 (i.e. κ = ω). Given G a ground set we deﬁne a tree complete set DG as follows. Deﬁnition III.2. The set DG is the minimal subset of c00 satisfying the following conditions. (i) G ⊂ DG . (ii) DG is symmetric (i.e. if f ∈ DG then −f ∈ DG ). (iii) DG is closed under the restriction of its elements on intervals of N (i.e. if f ∈ DG and E is an interval of N, then Ef ∈ DG ).

40 Chapter III. Hereditarily Indecomposable Extensions with a Schauder Basis (iv) DG is closed under the (An2j , m12j ) operations, i.e. if f1 < f2 < · · · < fn2j belong to DG , then the functional f = m12j (f1 + f2 + · · · + fn2j ) belongs also to DG . 1 (v) DG is closed under the (An2j−1 , m2j−1 ) operations on special sequences i.e. for every n2j−1 -special sequence (f1 , f2 , . . . , fn2j−1 ) of length n2j−1 the func1 tional f = m2j−1 (f1 + f2 + · · · + fn2j−1 ) belongs to DG .

(vi) The set DG is rationally convex. Next we deﬁne the n2j−1 -special sequences. For this we shall use a coding function σ deﬁned as follows. The coding function σ Let Qs denote the set of all ﬁnite sequences (φ1 , φ2 , . . . , φd ) such that φi ∈ c00 (N) , φi = 0 with φi (n) ∈ Q for all i, n and φ1 < φ2 < · · · < φd . We ﬁx a pair Ω1 , Ω2 of disjoint inﬁnite subsets of N. From the fact that Qs is countable we are able to deﬁne an injective coding function σ : Qs → {2j : j ∈ Ω2 } such that 1 mσ(φ1 ,φ2 ,...,φd ) > max{ |φi (e : l ∈ supp φi , i = 1, . . . , d} · max supp φd . l )| The n2j−1 -special sequences n

2j−1 A ﬁnite sequence (fi )i=1 is said to be a n2j−1 -special sequence provided that

(i) (f1 , f2 , . . . , fn2j−1 ) ∈ Qs and fi ∈ DG for i = 1, 2, . . . , n2j−1 . 1/2

(ii) w(f1 ) = m2k with k ∈ Ω1 , m2k > n2j−1 and w(fi+1 ) = mσ(f1 ,...,fi ) for each 1 ≤ i < n2j−1 . This completes the deﬁnition of DG and it is easy to check that the set DG , the deﬁnition of which depends on the coding function σ, is a tree complete extension of G. We denote by X[G, σ] the Banach space X[DG ]. As we have mentioned the weight w(f ) of a functional f ∈ DG of type I is n2j−1 not unique. However when we refer to a n2j−1 -special sequence (fi )i=1 by w(fi ) for 2 ≤ i ≤ n2j−1 we shall always mean w(fi ) = mσ(f1 ,...,fi−1 ) (∈ Ω2 ). Proposition III.3 (The tree-like property of n2j−1 -special sequences). Let Φ = n2j−1 n2j−1 (φ)i=1 , Ψ = (ψ)i=1 be two distinct n2j−1 -special sequences. Then (i) For 1 ≤ i < j ≤ n2j−1 we have that w(φi ) = w(ψj ). (ii) There exists kΦ,Ψ such that φi = ψi for i < kΦ,Ψ and w(φi ) = w(ψi ) for i > kΦ,Ψ . We leave the easy proof to the reader.

III.1. The HI Property in X[G, σ]

41 n

2j−1 with xk ∈ Deﬁnition III.4 (dependent sequences). A double sequence (xk , x∗k )k=1 ∗ X[G, σ] and xk ∈ DG is said to be a (θ, C, 2j−1) dependent sequence (for θ ∈ {0, 1}, n2j−1 C > 1 and j ∈ N) if there exists a sequence (2jk )k=1 of even integers such that the following conditions are fulﬁlled:

n

2j−1 (i) (x∗k )k=1 is a n2j−1 -special sequence with w(x∗k ) = m2jk for all k ≤ n2j−1 .

(ii) Each (xk , x∗k ) is a (θ, C, 2jk ) exact pair. Proposition III.5. Suppose that X[G, σ] is a strictly singular extension of YG and let ε > 0, j ∈ N. Then for every pair of block subspaces Z, W of X[G, σ] there n2j−1 exists a (1, 6, 2j − 1) dependent sequence (xk , x∗k )k=1 with xk G < ε, x2k−1 ∈ Z and x2k ∈ W for all k.

Proof. It follows easily from an inductive application of Proposition II.32. n

2j−1 be a (θ, C, 2j − 1) dependent sequence such Proposition III.6. Let (xk , x∗k )k=1 2 that xk G < m2 for all k. Then 2j−1

a) If θ = 1, it holds

n2j−1

1 n2j−1

(−1)k+1 xk ≤

k=1

8C . m22j−1

b) If θ = 0, it holds

1 n2j−1

n2j−1

xk ≤

k=1

8C . m22j−1 n

2j−1 Proof. a) It is easy to see that the sequence (xk )k=1 is a (2C, n21 ) R.I.S. 2j−1

The conclusion will follow from Proposition II.19 2) after showing that for every f ∈ D with w(f ) = m2j−1 and every interval E of positive integers we have that $ % 2 |f (−1)k+1 xk | ≤ 2C(1 + 2 #(E)). m2j−1 k∈E

1 Such an f is of the form f = m2j−1 (F x∗t−1 + x∗t + · · · + x∗r + fr+1 + · · · + fd ) ∗ ∗ for some special sequence (x1 , x2 , . . . , x∗r , fr+1 , . . . , fn2j−1 ) of length n2j−1 with x∗r+1 = fr+1 and w(x∗r+1 ) = w(fr+1 ), d ≤ n2j−1 and F an interval of the form [m, max supp x∗t−1 ] (see Proposition III.3). We estimate the quantity f (xk ) for each k.

• If k < t − 1, we have that f (xk ) = 0. • If k = t − 1, we get |f (xt−1 )| =

1 ∗ m2j−1 |F xt−1 (xt−1 )|

• If k ∈ {t, . . . , r}, we have that f (xk ) =

≤

1 ∗ m2j−1 xk (xk )

=

1 m2j−1 xt−1 1 m2j−1 .

≤

C m2j−1 .

42 Chapter III. Hereditarily Indecomposable Extensions with a Schauder Basis • If k > r + 1, Proposition III.3 yields that w(fi ) = m2jk for all i > r. Using the fact that (xk , x∗k ) is a (C, 2jk ) exact pair and taking in account that n22j−1 < m2j1 ≤ m2jk we get |f (xk )| = 1 = |(fr + . . . + fd )(xk )| m2j−1 1 ' ≤ |fi (xk )|+ m2j−1 '

1

≤

m2j−1

w(fi )<m2jk

2j−1

|fi (xk )|+

( |f2i (x2k−1 )|

2r+2≤2i≤d

w(fi )>m2jk

3C C C +n2j−1 2 ) ≤ 2 . ml m2jk m2j−1

• For k = r + 1, the same argument as in the previous case yields that C |f (xr+1 )| ≤ m2j−1 + m21 < mC+1 . 2j−1 2j−1

Let now E be an interval. From the previous estimates we get that

% $ 1 (−1)k+1 xk | ≤ |f (xt−1 )| + (−1)k+1 |f m2j−1 k∈E k∈E∩[t,r] +|f (xr+1 )| + |f (xk )| k∈E∩[r+2,n2j−1 ]

C 1 C +1 C + + + 2 #(E) m2j−1 m2j−1 m2j−1 m2j−1 2 < 2C(1 + 2 #(E)). m2j−1

≤

This completes the proof of the ﬁrst part. b) The proof of b) follows using similar arguments.

Theorem III.7. Every X[G, σ] which is a strictly singular extension of YG is a HI space. In other words, strictly singular extensions are HI extensions. Proof. Let Z, W be inﬁnite dimensional subspaces of X[G, σ]. We shall show that for every ε > 0 there exist z ∈ Z, w ∈ W with z − w < εz + w. It is easy to check that this yields the HI property of X[G, σ]. From the well known gliding hump argument we may assume that Z, W are block subspaces. Then for n2j−1 j ∈ N, using Proposition III.5, we select (xk , x∗k )k=1 a (6, 2j − 1) dependent 2 sequence with xk G < m2 , x2k−1 ∈ Z and x2k−1 ∈ W for all k. Observe that n2j−1 1

n2j−1

2j−1

xk ≥

k=1

Thus setting z = z − w ≤

1 m2j−1

n2j−1 /2 k=1

48 m2j−1 z

1 and n2j−1

n2j−1

(−1)k+1 xk ≤

k=1 n2j−1 /2

x2k−1 and w =

48 m22j−1

(Proposition III.6).

x2k we get that z ∈ Z, w ∈ W and

k=1

+ w which for suﬃciently large j yields the desired result.

III.2. The HI Property in X[G, σ]∗

43

Therefore the space X[G, σ] is HI.

The space X[G0 , σ] where G0 = {±e∗n : n < ω} is an example of a reﬂexive HI space. Theorem III.8. The space X[G0 , σ] where G0 = {±e∗n : n ∈ ω} is a reﬂexive HI space. Proof. It is easy to check that YG0 is the space c0 . Hence X[G0 , σ] is a strictly singular extension of YG0 and therefore HI. The reﬂexivity of X[G0 , σ] follows from Proposition II.28.

III.2 The HI Property in X[G, σ]∗ In this section we study the HI property of the predual X[G, σ]∗ . We are not able to establish this property for all strictly singular extensions. Actually we require an additional property of the space YG deﬁned as uniformly bounded averages (deﬁnition III.13). The HI property X[G, σ]∗ is basic for the existence of a non separable HI space. Notation III.9. We shall denote by X[G, σ]∗ the subspace of the dual space X[G, σ]∗ generated by the biorthogonal functionals (e∗n )n∈N of the Schauder basis (en )n∈N of the space X[G, σ]. Notice that the basis (en )n of X[G, σ] is boundedly complete hence X[G, σ] is the dual of X[G, σ]∗ . Proposition III.10. Assume that there exists a constant C > 0 such that for every inﬁnite dimensional subspace Z of X[G, σ]∗ , every ε > 0 and every j ∈ N there 5m 5m exists a (C, 2j) exact pair (x, φ) with dist(φ, Z) ≤ n2j2j and xG ≤ n2j2j . Then the space X[G, σ]∗ is HI. Proof. Let Z, W be a pair of inﬁnite dimensional subspaces of X[G, σ]∗ and let δ > 0. We shall ﬁnd fZ ∈ Z and fW ∈ W such that fZ + fW < δfZ − fW . For j ∈ N , using our assumption, we may inductively select a (C, 2j − 1) de5m n2j−1 m2j−1 pendent sequence (xk , x∗k )k=1 such that xk G < n2j2jk < n2j−1 , dist(x∗2k−1 , Z) <

5m2jk m2j−1 n2j2k n2j−1

and dist(x∗2k , W ) <

k

5m2j2k m2j−1 n2j2k n2j−1

for all k (where w(x∗k ) = m2jk ). n2j−1 1 From Proposition III.6 the vector x = n2j−1 (−1)k+1 xk satisﬁes x ≤ k=1

8C . m22j−1

We set hZ =

1 m2j−1

n2j−1 k=1

x∗2k−1 and hW =

1 m2j−1

n2j−1 k=1

x∗2k . The functional

1 hZ + hW belongs to the norming set DG , as it is the result of an (An2j−1 , m2j−1 ) ∗ ∗ ∗ operation on the n2j−1 −special sequence (x1 , x2 , . . . , xn2j−1 ). Hence hZ + hW ≤

1. On the other hand hZ − hW ≥

(hZ −hW )(x) x

≥

1 m2j−1 8C m2 2j−1

=

m2j−1 8C .

44 Chapter III. Hereditarily Indecomposable Extensions with a Schauder Basis n

2j−1 we also get that From our choice of the dependent sequence (xk , x∗k )k=1 1 1 dist(hZ , Z) < 2 and dist(hW , W ) < 2 . We may thus select fZ ∈ Z and fW ∈ W such that fZ +fW < hZ +hW + 12 + 12 ≤ 2 and fZ −fW > hZ −hW − 12 − 12 ≥ m2j−1 8C − 1. m2j−1 Hence fZ − fW > ( 16C − 2)fZ + fW which for large j yields the desired result, therefore the space X[G, σ]∗ is HI.

Deﬁnition III.11. Let k ∈ N. A ﬁnitely supported vector x∗ ∈ X[G, σ]∗ is said to be a C −ck0 vector if there exist x∗1 < · · · < x∗k such that x∗i > C −1 , x∗ = x∗1 +· · ·+x∗k and x∗ ≤ 1. Lemma III.12. Let Z be a block subspace of X[G, σ]∗ and let k ∈ N. Then there exists a block sequence (zi∗ )i∈N in Z such that for every i1 < i2 < · · · < ik the vector zi∗1 + zi∗2 + · · · + zi∗k is a 2 − ck0 vector. Proof. Assume that the conclusion of the lemma fails. We choose j, s ∈ N with ks ≤ n2j and 2s > m2j . Let (fi )i∈N be a normalized block sequence in Z. We set " # B1 = {l1 < l2 < · · · < lk } : fl1 + fl2 + · · · + flk > 2 . The Ramsey theorem yields that there exists L1 ∈ [N] such that either [L1 ]k ⊂ B1 or [L1 ]k ∩ B1 = ∅. From our assumption on the failure of the lemma the second alternative can not hold thus [L1 ]k ⊂ B1 . We may assume that L1 = N. We lk set f1,l = fl for l = 1, 2, . . .. As above we may assume, passing to a i=(l−1)k+1

subsequence, that f1,l1 + f1,l2 + · · · + f1,lk > 22 for every l1 < · · · < lk . After s steps, using the same argument, we arrive at a functional f ∈ Z which is of the ks form f = fli with f > 2s . Since ks ≤ n2j we get that the functional m12j f is i=1

the result of a (An2j , m12j ) on norm 1 functionals and hence f ≤ m2j . Therefore m2j < 2s , a contradiction which completes the proof of the lemma. Deﬁnition III.13. Let G be a ground set. The space YG has uniformly bounded averages if 1 → YG and for every ε > 0 and n0 ∈ N there exists k ∈ N such that for every weakly null block sequence (zn )n∈N in YG with zn G ≤ 1 there exist i1 < i2 < · · · < ik such that for every g ∈ G with min supp g < n0 we have that z +z +···+zik )| < ε. |g( i1 i2k Proposition III.14. Assume that YG has uniformly bounded averages and X[G, σ] is a strictly singular extension of YG . Let Z be an inﬁnite dimensional subspace of X[G, σ]∗ and j ∈ N. Then there exists (1, 12, 2j) exact pair (y, φ) with dist(φ, Z) < 5m2j 5m2j n2j and yG < n2j . Proof. We may assume that Z is a block subspace of X[G, σ]∗ . We select 0 < ε < 1 n2j .

III.2. The HI Property in X[G, σ]∗

45

∗ ∗ + · · · + z1,k in Z and Let k1 ∈ N. We choose a 2 − ck01 vector f1 = z1,1 1 k1 1 x1 = k1 (z1,1 + · · · + z1,k1 ) a 21 with ran f1 = ran x1 and f1 (x1 ) > 1. We set t1 = max supp f1 . Since YG has uniformly bounded averages there exists k2 = k( 4ε , t1 ) satisfying the property of Deﬁnition III.13; we may also as∗ ∗ sume that k2 > k1 . We choose a block sequence (z2,l )l in Z with min supp z2,1 > k2 ∗ ∗ > 12 and z2,i ≤ 2 for every i1 < · · · < ik2 in max supp f1 such that z2,l l l=1

∗ , z2,l < 2 and N. For each l we select z2,l ∈ X[G, σ] such that ran z2,l = ran z2,l ∗ z2,l (z2,l ) > 1. Since 1 → YG and (z2,l )l is a bounded sequence (in X[G, σ] and thus) in YG , we may assume that (z2,l )l is a weakly Cauchy sequence in YG . Thus the sequence (w2,l )l deﬁned by w2,l = z2,2l−1 −z2,2l is a weakly null sequence in YG with w2,l ≤ 4. From our choice of k2 we may assume, passing to a subsequence k2 of (w2,l )l that |g( k12 w2,l )| < ε for every g ∈ G with min supp g ≤ t1 .

We set x2 =

1 k2

l=1 k2

w2,l and f2 =

k2

∗ z2l−1 . Observe that f2 ∈ Z with f2 ≤ 1

l=1

l=1

x2 ≥ f2 (x2 ) =

k2 1 ∗ z2l−1 (z2l−1 ) > 1 k2

while

l=1

which, in particular yields that x2 is a 4 − k12 average. We select k3 = k(ε, t2 ) with k3 > k2 satisfying the property of Deﬁnition III.13 where t2 = max supp f2 and we deﬁne x3 ∈ X[G, σ] and f3 ∈ Z similarly to the second step. Following this procedure we may construct a block sequence (xr )r∈N in X[G, σ] and a sequence (fr )r∈N in Z such that the following conditions are satisﬁed. (i) Each xr is a 4 − k1r average and each fr is a 4 − ck0r vector with fr (xr ) > 1 and ran ff = ran xr . (ii) For every g ∈ G with min supp g ≤ max supp xr−1 we have that |g(xr )| < ε. Thus for every g ∈ G we have that |g(xr )| > ε for at most one r. Passing to a subsequence we may additionally assume (Lemma II.24) that (xr )r∈N is a (6, ε) R.I.S. We set φ=

n2j n2j 1 m2j fr and x = xr . m2j r=1 n2j r=1

We have that f ∈ Z with w(f ) = m2j . Proposition II.19 yields that x ≤ 2·6 1 m2j · m = 12. Thus 1 ≤ φ(x) ≤ x ≤ 12 hence we may select θ with 12 ≤θ≤1 2j such that φ(θx) = 1. We set y = θx. Using Proposition II.19 we easily get that (y, φ) is a (12, 2j) exact pair.

46 Chapter III. Hereditarily Indecomposable Extensions with a Schauder Basis It only remains to show that yG ≤ at most one r we get that |g(y)| ≤

5m2j n2j .

Let g ∈ G. Since |g(xr )| > ε for

m2j m2j (|g(x1 )| + · · · + |g(xn2j )|) ≤ (max xr + (n2j − 1)ε) n2j n2j r 4m2j 5m2j < + 4ε < . n2j n2j

Theorem III.15. If YG has uniformly bounded averages and X[G, σ] is a strictly singular extension of YG , then the space X[G, σ]∗ is HI. Proof. It follows from Proposition III.10 and III.14.

Deﬁnition III.16. Let G be a ground set. The space YG has the uniform weak Banach–Saks property if 1 → YG and for every ε > 0 there exists k ∈ N such that for every normalized weakly null block sequence (zn )n∈N in YG there exist z +z +···+zik i1 < i2 < · · · < ik such that i1 i2k G < ε. Corollary III.17. If YG has the uniform weak Banach–Saks property, then X[G, σ] is a strictly singular HI extension of YG and the space X[G, σ]∗ is HI. Notes and Remarks. There exists the unconditional counterpart of the reﬂexive HI space X[G0 , σ] (Theorem III.8). This space is a variation of the space presented by W.T. Gowers in [31]. To deﬁne this space we modify (v) of Deﬁnition III.2 by adding that for each spacial functional f and each E subset (not necessarily interval) of N, Ef ∈ DG . Then the resulting space denoted as Xu [G0 , σ] has an unconditional basis and it has a remarkable space of operators. Namely as it is shown in [34] every T ∈ L(Xu [G0 , σ]) is of the form D + S with D a diagonal operator and S strictly singular. This property yields that Xu [G0 , σ] is not isomorphic to any of its proper subspaces and every bounded linear projection is a strictly singular perturbation of a diagonal one. Other variants of the special sequences lead to indecomposable and unconditionally saturated Banach spaces (e.g. [14], [15]). The dual of a HI space X, even if X is reﬂexive, does not need to be HI. For example there exists a reﬂexive HI space X such that 2 is isomorphic to a subspace of X ∗ (e.g. [10]). More extreme is a recent result of a reﬂexive HI space X with X ∗ unconditionally saturated [17]. It is not known if such a divergent could be preserved in the subspaces of X. The following is open. Assume that X is a reﬂexive HI space. Does there exist a subspace Y of X such that Y ∗ is also HI? The HI constructions with the use of higher complexity saturation methods which appeared in [6] and [16] yield asymptotic 1 HI spaces. Variants of them could derive asymptotic p HI spaces for 1 < p < +∞ (e.g. [22]).

Chapter IV

The Space of the Operators for Hereditarily Indecomposable Banach Spaces IV.1

Some General Properties of HI Spaces

An important property of the HI spaces concerns the structure of the spaces of their operators. As we will see these spaces have few operators and also remarkable properties of the spaces itself are obtained as consequence of the structure of their operators. We start with the following fundamental results due to Gowers and Maurey [33]. Theorem IV.1. Every bounded linear operator T : X → X with X a complex HI space, is of the form T = λI + S with S a strictly singular operator. This result is not extendable in real HI spaces. However, as we will see, in the case of the HI extensions X[G, σ] the above property remains valid. For an arbitrary HI space the following holds. Theorem IV.2. Let X be a HI space (real or complex). Then every Fredholm operator is of index zero. For complex HI this is a consequence of the above Theorem and Fredholm’s theory. For real HI spaces X it uses the complexiﬁcation of X. Corollary IV.3. Let X be a HI space. Then X is not isomorphic to any proper subspace of it. In particular X is not isomorphic to its hyperplanes. For the proof of the above results we refer the reader to the Gowers–Maurey paper and also to Maurey’s survey in the Handbook of the Geometry of Banach spaces [46]

48

Chapter IV. The Space of the Operators for HI Banach Spaces

This rest of the section contains some results obtained from the deﬁnition of HI Banach spaces and equivalent reformulations on it. Corollary IV.8 describes the frame on which we shall build the non separable HI space which will be presented in the next chapters. Also, Theorem IV.6 shows that every HI space is a subspace of ∞ (N). The next proposition summarizes the equivalent reformulations of the deﬁnition of HI Banach spaces. Proposition IV.4. Let X be a Banach space. The following assertions are equivalent: (1) The space X is HI. (2) For every pair of inﬁnite dimensional closed subspaces Y ,Z of X, dist(SY , SZ ) = 0. (3) For every pair of inﬁnite dimensional closed subspaces Y ,Z of X and δ > 0 there exist y ∈ Y and z ∈ Z such that y − z ≤ δy + z. (4) [V.D. Milman] For every inﬁnite dimensional closed subspace Y of X, ε > 0 and W ⊂ BX ∗ such that W ε-norms Y , the space W⊥ = {x ∈ X : f (x) = 0 for every f ∈ W } is a ﬁnite dimensional subspace of X. Proof. The assertions (1), (2) and (3) are trivially equivalent by the open mapping theorem and the triangle inequality. We ﬁrst show that (1) implies (4). If Y is an inﬁnite dimensional closed subspace of X, ε > 0 and W ⊂ BX ∗ are such that W ε-norms Y then Y ∩ W⊥ = {0} and Y ⊕ W⊥ is a closed subspace of X. The HI property of X yields that W⊥ is necessarily ﬁnite dimensional. It remains to show that (4) implies (1). Suppose that the space X is not HI Then X has two inﬁnite dimensional closed subspaces Y, Z such that Y ∩ Z = {0} and Y + Z is a closed subspace of X. Then the projection P : Y + Z −→ Y is continuous; set ε = P1 . For every y ∗ ∈ εBY ∗ we may select, by the Hahn– Banach theorem, a functional y∗ ∈ BX ∗ such that y∗ extends y ∗ ◦ P and y ∗ X ∗ = y ∗ ◦ P (Y +Z)∗ . We set W = { y ∗ : y ∗ ∈ εBY ∗ }. Then W ε-norms Y thus by (4) the space W⊥ should be ﬁnite dimensional. This leads to a contradiction since clearly W⊥ contains the inﬁnite dimensional space Z. Therefore X is HI. Proposition IV.5. Let X be a HI Banach space, and let T : X −→ Y be a bounded linear operator where Y is any Banach space. Then exactly one of the following two holds: (i) The operator T is strictly singular. (ii) Ker T is a ﬁnite dimensional subspace of X and X can be written as the direct sum of Ker T and a subspace Z of X such that T|Z is an isomorphism. If Y = X and T is one-to-one, then either T is strictly singular or T is an onto isomorphism.

IV.1. Some General Properties of HI Spaces

49

Proof. Assume that T = 1. Suppose that (i) does not hold. Then there exists an inﬁnite dimensional closed subspace W of X and an ε > 0 such that T w ≥ ε for every w ∈ SW . Suppose also that (ii) does not hold. Then the restriction of T to any subspace of X of ﬁnite codimension is not an isomorphism, thus, by Proposition II.3 there ε exists an inﬁnite dimensional subspace Z of X such that T|Z ≤ . 2 Then for every z ∈ SZ and w ∈ SW we have that z − w ≥ T z − T w ≥ T w − T z ≥ ε −

ε ε = , 2 2

ε so we get that dist(SZ , SW ) ≥ which contradicts the HI property of X. Hence 2 Ker T is ﬁnite dimensional and has a complement Z such that T|Z is an isomorphism. If Y = X and T is one-to-one and not strictly singular, then T is an isomorphism. As we will see later no HI space is isomorphic to any proper subspace of it. Therefore T is onto. Theorem IV.6. Every HI Banach space X embeds into ∞ . Proof. Let Y be any separable inﬁnite dimensional closed subspace of X and select a countable subset D of the unit ball of X ∗ such that D 12 norms Y . Since X is HI Proposition IV.4 yields that D⊥ is ﬁnite dimensional. We enlarge D by a ﬁnite set F such that (D ∪ F )⊥ = {0}. Let D ∪ F = {x∗n : n ∈ N}. We deﬁne the operator T : X −→ ∞ by the rule T (x) = (x∗n (x))n∈N . Observe that T is one-toone and T restricted to Y is an isomorphism. Proposition IV.5 yields that T is an isomorphism. Theorem IV.7. Let X be a Banach space and let Z be an inﬁnite dimensional closed subspace of X such that Z is HI and the quotient map Q : X −→ X/Z is strictly singular. Then the space X is HI. Proof. We begin the proof with the next two claims. Claim 1. If Z0 is a ﬁnite dimensional subspace of Z, Y is an inﬁnite dimensional closed subspace of X and δ > 0, then there exist y ∈ SY and z ∈ SZ with y − z < δ such that z0 < (1 + δ)z0 + λz for every z0 ∈ Z0 and every scalar λ. Proof of Claim 1. We may assume that δ < 1. Let {x1 , x2 , . . . , xk } be a SZ0 and pick for each i = 1, 2, . . . , k an fi ∈ SX ∗ with fi (xi ) = 1. Then ﬁnite codimensional in X, thus the subspace Y ∩(

k i=1

k

δ net in 4 Ker fi is

i=1

Ker fi ) is inﬁnite dimensional.

From the fact that the operator Q : X −→ X/Z is strictly singular, we may choose

50

Chapter IV. The Space of the Operators for HI Banach Spaces k

δ δ . If z ∈ Z with y − z < 16 16 i=1 δ z then setting z = we get that z ∈ SZ and y − z < . z 8 1 for To ﬁnish the proof of the claim it is enough to show that z0 +λz > 1+δ every z0 ∈ SZ0 and every scalar λ. It is also enough to consider only λ with |λ| < 2. δ Let z0 ∈ SZ0 and |λ| < 2. We choose i ∈ {1, 2, . . . , k} such that z0 − xi < . 4 Then a y ∈ Y ∩(

Ker fi ) with y = 1 and Qy <

z0 + λz

≥ xi + λz − z0 − xi > |fi (xi + λz)| −

δ 4

δ δ > 1 − |λ| · (|fi (y)| + y − z) − 4 4 δ δ 1 δ . > 1 − 2(0 + ) − = 1 − > 8 4 2 1+δ

≥ 1 − |λ| · |fi (z)| −

Claim 2. For every inﬁnite dimensional closed subspace Y of X and every ε > 0 there exists an inﬁnite dimensional closed subspace W of Z such that dist(w, SY ) < ε for every w ∈ SW . Proof of Claim 2. Let (εn )n∈N be a sequence of positive reals with and SY

∞

∞ &

(1 + εn ) ≤ 2

n=1

ε . From Claim 1 we may inductively select a sequence (yn )n∈N in 8 n=1 n and a sequence (zn )n∈N in SZ such that yn − zn < εn and ai zi < εn <

(1 + εn+1 )

n+1

i=1

ai zi for every choice of scalars (ai )i∈N and all n. Then (zn )n∈N

i=1

is a Schauder basic sequence with basis constant less or equal to 2. We set W = span{zn : n ∈ N}. The space W satisﬁes the conclusion of the claim. Indeed, let w ∈ SW , ∞ ∞ w = an zn . Then |an | ≤ 4 for all n. The series an yn converges to some n=1

y ∈ Y and w − y ≤

∞ n=1

|an |yn − zn ≤ 4

∞

n=1

εn <

n=1

y ε . Setting y = we 2 y

obtain that dist(w, SY ) ≤ w − y ≤ w − y + y − y ≤

ε + | y − 1| < ε. 2

We pass now to the proof that the space X is HI. Let Y1 , Y2 be a pair of inﬁnite dimensional closed subspaces of X ∗ and we will show that dist(SY1 , SY2 ) = 0. Let ε > 0. From Claim 2, there exist two inﬁnite dimensional closed subspaces ε W1 , W2 of Z such that for every w1 ∈ SW1 we have that dist(w1 , SY1 ) < and 3

IV.1. Some General Properties of HI Spaces

51

ε . Since the space Z is HI we 3 ε may choose w1 ∈ SW1 and w2 ∈ SW2 with w1 − w2 < . Let y1 ∈ SY1 such that 3 ε ε w1 − y1 < and w2 ∈ SW1 such that w2 − y2 < . We deduce that 3 3 for every w2 ∈ SW2 we have that dist(w2 , SY2 ) <

dist(SY1 , SY2 )

≤ y1 − y2 ≤ y1 − z1 + z1 − z2 + z2 − y2 ε ε ε < + + = ε. 3 3 3

Thus dist(SY1 , SY2 ) = 0. Therefore the space X is HI.

With the next result we provide some suﬃcient conditions yielding the HI property for dual Banach spaces. Corollary IV.8. Let X be a Banach space and let Z a subspace of X ∗ such that the following conditions are fulﬁlled. (i) The space X contains no isomorphic copy of 1 . (ii) The space Z is HI. (iii) The space X ∗ /Z is isomorphic to c0 (Γ) for some set Γ. Then the space X ∗ is HI. Proof. From Theorem IV.7 it is enough to observe that the quotient map Q : X ∗ −→ X ∗ /Z is strictly singular. If Γ is ﬁnite there is nothing to be proved. Let Γ be inﬁnite and suppose that Q is not strictly singular. Let Y be an inﬁnite dimensional subspace of X ∗ such that Q restricted to Y is an isomorphism. Since X ∗ /Z is isomorphic to c0 (Γ) for an inﬁnite set Γ we may assume, passing to a subspace, that Q(Y ) is isomorphic to c0 . It follows that X ∗ contains isomorphically c0 , thus, by Bessaga Pelczynski’s Theorem, the space X contains isomorphically 1 , a contradiction. Similar arguments also yield the next. Corollary IV.9. Let X be a Banach space and Y a closed subspace of X. If the spaces Y and X/Y are HI saturated, then the same holds for the space X. A similar result also holds for somewhat reﬂexive (or reﬂexive saturated) Banach spaces. Proposition IV.10. Let X be a Banach space such that (i) X does not contain isomorphically 1 . (ii) X ∗∗ is isomorphic to X ⊕ 1 (Γ) for some inﬁnite set Γ. Then every bounded linear operator T : X ∗ −→ X ∗ is of the form T = Q∗ + K where Q is an operator on X and K is a compact operator on X ∗ .

52

Chapter IV. The Space of the Operators for HI Banach Spaces

Proof. Let T : X ∗ −→ X ∗ be a bounded linear operator. Consider the conjugate operator T ∗ : X ∗∗ −→ X ∗∗ and the projections P1 : X ⊕ 1 (Γ) −→ X and P2 : X ⊕ 1 (Γ) −→ 1 (Γ). Let Q : X −→ X be the operator deﬁned as Q = P1 ◦ (T ∗ |X) and S : X −→ X ∗∗ deﬁned by S = P2 ◦ (T ∗ |X). Observe that S(X) ⊂ 1 (Γ). The operator S is compact. Indeed, if S was not compact we could construct an operator from a subspace of X onto 1 , thus by the lifting property of 1 , X contains isomorphically 1 , a contradiction. It follows that the operator S ∗ : X ∗∗∗ −→ X ∗ is compact so the same holds for the operator K = S ∗ |X ∗ . An easy computation yields that T = Q∗ + K.

IV.2

The Space of Operators L(X[G, σ]), L(X[G, σ]∗ )

We pass now to discuss the structure of L(X[G, σ]) when X[G, σ] is a HI extension of a ground norm. We begin with the following. Lemma IV.11. Let Y be a subspace of X[G, σ] and let T : Y → X[G, σ] be a bounded linear operator. Let (yl )l∈N be a block sequence of 2 − n1 l averages with increasing lengths in Y such that (T yl )l∈N is also a block sequence and lim yl G = 0. Then l

lim dist(T yl , Ryl ) = 0. l

Proof. Assume on the contrary that there exist δ > 0 and L ∈ [N] such that dist(T yl , Ryl ) > δ for all l ∈ L. The Hahn–Banach theorem yields that there exists φl ∈ BX[G,σ]∗ such that a) φl (yl ) = 0, φl (T yl ) > δ and b) ran φl ⊂ ran(supp yl ∪ supp T yl ). For simplicity we may assume that φl ∈ DG (the precise argument asserts that we can choose φl ∈ DG such that φl (yl ) ≥ δ and |φl (yl )| being as small as we wish). First we observe that for every ε > 0 and every j ∈ N there exist l1 < . . . < lnj ∈ L such that setting x=

yl1 +...+yln

j

nj

and φ =

φl1 +...+φln

j

m2j

gives (m2j x, φ) is a (0, 3C, 2j) exact pair such that m2j xG < ε (Lemmas II.22– n2j−1 II.24, Proposition II.25). Then for a given j ∈ N we may deﬁne {(xk , x∗k )}k=1 to 2 ∗ be (0, 18C, 2j − 1)-dependent sequence with xk G < 1/m2j−1 and xk (T xk ) ≥ δ for k = 1, . . . , n2j−1 . Proposition III.6 yields that

1 n2j−1

n2j−1

k=1

xk ≤

104C . m22j−1

(IV.1)

IV.2. The Space of Operators L(X[G, σ]), L(X[G, σ]∗ ) On the other hand setting x∗ =

1 n2j−1

1 m2j−1

n2j−1 k=1

n2j−1

k=1

1

∗

T xk ≥ x (

n2j−1

53

x∗k we get,

n2j−1

k=1

xk ) ≥

δ m2j−1

For suﬃciently large j, (IV.1) and (IV.2) derive a contradiction.

.

(IV.2)

Theorem IV.12. Let Y be an inﬁnite dimensional closed subspace of X[G, σ]. Every bounded linear operator T : Y → X[G, σ] takes the form T = λIY + S with λ ∈ R and S a strictly singular operator (IY denotes the inclusion map from Y to X[G, σ]). Proof. Assume that T is not strictly singular. We shall determine a λ = 0 such that T − λIY is strictly singular. Let Y be an inﬁnite dimensional closed subspace of Y such that T : Y → T (Y ) is an isomorphism. By standard perturbation arguments and the fact that X[G, σ] is a strictly singular extension of YG , we may assume, passing to a subspace, that Y is a block subspace of X[G, σ] spanned by a normalized block sequence ∞ (yn )n∈N such that (T yn )n∈N is also a block sequence and yn G < 1. From n=1

Lemma II.22 we may choose a block sequence (yn )n∈N of 2 − n1 i averages of increasing lengths in span{yn : n ∈ N} with yn G → 0. Lemma IV.11 yields that lim dist(T yn , Ryn ) = 0. Thus there exists a λ = 0 such that lim T yn − λyn = 0. n

n

Since the restriction of T − λIY to any ﬁnite codimensional subspace of span{yn : n ∈ N} is clearly not an isomorphism and since also Y is a HI space, it follows from Proposition IV.5 that the operator T − λIY is strictly singular. Theorem IV.13. Let YG have uniformly bounded averages and X[G, σ] be the strictly singular extension of YG . Then every T ∈ L(X[G, σ]∗ ) is of the form T = λI + S with S strictly singular operator. We start with the following lemma. Lemma IV.14. Let X be an HI space with a Schauder basis (en )n . Assume that T : X → X is a bounded linear operator not of the form T = λI + S with S strictly singular. Then there exists n0 and δ > 0 such that for every z ∈ Xn0 = span{en : n ≥ n0 }, dist(T z, Rz) ≥ δz. Proof. If not, then there exists a normalized block sequence (zn )n such that dist(T zn , Rzn ) ≤ n1 . Choose λ ∈ R such that T zn − λzn n∈L → 0 for a subsequence (zn )n∈L . Then for a further subsequence (zn )n∈M , M ∈ [L], T − λI|span{zn : n∈N} is a compact operator. The HI property of X easily yields that T − λI is a strictly singular operator, contradicting our assumption. In the following lemma we assume that YG and X[G, σ] are as in the statement of the theorem.

54

Chapter IV. The Space of the Operators for HI Banach Spaces

Lemma IV.15. Let T : X[G, σ]∗ → X[G, σ]∗ , Z be a block subspace of X[G, σ]∗ such ∗ ∗ that Z = < (zn∗ )n > and for each n < m ran zn∗ ∪ ran T zn∗ < ran zm ∪ ran T zm . ∗ ∗ ∗ ∗ ∗ Assume further that for each z ∈ Z , with z = 1, dist(T z , Rz ) > δ. Then the following holds: (i) For each k ∈ N there exists a normalized block sequence (wl∗ )l∈N in Z satisfying the following properties: k ∗ ∗ For every l1 < . . . < lk setting w∗ = q=1 wlq we have that w ≤ 2. Further there exists an 2−k1 -average w ∈ X[G, σ] such that w∗ (w) = 0, T w∗ (w) > 1 and ran(w) ⊂ ran(supp w∗ ∪ supp T w∗ ). (ii) For every j ∈ N there exists (0, 18 δ , 2j) exact pair (x, φ) with φ ∈ Z, 18m x ∈ X[G, σ] and T φ(x) > 1, xG < δn2j2j . Proof. (i) The proof follows the arguments of Lemma III.12 using the following observation: Assume that w1∗ , . . . , wk∗ is a 2 − ck0 -vector. Our assumptions for the operator T yield that for q = 1, . . . , k there exists wq ∈ X[G, σ] such that wq∗ (wq ) = 0, T wq∗ (wq ) > 1 and wq < 1δ. Now using this and the arguments of Lemma III.12 we obtain the desired sequence. (ii) It only requires the adaptation described before the proof of Proposition III.14. Proof of Theorem IV.13. On the contrary assume that there exists T ∈ L(X[G, σ]∗) not of the desired form. Assume further that T = 1 and T e∗n is ﬁnitely supported with lim inf min supp T e∗n = ∞. (We may assume the later conditions from the fact that the basis (e∗n )n of (XG )∗ is weakly null.) In particular for every (zn∗ )n block sequence in X[G, σ]∗ there exists a subsequence (zn∗ )n∈L such that (ran zn∗ ∪ ran T zn∗ )n is a sequence of successive subsets of N. Let δ > 0 and n0 ∈ N as in Lemma IV.14. n2j−1 Then Lemma IV.15 yields that for every j ∈ N there exists (zk , zk∗ )k=1 ,a ∗ ∗ (0, 18 , 2j − 1) dependent sequence such that z (z ) = 0, T z (z ) > 1, ran z ⊂ k k k k k δ n2j−1 ran zk∗ ∪ran T zk∗ , (ran zk∗ ∪ran T zk )k=1 successive subsets of N and zk G ≤ m21 . 2j−1

Proposition III.6 yields that

1 Finally m2j−1

1≥

1 m2j−1

n2j−1 k=1

1 n2j−1

k=1

zk ≤

144 . m22j−1 δ

T zk∗ ≤ 1 (since T ≤ 1 ) and also

n2j−1

k=1

n2j−1

T zk∗

n2j−1 m22j−1 δ 1 m2j−1 δ . ≥ T zk∗ (zk ) ≥ 144m2j−1 n2j−1 144 k=1

This yields a contradiction for suﬃciently large j ∈ N.

IV.2. The Space of Operators L(X[G, σ]), L(X[G, σ]∗ )

55

Notes and Remarks. Our approach showing that the spaces X[G, σ] have few operators follows the lines of Gowers and Maurey [34]. In a recent paper ([4]) is shown that there exists a HI space with no reﬂexive subspace such that every operator T is of the form λI + W with W a weakly compact operator. The central remaining open problem is the existence of a Banach space with very few operators (i.e. T = λI + K with K compact). The recent results [14], [15] show that a Banach space X could admit few operators and also could have rich unconditional structure. It is not known if there exists a subspace Y of a space X with an unconditional basis, such that Y admits few operators. Actually it is unknown if there exists such a Y which is indecomposable.

Chapter V

Examples of Hereditarily Indecomposable Extensions In the present chapter we shall present HI extensions of YG when G is of speciﬁc form. Thus we shall show how we can obtain a quasi-reﬂexive HI space Xqr , a HI space Xp which has p as a quotient (1 < p < ∞) and also a non-separable HI space.

V.1

A Quasi-reﬂexive HI Space

We start with the quasi-reﬂexive HI space. We recall that a Banach space X is said to be quasi-reﬂexive if dim(X ∗∗ /X) < ∞. The set Gqr We consider the set Gqr = {±I ∗ : I is a ﬁnite interval of N}. Here we denote by I ∗ the function χI ∈ c00 . It follows readily that Gqr is a ground set. Moreover the basis of YGqr is equivalent to the summing basis of c0 , hence YGqr is c0 saturated and X[Gqr , σ] is a HI reﬂexive extension of YGqr . Theorem V.1. The space X[Gqr , σ] is a quasi-reﬂexive and HI space. Proof. Since X[Gqr , σ] is a reﬂexive extension of YGqr it follows that X[Gqr , σ]∗ = p span(Gqr ) (Proposition II.26). This yields that X[Gqr , σ]∗ = span({e∗n }n ∪ {N∗ }) from which we obtain that X[Gqr , σ]∗ /X[Gqr , σ]∗ ∼ = R. Since the basis of X[Gqr , σ] is boundedly complete we obtain that X[Gqr , σ]∗∗ /X[Gqr , σ] ∼ = R.

58

Chapter V. Examples of Hereditarily Indecomposable Extensions

Remark. Since every normalized weakly null sequence in YGqr contains a subsequence equivalent to the c0 basis, we conclude that YGqr has the uniform weak Banach–Saks property which yields that the predual of X[Gqr , σ] is also HI.

V.2

The Spaces p , 1 < p < ∞, are Quotients of HI Spaces

Next we deﬁne the HI space Xp which has p (1 < p < +∞) as a quotient. Let {Mi }i∈N be a disjoint partition of N into inﬁnite sets, Mi = {mi1 < mi2 < · · · < min < · · · }. We consider the partially ordered set (L, ≺) where L = N and l ≺ d iﬀ l < d and there exists i ∈ N such that l, d ∈ Mi . Clearly (L, ≺) is a tree. A segment s of L has the property that it is a segment of Mi for some i ∈ N. The set Gp is deﬁned as follows. Gp =

n ) i=1

ai s∗i :

n

* |ai |q ≤ 1 and each si is a ﬁnite segment of L .

i=1

Here q is the conjugate of p. Clearly Gp is a ground set and the space YGp has the following properties. Proposition V.2. Let Z be a closed inﬁnite dimensional subspace of YGp . Then either c0 → Z or p → Z. Moreover the space YGp has the uniform weak Banach– Saks property. Proof. It follows easily from the deﬁnition of YGp that YGp = ( ⊕Yi )p where Yi = < en >n∈Mi and (en )n∈Mi is equivalent to the summing basis of c0 . In particular n each Yi is isomorphic to c0 . Clearly for each n ∈ N, ( ⊕Yi )p remains isomorphic i=1

to c0 hence it is c0 saturated. Let Z be a subspace of YGp . Then one of the following two alternatives holds: 0 (a) There exists n0 ∈ N such that Pn0 |Z : Z → ni=1 ⊕Yi is not strictly singular. (b) For every n ∈ N, Pn |Z is a strictly singular operator. The preceding remarks immediately yield that in the ﬁrst case c0 is isomorphic to a subspace of Z. If (b) occurs then a “sliding hump” argument derives a normalized sequence (z ) in Z of ﬁnite supported vectors and a sequence (wn )n∈N in YGp such that n n z n − wn < 1, and if we denote In = {i ∈ N : supp wn ∩ Mi = ∅}, then (In )n n consists of successive ﬁnite subsets of N. The later property yield that (wn )n is equivalent to the usual p basis and hence, (zn )n is equivalent to the p basis. This completes the proof of the ﬁrst part. Next we show that YGp satisﬁes the uniform weak Banach–Saks property (Deﬁnition III.16). First we prove the following:

V.2. The Spaces p , 1 0 and every M ∈ [N] there exists L ∈ [M ] such that the following holds: For every s segment of the tree L, #{ ∈ L : |s∗ (y )| ≥ ε} ≤ 2 . Proof of the claim. Assume on the contrary that for some M no such L exists. Then applying the classical Ramsey theorem for triples we obtain an L ∈ [M ] such that for 1 < 2 < 3 in L there exists a segment s of L with |s∗ (yi )| ≥ ε for i = 1, 2, 3. Let L = {1 < 2 < . . .}. For each n ∈ N and 1 < k < n choose s∗k,n with |s∗k,n (i )| ≥ ε for i = 1, . . . , n. Observe that there exists n0 ∈ N such that for each n ∈ N and each k < n, sk,n ⊂ Mi for some i < n0 . Indeed, sk,n ∩ supp y1 = ∅; now choose n0 such that for every i > n0 , max supp x1 < min Mi . To ﬁnish the proof, passing if required to a subsequence, we may assume that for every n ∈ N, w∗ − limn s∗k,n = s∗k with min sk < max supp x1 . Clearly sk is an inﬁnite segment and sk ⊂ Min for some i ≤ n0 . Hence there exists Q ∈ [N] and i < n0 such that for each k ∈ Q, |Mi∗ (yk )| > ε, a contradiction since (y ) is weakly null. The uniform weak Banach–Saks property of YGp is obtained by the above claim in a similar manner as this property is established in p . As consequence we obtain the following. Proposition V.3. (i) The space X[Gp , σ] is a HI reﬂexive extension of YGp . (ii) The predual of X[Gp , σ]∗ is also HI. The next theorem describes the basic properties of X[Gp , σ]. Theorem V.4. There exists a surjective bounded linear operator Q : X[Gp , σ] → p . Additionally Q∗ [q ] is a complemented subspace of X[Gp , σ]∗ . In particular the quotient map Q is described by the rule Q(emik ) = ei for all i, k ∈ N. Also Q∗ (e∗i ) = Mi∗ and X[Gp , σ]∗ ∼ = X[Gp , σ]∗ ⊕ q . Proof. The proof follows easily from the next equation. For all {ai }ni=1 the following holds: n n + ' + (1/q + + ai Ii∗ + = |ai |q , (∗) + i=1

{Ii }ni=1

where next lemma.

i=1

are inﬁnite subsegments of {Mi }ni=1 . This is a consequence of the

n Lemma V.5. For all {ai }ni=1 , ε > 0, n0 ∈ N there exist {Ji }i=1 ﬁnite segments of 1 (L, ≺) such that n0 < Ji ⊂ Mi , and setting xi = #Ji m∈Ji em , we have that d + + + + a i xi + + i=1

[Gp ,σ]

≤ (1 + ε)

d ' i=1

|ai |p

(1/p .

60

Chapter V. Examples of Hereditarily Indecomposable Extensions

Let us see how we ﬁnish the proof of the theorem. First we establish the equality (∗). Indeed the deﬁnition of Gp yields that d d + ' + (1/q + + ai Ii∗ + ≤ |ai |q . + i=1

i=1

The inverse inequality is obtained by the lemma and a simple duality argument. As consequence we have that span{Mi∗ : i ∈ N} is isometric to q and further X[Gp , σ]∗ /X[Gp , σ]∗ = q . This yields the proof of the theorem. d Proof of Lemma V.5. Assume that i=1 api = 1 and ε, n0 ∈ N are given. Choose j ∈ N such that (i)

1 mj

<

ε 2d

and

log (m −1)

(ii)

nj−12 j nj

<

ε 2d .

Next for i ≤ d choose n0 ≤ i1 < . . . < inj with (iii) Ji = {i1 , . . . , inj } ⊂ Mi (iv) 1t < 2t < . . . < dt < 1t+1 < . . ., t = 1, . . . , nj . nj eik and we show that We set xi = n1j k=1

d

a i xi ≤ 1 + ε .

i=1

Since the norming set D of the space X[Gp , σ] is a subset of the set Wp = d 1 W [Gp , Anj , ] it suﬃces to show that f ( ai xi ) ≤ 1 + ε for every f ∈ Wp mj j i=1 of type I. Let f ∈ Wp of type I. Using similar arguments as in Lemma II.9 we may assume that there exists a tree Tf = (fa )a∈A of f such that each fa is not 0 of type II (Deﬁnition II.12). Let (gas )ss=1 be the functionals corresponding to the maximal elements of the tree A. We denote by the ordering of the tree A. Let 1 1 ≤ A = s ∈ {1, 2, . . . , s0 } : w(fγ ) mj γ≺a s

B = {1, 2, . . . , s0 } \ A and set fA = f |

s∈A

supp gas ,

fB = f |

s∈B

supp gas .

V.2. The Spaces p , 1 < p < ∞, are Quotients of HI Spaces We have fA (xi ) ≤

61

1 for each i thus mj d

d 1 1 ε fA ( a i xi ) ≤ |ai | ≤ ·d< . m m 2 j i=1 j i=1

It remains to estimate the value fB (

d

(V.1)

ai xi ). We observe that

i=1 d

a i xi =

i=1

Set

d

ai (

i=1

nj eli

nj d 1 )= ( ai elti ). nj n t=1 j i=1 t

t=1

) t ∈ {1, 2, . . . , n} : the set {lt1 , lt2 , . . . , ltd } is contained in ran gas for * some s ∈ B or does not intersect any ran gas , s ∈ B

E1

=

E2

= {1, 2, . . . , n} \ E1 .

For each s = 1, 2, . . . , s0 set θs = θs ≤ 1. that

)

1 nj

: {lt1 , lt2 , . . . , ltd } ⊂ ran gas

* and observe

s∈B

d 1 We ﬁrst estimate the quantity gas ( ( ai elti )) for s ∈ B. We may nj t∈E1 ∗ i=1 assume that gas is of the form gas = ci,j si,j where {si,j : i, j} is a family of i j pairwise disjoint segments with each si,j ⊂ bi and |ci,j |q ≤ 1. For each i = i

1, 2, . . . , d we get that (

j

ci ,j s∗i ,j )(

t∈E1

≤ |ai | max |ci ,j |θs . Thus

1 nj (

d i=1

j

ai elti )) = ai (

j

ci ,j s∗i ,j )(

t∈E1

1 nj elti )

j

gas (

d d d d 1 1 1 ( ai elti )) ≤ θs (max |ci,j |)|ai | ≤ θs ( max |ci,j |q ) q ( |ai |p ) p ≤ θs . j j nj i=1 i=1 i=1 i=1

t∈E1

Therefore d d 1 1 i ( ai elt )) ≤ ( gas )( ( ai elti )) ≤ θs ≤ 1. fB ( nj i=1 nj i=1 t∈E1

s∈B

t∈E1

(V.2)

s∈B

By the deﬁnition of the set B we may prove, as in the proof of Lemma II.9, that the family of functionals {gas : s ∈ B} has cardinality less or equal log (m −1) nj−12 j . By the deﬁnition of the set E2 for t ∈ E2 the set {lt1 , lt2 , . . . , ltd }

62

Chapter V. Examples of Hereditarily Indecomposable Extensions

intersects at least one but is not contained in any ran gas , s ∈ B. This easily yields that log (m −1) #{lt1 : t ∈ E2 } ≤ nj−12 j and thus

log (m −1) 1 nj−12 j ε . < < nj nj 2d

t∈E2

Therefore fB (

d d 1 1 ε ε ·d= . ( ai elti )) ≤ ( )( |ai |) < nj i=1 nj i=1 2d 2

t∈E2

(V.3)

t∈E2

From (V.1),(V.2) and (V.3), we conclude that f(

d

a i xi ) ≤

i=1

d d 1 1 i fA ( ai xi ) + fB ( ( ai elt )) + fB ( ( ai elti )) nj i=1 nj i=1 i=1 d

t∈E1

≤

t∈E2

ε ε + 1 + = 1 + ε. 2 2

Remark. In a similar manner we could show that c0 is a quotient of a HI space. The corresponding ground set GL 0 is deﬁned as ∗ GL 0 = {±s : s is a segment of (L, ≺)} ∗ L L ∼ and X[GL 0 , σ] has as quotient c0 and also X [G0 , σ] = X∗ [G0 , σ] ⊕ 1 . The wellknown lifting property of 1 does not permit us to have a similar result for 1 .

V.3

A Non Separable HI Space

In this part we shall provide a ground set Gns with the property that X[Gns , σ]∗ is a non separable HI space. This requires more eﬀort than the previous examples. The ground set Gns and the space X[Gns , σ] Let (D, ≺) denote a reorder of N as a dyadic tree with the property n ≺ m implies that n < m. We shall denote by s the segments of (D, ≺). We deﬁne Gns =

d ) i=1

εi s∗i : (εi )di=1 ∈ {−1, 1}d and {si }di=1 are pairwise disjoint * ﬁnite segments of (D, ≺) and min{min si : i = 1, . . . , d} ≤ d .

The set Gns is a ground set. The following theorem describes the properties of YGns .

V.3. A Non Separable HI Space

63

Theorem V.6. (i) The space YGns is c0 saturated. (ii) The space YGns has uniformly bounded averages. The proof of this theorem requires some steps described by the next lemmas. Lemma V.7. The closure of the set Gns in the topology of pointwise convergence is Gns

p

=

d )

εi s∗i : (ε)di=1 ∈ {−1, 1}d , (si )di=1 are pairwise disjoint

i=1

* segments with {min s1 , min s2 , . . . , min sd } ∈ S, d ∈ N . p

Proof. Let g ∈ Gns and we show that it is of the form described above (the other p inclusion is trivial). Let (gk )k∈N be a sequence in Gns such that gk −→ g. Set k→∞

k→∞

d = min supp g. Since gk (n) −→ 0 for n = 1, . . . , d − 1 and gk (d) −→ g(d) = 0 and from the fact that each gk (n) belongs to {−1, 0, 1} we may assume, passing d to a subsequence, that min supp gk = d for all k ∈ N. Let gk = εki (ski )∗ for i=1

k = 1, 2, . . .. Observe that the limit of a sequence of ﬁnite segments in the pointwise topology is a segment which can be either ﬁnite or inﬁnite. We thus may select L1 ∈ [N], ε1 ∈ {−1, 1} and a segment s1 such that ε1 s∗1 is the pointwise limit of the sequence (εk1 (sk1 )∗ )k∈L1 . After d consecutive applications of the same argument we may select inﬁnite sets of natural numbers L1 ⊃ L2 ⊃ · · · ⊃ Ld , ε1 , ε2 , . . . εd ∈ {−1, 1} and p disjoint segments s1 , s2 , . . . , sd such that (εki (ski )∗ )k∈Li −→ εi s∗i for i = 1, 2, . . . , d. d We deduce that g = εi s∗i . i=1

We remind the reader at this point that YGns denotes the completion of the space (c00 , Gns ). Proposition V.8. The space YGns does not contain isomorphically 1 . Proof. It is enough to show that every bounded sequence in YGns has a weakly Cauchy subsequence. Let (xn )n∈N be a bounded sequence in YGns . By Rainwater’s Theorem [54] is enough to ﬁnd a subsequence (xn )n∈M such that (f (xn ))n∈M is w∗

convergent for every f ∈ Ext(BYG∗ns ). Since BYG∗ns = conv(Gns ) Theorem we get that Ext(BYG∗ns ) ⊂ Gns

w∗

by Milman’s

. Therefore, by the form of elements of

w∗

Gns it is enough to show that there exists M ∈ [N] such that (s∗ (xn ))n∈M is convergent for every segment s. This is done in the following two lemmas. Lemma V.9. Let (xn )n∈N be a bounded sequence in YGns and ε > 0. Then there exists a ﬁnite set {s1 , s2 , . . . , sk } of pairwise disjoint segments and an L ∈ [N] such

64

Chapter V. Examples of Hereditarily Indecomposable Extensions

that

lim sup |s∗ (xn )| ≤ ε n∈L

for every segment s with s ∩ (

k

si ) = ∅.

i=1

Proof. Assume the contrary. Then for every ﬁnite set {t1 , t2 , . . . , tm } of pairwise m disjoint segments and every L ∈ [N] there exists a segment t with t ∩ ( ti ) = ∅ i=1

such that lim sup |t∗ (xn )| > ε. n∈L

Using this fact we may inductively construct a sequence (sj )j∈N of pairwise disjoint segments of D and a decreasing sequence (Lj )j∈N of inﬁnite subsets of the natural numbers such that |s∗j (xn )| > ε

∀j ∈ N

∀n ∈ Lj .

r . Since the ε segments s1 , s2 , s3 , . . . are pairwise disjoint we may choose an i0 ∈ N such that min si ≥ k for every i > i0 . Let n ∈ Li0 +k . Then n ∈ Li0 +t for each t ∈ {1, 2, . . . , k} i.e. |s∗i0 +t (xn )| > ε for each t ∈ {1, 2, . . . , k}. Setting εt = sgn(s∗i0 +t (xn )) we have that We set r = sup{xn : n ∈ N} and choose k ∈ N with k >

f=

k

εt s∗i0 +t ∈ Gns

w∗

t=1

thus f ∈ BYG∗ns . It follows that r ≥ xn ≥ f (xn ) =

k

εt s∗i0 +t (xn ) =

t=1

k

|s∗io +t (xn )| > kε,

t=1

a contradiction of the choice of k.

Lemma V.10. Let (xn )n∈N be a bounded sequence in YGns . There exists a subsequence (xn )n∈M of (xn )n∈N such that for every segment s the sequence (s∗ (xn ))n∈M is convergent. Proof. From Lemma V.9 we may inductively construct a decreasing sequence (Lk )k∈N of inﬁnite subsets of the natural numbers and a sequence (Fk )k∈N of ﬁnite sets of pairwise disjoint segments, Fk = {sk1 , sk2 , . . . , skmk }, such that for each m k k ∈ N and segment s with s ∩ ( ski ) = ∅ we have that i=1

lim sup |s∗ (xn )| < n∈Lk

1 . k

V.3. A Non Separable HI Space

65

Let F be the countable set consisting of all the ﬁnite segments of D and all ∞ Fk . We choose a diagonal set L of the subsegments of segments contained in k=1

decreasing sequence (Lk )k∈N . Since {s∗ : s ∈ F } is a countable subset of BYG∗ns and the sequence (xn )n∈L is bounded we may choose, by a diagonal argument, an M ∈ [L] such that the sequence (s∗ (xn ))n∈M is convergent for every s ∈ F . It remains to show that the sequence (s∗ (xn ))n∈M is convergent for every segment s. Let s be a segment. We show that (s∗ (xn ))n∈M is a Cauchy sequence. Let ε 1 ε > 0 and choose k ∈ N with < . We have k 4 s=

m k

(ski ∩ s) ∪ (s \

i=1

and s \

m k i=1

m k

ski )

i=1

ski can be written as the ﬁnite union of pairwise disjoint segments with

at most one of them being inﬁnite. We assume that one of them is inﬁnite and let s\

m k i=1

ski

=(

nk

tj ) ∪ t

j=1

where t1 , t2 , . . . , tnk , t are pairwise disjoint segments, t is inﬁnite while t1 , t2 , . . . , tnk are ﬁnite. m k 1 It is clear that t ∩ ( ski ) = ∅ thus lim sup |t∗ (xn )| < and by the construck n∈Lk i=1 1 tion of M as a (subset of a) diagonal set we have that lim sup |t∗ (xn )| < . Thus k n∈M we may choose an ma ∈ M such that |t∗ (xn )| <

1 k

∀n ∈ M with n ≥ ma .

Since ski ∩ s ∈$ F for each i %= 1, 2, . . . ,$mk and % tj ∈ F for each j = 1, 2, . . . , nk , the sequences (ski ∩ s)∗ (xn ) n∈M and t∗j (xn ) n∈M are convergent. Thus we may select an mb ∈ M such that mk

|(ski ∩ s)∗ (xm ) − (ski ∩ s)∗ (xn )| +

i=1

nk

|t∗j (xm ) − t∗j (xn )| <

j=1

ε 2

for each m, n ∈ M with m > n ≥ mb . Therefore for each m, n ∈ M with m > n ≥ max{ma , mb } we have that |s∗ (xm ) − s∗ (xn )| <

2 ε + < ε. 2 k

Thus (s∗ (xn ))n∈M is a Cauchy, and hence convergent, sequence.

66

Chapter V. Examples of Hereditarily Indecomposable Extensions

y ] + 1. If ε∗ s1 , s2 , . . . , st are pairwise disjoint segments such that si ⊂ ran y and |si (y)| > ε y . for each i = 1, 2, . . . , t then t ≤ ε y y Proof. Suppose that t > and choose A ⊂ {1, 2, . . . , t} with #A = [ ] + 1. ε ε p Set εj = sgn(s∗j (y)) for each j ∈ A and f = εj s∗j . We have f ∈ Gns ⊂ BYG∗ns . Lemma V.11. Let ε > 0 and y ∈ YGns such that min supp y ≥ [

j∈A

Indeed, the segments sj , j ∈ A are pairwise disjoint and min sj ≥ min supp y ≥ y ] + 1 = #A. Thus [ ε y y ≥ f (y) = ] + 1) > y, εj s∗j (y) = |s∗j (y)| > ε(#A) = ε([ ε j∈A

j∈A

a contradiction.

y . ε The following also holds.

Therefore t ≤

Lemma V.12. Let (xn )n be a bounded block sequence such that b∗ (xn ) → 0 for every b branch of D. Then for every ε > 0 there exists L ∈ [N] such that for every segment s of D the following holds: #{n ∈ L : |s∗ (xn )| ≥ ε} ≤ 2. Proof. We set A

) =

{k1 , k2 , k3 } ∈ [N]3 : there exists a segment s * such that |s∗ (ykl )| > ε for l = 1, 2, 3

and B = [N]3 \ A. It follows from Ramsey’s theorem that there exists an L ∈ [N] such that [L]3 ⊂ A or [L]3 ⊂ B. The conclusion of the lemma is exactly that there exists an L ∈ [N] such that [L]3 ⊂ B. Therefore it is enough to exclude the possibility of existing a L ∈ [N] with [L]3 ⊂ A. Suppose that there exists an L ∈ [N], L = {l1 < l2 < l3 < · · · } such that 2r 3 [L] ⊂ A. We set r = sup yn . We may assume that min L > . ε n For each k ≥ 3 we apply the following procedure. We consider the set ) Tk = ran(yl2 + ylk−1 ) ∩ s : s is a segment , such that |s∗ (yl1 )| > ε, * |s∗ (ylk )| > ε, and there exists a t ∈ {2, . . . , k−1} such that |s∗ (ylt )| > ε .

V.3. A Non Separable HI Space

67

r . Indeed, let t1 , t2 , . . . , tm be pairwise diﬀerent eleε ments of Tk and for each i = 1, 2, . . . , m let si be a segment such that ran(yl2 + ylk−1 ) ∩ si = ti and |si (ylk )| > ε. The segments ri = (ran ylk ) ∩ si i = 1, 2, . . . , m are pairwise disjoint. We also have that |ri∗ (ylk )| > ε and ri ⊂ ran ylk for each i = 1, 2, . . . , m. From the fact that ylk ≥ |ri∗ (ylk )| > ε we get that We claim that #Tk ≤

min supp ylk ≥ lk > l1 > We easily get that m ≤ r m0 = [ ]. Then ε

2ylk yl 2r ≥ > [ k ] + 1. ε ε ε

ylk r r , thus m ≤ . We conclude that #Tk ≤ . We set ε ε ε Tk = {sk1 , sk2 , . . . , sktk }

for some tk ≤ m0 . For k ≥ 3 and i = 1, 2, . . . , tk we set ) * Aki = j ∈ {2, . . . , k − 1} : |(ski )∗ (ylj )| > ε while for tk < i ≤ m0 we set Aki = ∅. Hence {2, . . . , k − 1} =

m0

Aki .

(V.4)

i=1

We consider the following partition of [N \ {1}]2 . For j = 1, 2, . . . , m0 we set * ) Bj = {p, q} ∈ [N \ {1}]2 : p < q and p ∈ Aqj . Observe that (V.4) yields that [N \ {1}]2 =

m 0

Bj ; thus by Ramsey’s theorem

j=1

there exist an M ∈ [N \ {1}] and a j0 ∈ {1, 2, . . . , m0 } such that [M ]2 ⊂ Bj0 . Let M = {m1 < m2 < m3 < · · · }. Then for each t ∈ N and i ≤ t we have m that {mi , mt+1 } ∈ Bj0 , thus mi ∈ Aj0 t+1 . So there exists a segment st such that |s∗t (ymi )| > ε

∀i = 1, 2, . . . , t.

The sequence (s∗t )t∈N has a w∗ convergent subsequence; its w∗ limit is of the form s∗ for some inﬁnite segment s. We deduce that |s∗ (ymi )| > ε for all i which contradicts to the assumption that the sequence (yn )n∈N is weakly null. The next result uses the above lemmas. Lemma V.13. For every Z block subspace of YGns and every ε > 0 there exists z ∈ Z with z = 1 and |s∗ (z)| < ε for every s segment of (D, ≺).

68

Chapter V. Examples of Hereditarily Indecomposable Extensions

Proof. Assume that the conclusion fails. Then there exists a Z block subspace of YGns and ε0 > 0 such that for every z ∈ Z there exists a segment s of D with s∗ (z) ≥ ε0 z. Choose n0 > 4/e0 and let (yn )n∈N be a normalized weakly null block sequence in Z. Lemma V.12 yields that we may assume that for every segment s of D ε0 #{n ∈ N : s∗ (xn ) ≥ } ≤ 2 . (V.5) n Select ym , . . . , ym+n0 with n0 < ym . Further for every m ≤ k ≤ m+n0 choose a segment sk of D with supp sk ⊂ ran(yk ) and s∗k (yk ) ≥ ε0 . Since min{min s∗k } > n0 supp ∗ 0 n0 the family {min supp s∗k }nk=1 is Schreier admissible, hence s ∈ Gns . k=1 k Therefore n n0 n0 0 ∗ ym+k ≥ ( sk ) ym+k ≥ n0 ε0 > 4 . (V.6) k=1

k=1

k=1

Also from (V.5) for a segment s of D |s∗ (

n0

ym+n )| ≤ 3ε0 .

(V.7)

k=1

(V.6) and (V.7) derive a contradiction completing the proof.

Proof of Theorem V.6. (i) Let Z be a block subspace of YGns and let ε > 0. Using Lemma V.13 we may inductively select a normalized block sequence (yn )n∈N in Y such that, setting dn = max supp yn for each n and d0 = 1, |x∗ (yn )| < 2n dεn−1 for every σF special functional x∗ . We claim that (yn )n∈N is 1 + ε isomorphic to the standard basis of c0 . Indeed, let (βn )N n=1 be a sequence of scalars. We shall show that max |βn | ≤ 1≤n≤N

N n=1

βn yn Gns ≤ (1+ε) max |βn |. We may assume that max |βn | = 1. The left 1≤n≤N

1≤n≤N

inequality follows directly from the bimonotonicity of the Schauder basis (en )n∈N d of YGns . To see the right inequality we consider a g ∈ Gns , g = ai s∗i , where i=1

(si )di=1 are ﬁnite segments with min supp si ≥ d. Let n0 be the minimum integer n such that d ≤ dn . Since min supp g ≥ d > dn0 −1 we get that g(yn ) = 0 for n < n0 . Therefore g(

N

βn yn )

N

≤ |g(yn0 )| +

N d

|g(yn )| ≤ 1 +

n=n0 +1

n=1

< 1+

N n=n0 +1

d

|s∗i (yn )|

n=n0 +1 i=1 N

ε < 1+ 2n dn−1 n=n

0

ε < 1 + ε. n 2 +1

(ii) We pass now to show that YGns has uniformly bounded averages. Let 0 < ε < 1 and n ∈ N. We set k0 = k(n, ε) > 3n/ε and we claim that k0 satisﬁes the requirements of Deﬁnition III.13.

V.3. A Non Separable HI Space

69

Indeed, let (yl )l be a normalized weakly null block sequence in YGns . It follows from Lemma V.12 that we can assume the following: For every segment s of D, #{l : |s∗ (yl )| >

ε } ≤ 2. k02

Consider g ∈ Gns , g = dj=1 s∗j with min g < n. This yields that d < n. Consider k0 yli , l1 < . . . < lk0 . Then also x = k10 i=1 |g(x)| ≤

d j=1

|s∗j (x)| ≤

d 3 3d 3n = < < ε, k k k0 0 i=1 0

and this completes the proof of (ii) and the entire proof of the theorem.

Proposition V.14. Let B denote the set of all branches of the binary tree D. Then we have the following: (i) X[Gns , σ]∗ = span({e∗n : n ∈ N} ∪ {b∗ : b ∈ B}). (ii) X[Gns , σ]∗ /X[Gns , σ]∗ = span{b∗ + X[Gns , σ]∗ : b ∈ B} = c0 (B). (iii) The space X[Gns , σ]∗∗ is isomorphic to X[Gns , σ] ⊕ 1 (B). Proof. (i) It is easy to see that the set Gns Gns

p

=

d "

w∗

is equal to the set

εi s∗i : εi ∈ {−1, 1}, i = 1, 2, . . . , d, (si )di=1 are pairwise

i=1

# disjoint segments and {min s1 , min s2 , . . . , min sd } ∈ S1 , d ∈ N . Therefore, from Proposition II.26 we get that X[Gns , σ]∗ = span({e∗n : n ∈ N} ∪ {b∗ : b ∈ B}). (ii) By Remark II.29 the basis (en )n∈N is boundedly complete and the space X[Gns , σ]∗ = span{e∗n : n ∈ N} is the predual of X[Gns , σ]. It follows from (i) that X[Gns , σ]∗ /X[Gns , σ]∗ = span{b∗ + X[Gns , σ]∗ : b ∈ B} so it remains to show that span{b∗ + (X[Gns , σ])∗ : b ∈ B} = c0 (B). It is enough to show that if b1 , b2 , . . . , bn are pairwise diﬀerent elements of B and ε1 , ε2 , . . . , εn ∈ {−1, 1} then ε1 (b∗1 + X[Gns , σ]∗ ) + ε2 (b∗2 + X[Gns , σ]∗ ) + · · · + εn (b∗n + X[Gns , σ]∗ ) = 1.

70

Chapter V. Examples of Hereditarily Indecomposable Extensions

For k = 1, 2, . . . we denote by Ek the interval Ek = {n ∈ N : n ≥ 2k }. Since b1 , b2 , . . . , bn are pairwise diﬀerent we may select a k0 ≥ n such that b1 ∩ Ek , b2 ∩ Ek , . . . , bn ∩ Ek are incomparable segments for all k ≥ k0 . Thus ε1 (b∗1 + X[Gns , σ]∗ ) + · · · + εn (b∗n + X[Gns , σ]∗ ) = dist(ε1 b∗1 + · · · + εn b∗n , X[Gns , σ]∗ ) = dist(ε1 b∗1 + · · · + εn b∗n , span{e∗j : j ∈ N}) = dist(ε1 b∗1 + · · · + εn b∗n , span{e∗j : j ∈ N}) = lim dist(ε1 b∗1 + · · · + εn b∗n , span{e∗j : j ≤ 2k − 1}) k

= lim ε1 (b1 ∩ Ek )∗ + · · · + εn (bn ∩ Ek )∗ k

= 1. The last equality holds since for every k ≥ k0 the functional gk =

n

εi (bi ∩ Ek )∗

i=1

belongs to Gns , thus gk ≤ 1, while for n ∈ b1 ∩ Ek we have that |gk (en )| = 1. (iii) As is well known, for every Banach space Z the space Z ∗∗∗ is isomorphic to Z ∗ ⊕(Z ∗∗ /Z)∗ . For Z = X[Gns , σ]∗ we get that the space X[Gns , σ]∗∗ is isomorphic to the space X[Gns , σ] ⊕ (X[Gns , σ]∗ /X[Gns , σ]∗ )∗ = X[Gns , σ] ⊕ (c0 (B))∗ = X[Gns , σ] ⊕ 1 (B). Theorem V.15. The space X[Gns , σ]∗ is a non separable HI Banach space. Proof. The space X[Gns , σ], being HI, contains no isomorphic copy of 1 , the space X[Gns , σ]∗ = span{e∗n : n ∈ N} is HI and the quotient space X[Gns , σ]∗ /X[Gns , σ]∗ is isometric to c0 (B). From Corollary IV.8 we deduce that X[Gns , σ]∗ is HI while it is clear that X[Gns , σ]∗ is non separable. Remark. The three examples presented in this chapter are from [16] where they have been constructed with the use of higher complexity saturation methods and have the additional property that they are asymptotic 1 spaces. In particular for the space corresponding to X[Gns , σ]∗ , it is shown that every bounded linear operator is of the form λI +W with W weakly compact. Since the space X[Gns , σ]∗ is the dual of a separable space we conclude that the weakly non-compact operator W has separable range. We do not include this property here as it requires much more eﬀorts.

Chapter VI

The Space Xω1 In this part we present a reﬂexive Banach space Xω1 with a transﬁnite basis (eα )α<ω1 not containing any unconditional basic sequence. As it is well known every non separable reﬂexive Banach space is decomposable. In particular it admits a resolution of the identity. Hence any non separable reﬂexive space not containing an unconditional basic sequence is not HI. On the other side Gowers dichotomy yields that such a space is HI saturated. Constructing the space Xω1 we shall follow the general approach we have developed in the previous parts, namely Xω1 will be Xω1 [DG0 ], where G0 = {±eα : α < ω1 }, DG0 is a tree complete set and it 1 will be closed for all (An2j+1 , m2j+1 ) operations acting on n2j+1 -special sequences. The main diﬃculty that we have to overcome is that deﬁning the n2j+1 -special sequences we are not able to use a one-to-one coding σ, a crucial property for the ω constructions. To solve this problem we employ Todorcevic’s ρ-function. Thus changing one of the fundamental ingredients of the ω-construction (i.e. the injection of the coding σ) we arrive to a non separable reﬂexive space where we control the structure of all separable subspaces of it. The norming set Kω1 The space Xω1 will be deﬁned as the completion of (c00 (ω1 ), · ∗ ) under the norm · ∗ induced by a set of functionals Kω1 ⊆ c00 (ω1 ). The set Kω1 is the minimal subset of c00 (ω1 ) satisfying that: (1) It contains (e∗γ )γ<ω1 , is symmetric (i.e., φ ∈ K implies −φ ∈ K) and is closed under the restriction on intervals of ω1 . , n2j } ⊆ Kω1 with supp φ1 < · · · < supp φn2j , (2) For every {φi : i = 1, . . . n2j it holds that φ = (1/m2j ) i=1 φi ∈ Kω1 . We say that φ is a result of a (An2j , m12j )-operation. (3) For every special sequence (φ1 , . . . , φn2j+1 ) (see later for the deﬁnition), the

Chapter VI. The Space Xω1

72

n2j+1 functional φ = (1/m2j+1 ) i=1 φi is in Kω1 . We call such a φ special func1 tional and say that φ is a result of a (An2j+1 , m2j+1 )-operation. (4) It is rationally convex. Remark VI.1. From the deﬁnition of the norming set Kω1 it follows easily that (eα )α<ω1 is a bimonotone basis of Xω1 . Also, it is not diﬃcult to see using (2) from the deﬁnition of Kω1 that the basis (eα )α<ω1 is boundedly complete. Indeed, for x ∈ c00 (ω1 ) and E1 < · · · < En intervals of ω1 , property (2) of the norming set 2j n2j Kω1 yields that x ≥ (1/m2j ) i=1 Ei x. Also, from the choice of the sequence (mj )j , (nj )j , it follows that n2j /m2j increases to inﬁnity. From these observations it follows that the basis (eα )α<ω1 is boundedly complete. To prove that the space Xω1 is reﬂexive we need to show that the basis is shrinking. The deﬁnition of the special sequences, as in the spaces X[G, σ], depends crucially on a certain coding σ . The essential diﬀerence is that now σ is not an injection, a crucial property on which the proofs in the case of X[G, σ] rely (the injectivity of σ is required for the proof of Proposition III.3). Our proofs on the other hand will rely on a “tree-like property” of our coding which we now describe. First we notice that each 2j + 1-special sequence Φ = (φ1 , φ2 , . . . , φn2j+1 ) is of the form supp φ1 < · · · < supp φn2j+1 with each φi of type I. The tree-like property is the following: For any pair of 2j + 1-special sequences Φ = (φ1 , φ2 , . . . , φn2j+1 ), Ψ = (ψ1 , ψ2 , . . . , ψn2j+1 ) there exist 1 ≤ κΦ,Ψ ≤ λΦ,Ψ ≤ n2j+1 such that (i) If 1 ≤ k < κΦ,Ψ then φk = ψk and if κΦ,Ψ < k < λΦ,Ψ , then w(φk ) = w(ψk ). (ii) (∪κΦ,Ψ
Chapter VI. The Space Xω1

73

Deﬁnition VI.2. Recall that given a ﬁnite set F ⊆ ω1 , we let pF = p (F ) = maxα,β∈F (α, β). For a ﬁnite set F ⊆ ω1 and p ∈ N, let p

F = {α ≤ max F : there is β ∈ F s.t. α ≤ β and (α, β) ≤ p}. p

Notice that by condition 3., F is a ﬁnite set of countable ordinals. We say that p F is p-closed iﬀ F = F , and that F is -closed iﬀ it is pF -closed. Remark VI.3. 1. Note that ·p is a monotone and idempotent operator and so, in p p p particular, every F is a p-closed set: It is clear that if F ⊆ G, then F ⊆ G . pp p pp Let us show now that F = F . Let α ∈ F . This implies that (α, α0 ) ≤ p, for p some α0 ∈ F , α ≤ α0 . Choose α1 ≥ α0 , α1 ∈ F such that (α0 , α1 ) ≤ p. Then, (α, α1 ) ≤ max{(α, α0 ), (α0 , α1 )} ≤ p. 2. Suppose that F ⊆ ω1 is ﬁnite and suppose that p ≥ pF . Then pF p ≤ p: Fix F p ﬁnite, and p ≥ pF . Suppose that α < β are both in F . Let α ≥ α, β ≥ β such that α, β ∈ F and (α, α ), (β, β ) ≤ p. Then one of the following cases occurs: (a) If α ≤ α ≤ β ≤ β , then (α, β) ≤ max{(α, α ), (α , β)} ≤ max{(α, α ), (α , β ), (β, β )} ≤ p. (b) If α ≤ β ≤ α ≤ β , then (α, β) ≤ max{(α, α ), (β, α )} ≤ max{(α, α ), (β, β ), (α , β )} ≤ p. (c) If α ≤ β ≤ β ≤ α , use a similar proof to case (a). Proposition VI.4. Let F, G ⊆ ω1 be two ﬁnite sets and p ≥ pF , pG . Then: p

p

p

1. For every ordinal α ≤ ω1 , F ∩ α = F ∩ α and F ∩ α is an initial part of p F . Therefore, if F is p-closed, so is F ∩ α. p

p

2. For every α ∈ F ∩ G, we have that F ∩ (α + 1) = G ∩ (α + 1) . Hence, if F and G are in addition p-closed, then F ∩ (α + 1) = G ∩ (α + 1). p

p

p

3. F ∩ G = F ∩ G . Therefore, if F and G are p-closed then F ∩ G is also p-closed and it is an initial part of both F and G. p

p

p

Proof. 1. Since F ∩ α ⊆ F, α, it follows that F ∩ α ⊆ F ∩ α. Now let β ∈ F ∩ α. Then there is some γ ∈ F , γ ≥ be such that (β, γ) ≤ p. If γ < α, then we are done. If not, let δ = max F ∩ α ∈ F and since β ≤ δ < γ we have that (β, δ) ≤ max{(β, γ), (δ, γ)} ≤ max{p, pF } = p,

(VI.1) p

the last equality using our assumption that p ≥ pF . (VI.1) shows that β ∈ F ∩ α . p p Suppose now that F is p-closed. Then we have just shown that F ∩ α = F ∩ α = F ∩ α, and we are done.

Chapter VI. The Space Xω1

74 p

p

2. Fix α ∈ F ∩G. Let β ∈ F ∩ (α + 1) = F ∩(α+1). Let γ ∈ F ∩(α+1), γ ≥ β be such that (β, γ) ≤ p. Then (β, α) ≤ max{(β, γ), (γ, α)} ≤ max{p, pF } = p p. Since G is p-closed, and α ∈ G, we can conclude that β ∈ G ∩ (α + 1) . This p p shows that F ∩ (α + 1) ⊆ G ∩ (α + 1) . The other inclusion follows by symmetry. The last part of 2. follows easily. p p p 3. Let α = max F ∩G. Then by 2., F ∩ G = F ∩ G ∩ (α + 1) = F ∩ (α + 1) p p p = F ∩ (α + 1) and F ∩ G = G ∩ (α + 1). Combining the above equalities we p p p p p p p get F ∩ G = F ∩ G ∩ (α + 1) = F ∩ G , the last equality because F ∩ G ⊆ F ∩ G ⊆ max(F ∩ G) + 1 = α + 1. The σ -coding and the special sequences We denote by Qs (ω1 ) the set of ﬁnite sequences (φ1 ,w1 , p1 ,φ2 ,w2 ,p2 , . . . , φd , wd ,pd ) such that 1. for all i ≤ d, φi ∈ c00 (ω1 ) and φ1 < φ2 < · · · < φd , 2. (wi )di=1 , (pi )di=1 ∈ Nd are strictly increasing, and 3. pi ≥ p(∪ik=1 supp φk ) for every i ≤ d. Let Qs be the set of ﬁnite sequences (φ1 , w1 , p1 , φ2 , w2 , p2 , . . . , φd , wd , pd ) satisfying 1., and 2. above and in addition for every i ≤ d, φi ∈ c00 (N). Notice that Qs is a countable set. Fix a one-to-one function σ : Qs → {2j : j odd} such that σ(φ1 , w1 , p1 , φ2 , w2 , p2 , . . . , φd , wd , pd ) > max{p2d , ε12 , max supp φd }, where ε = min{|φk (eα )| : α ∈ supp φk , k = 1, . . . , d}. Given a ﬁnite subset F of ω1 , we denote by πF : {1, 2, . . . , #F } → F the natural order preserving map. Given Φ = (φ1 , w1 , p1 , φ2 , w2 , p2 , . . . , φd , wd , pd ) ∈ Qs (ω1 ) we denote GΦ = ∪di=1 supp φi

pd

and then we consider the family

πGΦ (Φ) = (πG (φ1 ), w1 , p1 , πG (φ2 ), w2 , p2 , . . . , πG (φd ), wd , pd ) ∈ Qs , where

πG (φk )(n) =

φk (πGΦ (n)) if n ∈ GΦ 0 otherwise.

Finally, σ : Qs (ω1 ) → {2j : j odd} is deﬁned as σ (Φ) = σ(πG (Φ)). A sequence Φ = (φ1 , . . . , φn2j+1 ) of functionals of Kω1 is said to be a 2j + 1special sequence if: (1) supp φ1 < supp φ2 < · · · < supp φn2j+1 , each φk is of type I, w(φk ) = m2jk and w(φ1 ) = m2j1 with j1 even and satisfying m2j1 > n22j+1 .

Chapter VI. The Space Xω1

75

Φ (2) There exists a strictly increasing sequence (pΦ 1 , . . . , pn2j+1 −1 ) of natural numbers such that for all 1 ≤ i ≤ n2j+1 − 1 we have that w(φi+1 ) = mσ (Φi ) where Φ Φ Φi = (φ1 , w(φ1 ), pΦ 1 , φ2 , w(φ2 ), p2 , . . . , φi , w(φi ), pi ).

Notice that for a given 2j + 1-special sequence Φ, both sequences (pΦ i )i and (w(φi ))i above are uniquely determined, and that w(φ1 ) = m2j1 with j1 even, while for every i > 1, w(φi ) = m2ji is such that ji is odd. Lemma VI.5 (Tree-like interference of a pair of special sequences). Let Φ = (φ1 , . . . , φn2j+1 ) and Ψ = (ψ1 , . . . , ψn2j+1 ) be two 2j + 1-special sequences. Then there are two numbers 0 ≤ κΦ,Ψ ≤ λΦ,Ψ ≤ n2j+1 such that the following conditions hold: Ψ TP.1 For all i ≤ λΦ,Ψ , w(φi ) = w(ψi ) and pΦ i = pi .

TP.2 For all i < κΦ,Ψ , it holds that φi = ψi , TP.3 For all κΦ,Ψ < i < λΦ,Ψ , it holds that pλΦ,Ψ −1

=∅

pλΦ,Ψ −1

= ∅.

supp φi ∩ supp ψ1 ∪ · · · ∪ supp ψλΦ,Ψ −1 supp ψi ∩ supp φ1 ∪ · · · ∪ supp φλΦ,Ψ −1

TP.4 {w(φi ) : λΦ,Ψ < i ≤ n2j+1 } ∩ {w(ψi ) : i ≤ n2j+1 } = ∅ and {w(ψi ) : λΦ,Ψ 1, w(φi ) = w(ψi ), and that for every 1 1. Suppose now that λΦ,Ψ > 0. Deﬁne κΦ,Ψ by κΦ,Ψ = min{i < λΦ,Ψ : φi = ψi }, if deﬁned and κΦ,Ψ = 0 if not. For this last case it is trivial to check our requirements. So assume that κΦ,Ψ > 0. (TP.2) and (TP.4) follow easily from the properties of the coding σ . Let us show now (TP.3). Let G=

λΦ,Ψ −1 i=1

supp φi

pλΦ,Ψ −1

and G =

λΦ,Ψ −1 i=1

supp ψi

pλΦ,Ψ −1

,

and let πG : G → {1, . . . , #G} and πG : G → {1, . . . , #G} be the unique orderpreserving bijections.

Claim.

1. #G = #G .

2. πG |(G ∩ G ) = πG |(G ∩ G ) and (G ∩ G )φκΦ,Ψ = (G ∩ G )ψκΦ,Ψ .

Chapter VI. The Space Xω1

76

3. max(G ∩ G ) < min{max supp φκΦ,Ψ , max supp ψκΦ,Ψ } Proof. 1: Notice that #G = max supp πG (φλΦ,Ψ −1 ) and #G = max supp πG (ψλΦ,Ψ −1 ). λ

−1

λ

(VI.2)

−1

Φ,Ψ Φ,Ψ Since σ ((φi , w(φi ), pi )i=1 ) = σ ((ψi , w(ψi ), pi )i=1 ), we have πG (φλΦ,Ψ −1 ) = πG (ψλΦ,Ψ −1 ) and hence #G = #G , as desired. 2: It follows from the properties of that πG |(G ∩ G ) = πG |(G ∩ G ). Fix now α ∈ G ∩ G . Since πG (α) = πG (α) we have that

φκΦ,Ψ (eα ) = ψκΦ,Ψ (eπG (π−1 α) ) = ψκΦ,Ψ (eα ),

(VI.3)

G

as desired. 3: W.l.o.g we assume that max G ∩ G ≥ max supp φκΦ,Ψ . Property 2. yields that φκΦ,Ψ = (G ∩ G )φκΦ,Ψ = (G ∩ G )ψκΦ,Ψ , (VI.4) and since # supp φκΦ,Ψ = # supp ψκΦ,Ψ we obtain that φκΦ,Ψ = ψκΦ,Ψ , a contradiction. To complete the proof choose κΦ,Ψ < i < λΦ,Ψ . Then previous Claim yields that supp φi ⊆ G \ (G ∩ G ) and hence supp φi ∩ G = ∅. The space Xω1 has no unconditional basic sequence Deﬁnition VI.6. Let j ∈ N. A sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) is said to be a (1, j)-dependent sequence if: DS.1 supp x1 ∪ supp φ1 < · · · < supp xn2j+1 ∪ supp φn2j+1 . DS.2 The sequence Φ = (φ1 , . . . , φn2j+1 ) is a 2j + 1-special sequence. DS.3 (xi , φi ) is a (6, 2ji )-exact pair for 1 ≤ i ≤ n2j+1 , with # supp xi ≤ m2ji+1 /n22j+1 for every 1 ≤ i ≤ n2j+1 . DS.4 For every Ψ = (ψ1 , . . . , ψn2j+1 ) (2j + 1)-special sequence we have that (VI.5) κΦ,Ψ
2j+1 as follows. Proof. Let (yn )n and j be given. We inductively produce {(xi , φi )}i=1 For i = 1 we choose a (6, 2j1 )-exact pair (x1 , φ1 ) such that m2j1 > m22j+1 , j1 even (see the deﬁnition of special sequences) and x1 ∈ yn n . Assume that {(xl , φl )}i−1 l=1 has been chosen such that there exists (pl )i−2 l=1 satisfying

Chapter VI. The Space Xω1

77

(a) supp x1 ∪ supp φ1 < · · · < supp xi−1 ∪ supp φi−1 , each xl ∈ yn n and (xl , φl ) being a (6, 2jl )-exact pair. (b) For 1 < l ≤ i − 1, w(φl ) = σ (φ1 , w(φ1 ), p1 , . . . ., φl−1 , w(φl−1 ), pl−1 ) l (c) For 1 ≤ l < i − 1, pl ≥ max{pl−1 , pFl }, where Fl = k=1 supp φk ∪ supp xk . To deﬁne (xi , φi ) we choose pi−1 ≥ max{pi−2 , pFi−1 , n22j+1 · # supp xi } and we set 2ji = σ (φ1 , w(φ1 ), p1 , . . . ., φi−1 , w(φi−1 ), pi−1 ). Choose a (6, 2ji )-exact pair (xi , φi ) such that xi ∈ yn n and supp xi−1 ∪supp φi−1 < supp xi ∪ supp φi . This completes the inductive construction. (DS.1)–(DS.3) easily holds, while (DS.4) follows from (c) and (TP.3) of Lemma VI.5. Remark VI.8. Suppose that (yn )n and (zn )n are block sequences such that supn max supp yn = supn max supp zn . Then for every j ∈ N there is a (1, j)dependent sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) with the property that x2i−1 ∈ yn n and x2i ∈ zn n for every i = 1, . . . , n2j+1 /2. Lemma VI.9. Fix a (1, j)-dependent sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ), and a n2j+1 sequence (λi )i=1 of scalars such that maxi |λi | ≤ 1. Suppose that for every ψ ∈ Kω1 with w(ψ) = m2j+1 , and every interval of integers E ⊆ [1, n2j+1 ] it holds that ). (VI.6) |ψ( i∈E λi xi )| ≤ 12(1 + n#E 2 2j+1

Then, 1 n2j+1

n2j+1 i=1

λ i xi ≤

1 . m22j+1

(VI.7)

Proposition VI.10. If (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) is a (1, j)-dependent sequence, then n2j+1 n2j+1 1 1 1 i+1 n2j+1 xi ≤ m21 . (VI.8) i=1 xi ≥ m2j+1 and n2j+1 i=1 (−1) 2j+1

n2j+1 Proof. The ﬁrst estimation is clear since the functional ψ = (1/m2j+1 ) i=1 φi ∈ n2j+1 Kω1 and ψ((1/n2j+1 ) i=1 xi ) = 1/m2j+1 . For the second, we use Lemma VI.9 applied to the sequence of scalars ((−1)i+1 )i , and the desired estimation will follow from (VI.7). Fix ψ ∈ Kω1 with w(ψ) = m2j+1 , and an interval E ⊆ [1, n2j+1 ]. Set Ψ = (ψ1 , . . . , ψn2j+1 ) and x = i∈E (−1)i+1 xi , where ψ = (1/m2j+1 ) i∈E ψi . Notice that |ψ(x)| = | ≤

1 m2j+1 1

m2j+1

κΦ,Ψ −1

φi (x) +

i=1

+|

1 m2j+1

We shall show that the following hold:

1 m2j+1

n2j+1

i=κΦ,Ψ

n2j+1

i=κΦ,Ψ

ψi (x)|.

ψi (x)| ≤

(VI.9)

Chapter VI. The Space Xω1

78 (a) |ψκΦ,Ψ (

i∈E (−1)

i+1

xi )| ≤ 1 + 12(#E − 1)/n22j+1 ,

(b) |ψλΦ,Ψ |( i∈E (−1)i+1 xi ) ≤ 1 + 12(#E − 1)/n22j+1 , and (c) |( l>κΦ,Ψ ,l=λΦ,Ψ ψl )(xi )| ≤ 12/n2j+1 for every 1 ≤ i ≤ n2j+1 . Let us show ﬁrst (a). Let 2ji be such that w(φi ) = m2ji . Notice that for i = κΦ,Ψ we have that 12 if i > κΦ,Ψ w(ψκΦ,Ψ ) (VI.10) |ψκΦ,Ψ (xi )| ≤ 6 if i < κΦ,Ψ . m2 2ji

By the properties of the sequences (ml )l , (nl )l and the fact that n22j+1 < w(ψκΦ,Ψ ), m2ji , (VI.10) yields that |ψκΦ,Ψ (xi )| ≤ n212 for i = κΦ,Ψ . Hence 2j+1

|ψκΦ,Ψ (

i∈E

xi )| ≤ |ψκΦ,Ψ (xκΦ,Ψ )| + |ψκΦ,Ψ (

i∈E, i=κΦ,Ψ

xi )| ≤ 1 +

12(#E−1) . n22j+1

(VI.11) (b) has a proof similar to the one of (a). We check now (c). Fix l > κΦ,Ψ , l = λΦ,Ψ . Suppose that l > λΦ,Ψ . Since w(ψl ) = w(φi ) for all i ≤ n2j+1 , we obtain that |ψl (xi )| ≤ n212 . Now suppose that κΦ,Ψ < l < λΦ,Ψ . By (DS.4) we have 2j+1

/ (κΦ,Ψ , λΦ,Ψ ), using that ψl (xi ) = 0 for every κΦ,Ψ κΦ,Ψ ,l=λΦ,Ψ ψl )(xi ) ≤ 12/n2j+1 for every 1 ≤ i ≤ n2j+1 , as desired. Combining (a), (b) and (c) we obtain that 1 | m2j+1

n2j+1 i=κΦ,Ψ

ψi (x)| ≤ 1 +

#E . n22j+1

(VI.12)

From (VI.9) and (VI.12) we conclude that |ψ(x)| ≤ 12(1 + #E/n22j+1 ), as desired. Proposition VI.11. Let (yn )n be a block sequence of vectors of Xω1 . Then the closed linear span of (yn )n is hereditarily indecomposable. Proof. Fix a block sequence (yn )n of Xω1 , two block subsequences (zn )n and (wn )n of (yn )n and ε > 0. Let j be large enough such that m2j+1 ε > 1. By Proposition VI.7 we can choose a (1, j)-dependent sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) such n2j+1 that x2i−1 ∈ zn n , and x2i ∈ wn n . Set z = (1/n2j+1 ) i=1,i odd xi and w = n2j+1 (1/n2j+1 ) i=1,i even xi . Notice that z ∈ zn n and w ∈ wn n . By Proposition VI.9, we know that z + w ≥ 1/m2j+1 and z − w ≤ 1/m22j+1 . Hence z − w ≤ εz + w. Corollary VI.12. (a) The distance between the unit spheres of every two normalized block sequences (xn ) and (yn ) in Xω1 such that supn max supp xn = supn max supp yn is 0. (b) There is no unconditional basic sequence in Xω1 .

Chapter VI. The Space Xω1

79

(c) Every inﬁnite dimensional closed subspace of Xω1 contains a hereditarily indecomposable subspace. (d) The distance between the unit spheres of two non separable subspaces of Xω1 is equal to 0. Proof. (b) follows from Proposition VI.11 and 4. of Proposition A.3. (c) This result follows from the previous corollary and Gowers’ dichotomy. Moreover, every subspace of Xω1 isomorphic to the closed linear span of a block sequence with respect to the basis (eα )α<ω1 is hereditarily indecomposable. (d) Fix two non separable closed subspaces X and Y of Xω1 . Now we can ﬁnd a sequence (zn )n of normalized vectors such that for every n (a) z2n−1 ∈ X, z2n ∈ Y and (b) supp zn < supp zn+1 . Notice that the supports supp zn are not necessarily ﬁnite. Now approximate (zn )n by a normalized block sequence (wn )n as closed as needed and we are done.

Chapter VII

The Finite Representability of JT0 and the Diagonal Space D(Xγ ) Since the space Xω1 has a transﬁnite basis (eα )α<ω1 it also admits a transﬁnite family of naturally deﬁned projections. Thus we could not expect that the operators on Xω1 or on the spaces Xγ , γ < ω1 , have small spaces of non-strictly singular operators. In this part we shall discuss the structure of the step diagonal operators which are the non strictly singular diagonal operators and we shall show that for all inﬁnite γ < ω1 , D(Xγ ) is isomorphic to JT∗0 (Aγ ), where Aγ is a closed subset of γ. This result, quite unexpectedly, is a consequence of the ﬁnite interval representability of JT0 in the transﬁnite block subspaces of Xω1 . James-like spaces Deﬁnition VII.1. Let X be a reﬂexive space with a 1-subsymmetric basis (xn )n , and let A be a set of ordinals. JX (A) is the completion of (c00 (A), · JX (A) ), where for x ∈ c00 (A), % l $ xJX (A) = sup{ n=1 i∈In x(i) xn X : I1 < · · · < In intervals of A}. The natural Hamel basis (vα )α∈A of c00 (A) is a bimonotone 1-subsymmetric transﬁnite basis interval I of A the functional I ∗ : JX (A) → R, of JX (A). Also, for every ∗ ∗ I (x) = α∈A x(α) belongs to JX (A) and I ∗ = 1. Remark VII.2. Spaces of the above form for A = N, have been introduced in [19]. The transﬁnite analogue of James quasi-reﬂexive space was deﬁned by G.A. Edgar in [23]. As we shall see next 1 does no embed into JX (A) hence the basis (vα )α∈A is not unconditional.

82

Chapter VII. Finite Representability of JT0 and the Diagonal Space D(Xγ )

Proposition VII.3. Let (yn )n be a semi-normalized block sequence in JX (A) with α∈A yn (α) = 0 for every n. Then (yn )n is equivalent to the basis (xn )n of X. Proof. Let 0 < c < C be such that c ≤ yn ≤ C for all n. It is easy to see that l−1 an xn X ≤ n an yn JX (A) ≤ supi1 ≤i2 ≤···≤il q=1 (|aiq | c n

+ |aiq+1 |)xq X ≤ (2CK)

n

an xn X ,

(VII.1)

where K is the unconditional constant of (xn )n . The ﬁrst inequality holds for any block sequence and the second uses our assumptions. Corollary VII.4. The space 1 does not embed into JX (A). Proof. Assume the contrary, then from Proposition A.3 we could ﬁnd a seminormalized block sequence (yn )n equivalent to the 1 -basis. Therefore, passing if necessary to a further block sequence, we may assume that for all n ∈ N, α∈A yn (α) = 0. Hence Proposition VII.3 yields that (yn )n is equivalent to (xn )n , a contradiction. Remark VII.5. Suppose that A and B are two sets of ordinals with the same order type. Then the unique order-preserving mapping deﬁnes naturally an f : A → B isometry between f : JT0 (A) → JT0 (B) by f( α∈H rα vα ) = α∈H rα vf (α) . ∗ Proposition VII.6. For every ordinal γ the space JX (γ) is generated in norm by ∗ {[0, α) }α<γ+1 .

Proof. We proceed by induction. It is clear that the successor ordinal case follows immediately from the inductive assumption. So we assume that γ is limit ordinal and for all λ < γ the conclusion holds. Assume on the contrary that ·

∗ ∗ Y = [0, α)∗ α<γ+1 JX (γ), then there exists x∗ ∈ JX (γ) with x∗ = 1 and ∗ ε > 0 such that d(x , Y ) > ε. Observe also that the inductive assumption yields that for all α < γ if x∗α denotes the functional deﬁned by

0 if β < α x∗α (vβ ) = x∗ (vβ ) if β ≥ α

satisﬁes that x∗α ≤ 1 and d(x∗α , Y ) > ε. In particular for all α < γ, d(x∗α , [α, γ)∗ ) > ε and from the Hahn–Banach and Goldstime theorems there exists a ﬁnitely supported y˜α ∈ JX (γ), ˜ yα ≤ 1, α ≤ min supp y˜α , x∗ (˜ yα ) > ε and | β<γ y˜α (β)| ≤ ε/4. Assuming further that α is a successor ordinal we consider the vector yα = y˜α − ( β≥α y˜α (β))vα− . Observe that α− ≤ min supp yα , x∗ (yα ) > ε − ε/4 > ε/2 and β<γ yα (β) = 0. Hence we may inductively choose a block sequence (zn )n such that ε/2 ≤ zn ≤ 1, α<γ zn (α) = 0 and x∗ (zn ) > ε/2. Observe that (zn )n is unconditional (Proposition VII.3) therefore equivalent to 1 -basis which yields a contradiction. ∗ Corollary VII.7. For every set of ordinals A we have that dim JX (A) = #A.

Chapter VII. Finite Representability of JT0 and the Diagonal Space D(Xγ )

83

Finite interval representability of JT0 and the space of diagonal operators Deﬁnition VII.8. Let X and Y be Banach spaces and let (xα )α<γ and (yn )n be a transﬁnite basis for X and a Schauder basis of Y respectively. We say that Y is ﬁnitely interval representable in X if there exists a constant C > 0 such that for every integer n and intervals I1 ≤ I2 ≤ · · · ≤ In successive, not necessarily distinct, intervals of γ there exists zi ∈ (xα )α∈Ii (i = 1, . . . , n) with supp z1 < supp z2 < · · · < supp zn and such that the natural order preserving isomorphism H : (yi )ni=1 → (zi )ni=1 satisﬁes H · H −1 ≤ C. Theorem VII.9. Let (yα )α<γ be a normalized transﬁnite block sequence in Xω1 , and Y its closed linear span. Then JT0 is ﬁnitely interval representable in the space Y , where T0 is the mixed Tsirelson space T [(An2j , m12j )j ]. We present the proof in Appendix B. Throughout all this section C will denote the ﬁnitely block representability constant of JT0 in Xω1 . We will see in Appendix B that C < 121. Remark VII.10. 1. Let us observe that since, as we will show, the basis of JT0 is not unconditional and it is ﬁnitely block representable in any block subsequence of the basis (eα )α<ω1 , Xω1 cannot have any unconditional basic sequence. In other words the ﬁnite interval representability of JT0 in the block subsequences of Xω1 must make use of the conditional structure of Xω1 . Indeed we get more. Suppose that X has a transﬁnite basis, and suppose that a Banach space Y with a conditional basis (yn )n is ﬁnite block representable in every block sequence of X. Then X does not contain unconditional basic sequences and from Gowers dichotomy, X is HI saturated. 2. The James like space JT0 has the following alternative description. It is the mixed Tsirelson extension T [G, (An2j , m12j )j ], where G = {I ∗ : I ⊆ N interval}. We recall that the norming set of this space is the minimal subset K0 of c00 (N) which is symmetric, contains G, and is closed under the (An2j , m12j )j -operations. Proposition VII.11. Let x1 < · · · < xnbe ﬁnitely supported, φ ∈ Kω1 and set n ri = φ(xi ) for each i = 1, . . . , n. Then i=1 ri vi JT0 ≤ x1 + · · · + xn . Proof. Fix a functional f of K0 with support contained in {1, . . . , n}, and a tree-analysis (ft )t∈T of f . We show by induction n over the tree T that for every t ∈ T there issome φt ∈ Kω1 such that ft ( i=1 ri vi ) = φt (x1 + · · · + xn ). In particular f0 ( i ri vi ) = φ0 (x1 + · · · + xn ), and hence the desired result holds. If t ∈ T is a terminal node, then ft = ±I ∗ , I ⊆ {1, . . . , n} interval. We set φt = ±φ|[min supp xmin I , max supp xmax I ]. It is clear that φt ∈ Kω1 , and (VII.2) φt (x1 + · · · + xn ) = ± i∈I φxi = ± i∈I ri = ft ( i ri vi ). If t ∈ T is not a terminal node, then ft = (1/m2j ) di=1 fsi , where St = {s1 , . . . , sd } d is ordered by fs1 < · · · < fsd . Then φt = (1/m2j ) i=1 φsi clearly satisﬁes our inductive requirements.

84

Chapter VII. Finite Representability of JT0 and the Diagonal Space D(Xγ ) The next result shows that JT0 is minimal in a precise sense.

Corollary VII.12. Suppose that X is a Banach space with a normalized Schauder basis (xn )n which dominates the summing basis of c0 and ﬁnitely block represented in Xω1 . Then (xn )n also dominates the basis (vn )n of JT0 . Proof. Fix scalars (ai )ni=1 . Choose a normalized block sequence (wi )ni of Xω1 , Cequivalent to (xi )ni=1 . Fix f ∈ K0 with supp f ⊆ {1, . . . , n} and a tree-analysis n (ft )t∈T of it. We are going to ﬁnd φ ∈ K such that |f ( t ω t 1 i=1 ai vi )| ≤ C|φt ( ai wi )|, for each t ∈ T . This will show that n n n (VII.3) i=1 ai vi JT0 ≤ C i=1 ai wi Xω1 ≤ C 2 i=1 ai wi X , as desired. If t ∈ T is a terminal node, then ft = ±I ∗ , I ⊆ [1, n] an interval. Since (xn )n dominates the summing basis of c0 , we can ﬁnd φt ∈ Kω1 such that φt (

n i=1

a i wi ) =

n i=1

ai wi Xω1

≥

n 1 ai xi X ≥ C i=1

≥

n 1 | ai | = |ft ( ai vi )|. C i=1

(VII.4)

i∈I

If t is not terminal node, then we use the appropriate (An2j , m12j )-operation.

Deﬁnition VII.13. Let (xα )α<γ be a normalized transﬁnite block sequence, X its closed linear span. We denote by D(X) the space of all bounded diagonal operators D : X → X satisfying the property that for all α < γ limit there exists some λα ∈ R such that D(xβ ) = λα xβ for every β ∈ [α, α + ω). We also denote by D(X) the space of all diagonal operators (not necessarily bounded) satisfying the above condition acting on xα α<γ . Notice the following (linear) decomposition of xα α<γ , . (VII.5) xα α<γ = α∈Λ(γ) xβ β∈[α,α+ω) . The canonical decomposition of y ∈ xα α<γ in X is y = y1 + · · · + yn given by (VII.5). Remark VII.14. D(X) is a closed subalgebra of L(X). For an ordinal µ we denote by Λ(µ)(0) the set of limit ordinals α < µ such that α = β +ω for a limit ordinal β. We denote this β by α− . For technical reasons, we assume that 0 is a limit ordinal. Remark VII.15. Notice that for γ limit, Λ(γ + 1)(0) is order isomorphic to Λ(γ) considering the predecessor map. Deﬁnition VII.16. Let D ∈ D(X). We deﬁne the map ξD : Λ(γ + 1)(0) → R via D(xα− ) = ξD (α)xα− . Namely, ξD (α) is the eigenvalue of D associated to the eigenvectors (xβ )β∈[α− ,α) .

Chapter VII. Finite Representability of JT0 and the Diagonal Space D(Xγ )

85

We consider the linear map Ξ : D(X) → c00 (Λ(γ + 1)(0) )# deﬁned by Ξ(D)(vα ) = ξD (α),

(VII.6)

where c00 (Λ(γ + 1)(0) )# denotes the algebraic conjugate of c00 (Λ(γ + 1)(0) . The main goal here is to show that Ξ deﬁnes an isomorphism between D(X) and JT∗0 (Λ(γ + 1)(0) ). For D ∈ D(X), let us denote D = sup{DxXω1 : x ∈ xα α<γ , xXω1 ≤ 1} ≤ ∞, and for f ∈ c00 (Λ(γ + 1)(0) )# , f = sup{f (x) : x ∈ c00 (λ(γ + 1)(0) ), xJT0 ≤ 1} ≤ ∞. Proposition VII.17. D ≤ Ξ(D) ≤ CD for every D ∈ D(X). Proof. Fix D ∈ D(X), and ε > 0. Let y ∈ xα α<γ with y ≤ 1 be such that |D − Dy| < ε. Let y = y1 + · · · + yn be the canonical decomposition of y in X, and α1 , . . . , αn be such that yi ∈ xβ β∈[α− ,αi ) for every 1 ≤ i ≤ n. Let φ ∈ K i be such that Dy = φ(Dy), and set r = φy i i for i = 1, . . . , n. By Proposition n VII.11, i=1 ri vi JT0 ≤ x, and since (vα )α is 1-subsymmetric we have that n i=1 ri vαi JT0 ≤ y ≤ 1. Hence Ξ(D) ≥

Ξ(D)(

n

ri vαi )JT0 =

i=1

≥

n

n

ξD (αi )ri vαi JT0

(VII.7)

i=1

ξD (αi ) = φ(Dy) ≥ D − ε.

i=1

n This shows that D ≤ Ξ(D). Fix v = i=1 ai vαi ∈ JT0 with vJT0 ≤ 1, and choose a ﬁnite normalized block sequence (wi )ni=1 C-equivalent to (vαi )ni=1 with wi ∈ xβ β∈[α− ,αi ) for every i = 1, . . . , n (indeed we may assume that the i natural isomorphism F : wi ni=1 → vi ni=1 satisﬁes that F ≤ 1, F −1 ≤ C; see Corollary B.15). Then, Ξ(D)(v)JT0 = ni=1 ξD (αi )ai vαi JT0 ≤ ni=1 ξD (αi )ai wi Xω1 = D(

n

ai wi )Xω1 ≤ D

n i=1

ai wi Xω1 ≤ CD.

i=1

(VII.8)

Theorem VII.18. The spaces D(X) and JT∗0 (Λ(γ + 1)(0) ) are isomorphic. Proof. By Proposition VII.17, Ξ|D(X) : D(X) → JT∗0 (Λ(γ + 1)(0) ) is an isomorphism. To see that it is also onto consider f ∈ JT∗0 (Λ(γ + 1)(0) ) and deﬁne Df ∈ D(X) as follows. For β ∈ [α− , α) set Df (xβ ) = f (vα )xβ . It is easy to check that Ξ(Df ) = f . This completes the proof.

86

Chapter VII. Finite Representability of JT0 and the Diagonal Space D(Xγ )

Corollary VII.19. Let X and Y be the closed linear span of two transﬁnite block sequences of the same length γ. Then the natural mapping ψγ : D(X) → D(Y ) deﬁned by ψγ (D) = DξD is an isomorphism. Our intention now is to compare D(X) and D(Xω1 ). A (Xω ) be the subalgebra Deﬁnition VII.20. 1. Given a closed A ⊆ Λ(ω1 +1), let D 1 of D(Xω1 ) consisting on all D ∈ D(Xω1 ) satisfying that for every α ∈ A(0) , there is some λα such that D|X[α− ,α) = λα iX[α− ,α),Xω and D|X[max A,ω1 ) = 0. Let DA (Xω1 ) 1 A (Xω ). be the subalgebra of bounded operators of D 1 2. Given a transﬁnite block sequence (xα )α<γ , let ΓX ⊆ Λ(ω1 + 1) be deﬁned as follows. Let Γ = {supn→∞ max supp xαn : (αn )n ↑, αn < γ}, (VII.9) and let ΓX = Γ ∪ {0, sup Γ }. Another interpretation of ΓX is to consider the map fX : Λ(γ + 1) → ω1 deﬁned by fX (α) = supβ<α max supp xβ and ΓX is nothing else but the image f (Λ(γ + 1)), and hence ΓX \ max{ΓX } and Λ(γ + 1)(0) are order isomorphic. Γ (Xω ) be the unique extension of D. 3. Given D ∈ D(X), let E(D) ∈ D X 1 Notice that D|X ∈ D(X) for every D ∈ DΓX (Xω1 ). Proposition VII.21. ED ≤ CD for every D ∈ D(X). Moreover the restriction DΓX (Xω1 ) → D(X), D → D|X is an isomorphism onto with inverse E : D(X) → DΓX (Xω1 ). Proof. We show that E(D) ≤ CD for every D ∈ D(X). Fix a ﬁnitely supported y ∈ Xω1 such that y ≤ 1 and E(D) = E(D)(y). Since I = (0) {[αΓ−X , α) : α ∈ ΓX } ∪ {[max ΓX , ω1 )} is a partition of ω1 , y has a unique decomposition y = y1 + · · · + yn for I1 < · · · < In in I and yi ∈ eα α∈Ii . Notice that E(D)|X[max ΓX ,ω1 ) = 0, so we may assume that In = [max ΓX , ω1 ). By deﬁnition −1 (αi ) for every of E(D) we have that E(D)(y) = ni=1 ξD (βi )yi where βi = fX i = 1, . . . , n. Choose φ ∈ Kω1 such that E(D)(y) = φ(E(D)(y)). By Proposition VII.17, E(D) =φ( ni=1 ξD (βi )yi ) = ni=1 ξD (βi )φ(yi ) = Ξ(D)( ni=1 φ(yi )vβi ) ≤Ξ(D)JT∗

0

(Λ(γ+1)(0) )

n i=1

(VII.10) φ(yi )vi JT0 ≤ CD.

(VII.11)

Chapter VIII

The Spaces of Operators L(Xγ ), L(X, Xω1 ) This is a continuation of the previous part, and the main result is that for every subspace X of Xω1 there exists a closed subset ΓX of ω1 such that L(Xω1 ) is isomorphic to JT∗0 (ΓX ) ⊕ S(X, Xω1 ). This representation yields some further properties for the space Xω1 , for example for I, J disjoint intervals of ω1 , the corresponding spaces XI , XJ are totally incomparable, which yields that Xω1 is arbitrarily distortable. The spaces L(Xγ ) Deﬁnition VIII.1. A sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) is called a (0, j)-dependent sequence if the following conditions are fulﬁlled: DS0.1 Φ = (φ1 , . . . , φn2j+1 ) is a 2j + 1-special sequence and φi (xi ) = 0 for every 1 ≤ i, i ≤ n2j+1 . DS0.2 There exists {ψ1 , . . . , ψn2j+1 } such that w(ψi ) = w(φi ), # supp xi ≤ ≤ w(φi+1 )/n22j+1 and (xi , ψi ) is a (6, 2ji )-exact pair for every 1 ≤ i ≤ n2j+1 . DS0.3 If H = (h1 , . . . , hn2j+1 ) is an arbitrary 2j + 1-special sequence, then ' κΦ,H
( ' ( supp xi ∩ κΦ,H
(VIII.1)

Proposition VIII.2. For every (0, j)-dependent sequence (x1 , φ1 , . . . , xn2j+1 , φn2j+1 ) we have that 1 1 (x1 + · · · + xn2j+1 ) ≤ 2 . n2j+1 m2j+1

88

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

Proof. The proof is rather similar to the proof of Proposition VI.9. One ﬁrst shows that xi )| ≤ 12(1 + #E/n22j+1 ) |ψ(1/n2j+1 i∈E

for every special functional ψ with w(ψ) = m2j+1 , and then the result follows n2j+1 from the basic inequality, since, by condition (DS0.2), (xi )i=1 is a (12, 1/n22j+1 )RIS. Proposition VIII.3. Suppose that (yk )k is a (C, ε)-RIS, and suppose that T : yk k → Xω1 is a linear function (not necessarily bounded) such that limn→∞ d(T yn , Ryn ) = 0. Then for every ε > 0 there is some z ∈ yk k such that z < εT z. Proof. We may assume that there is some δ > 0 such that inf n d(T yn , Ryn ) > δ > 0, and also that (T yn )n is a block sequence (hint: consider the following limit ordinal γ0 = min{γ < ω1 : ∃A ∈ [N]∞ inf n∈A d(Pγ T yn , Ryn ) > 0}, and pass, if necessary, to a subsequence of (yn )n and replace T by Pγ0 T ). Claim. There exist an inﬁnite set A ⊆ N and a block sequence (fn )n∈A of functionals in K such that: (a) For every n ∈ A, fn T yn ≥ δ, fn yn = 0, ran fn ⊆ ran T yn and supp fn ∩ supp ym = ∅ for every m = n. (b) Either for every n ∈ A, max supp yn ≥ max supp fn or for every n ∈ A, max supp yn ≤ max supp fn . Proof of the claim. By the Hahn–Banach theorem, for each n ∈ N we can ﬁnd a functional fn of norm 1 such that fn (T yn ) ≥ δ and fn (yn ) = 0. Since the w∗ closure of K is BX∗ω (notice that K by deﬁnition is closed under rational convex 1 combinations) and K is closed under intervals, we may assume that fn ∈ K and ran fn ⊆ ran T yn . Let α = maxn supp yn and β = maxn supp fn . If α = β, it is rather easy to achieve the desired result. If α = β, then we can pass to a subsequence A and distort fn such that for every n ∈ A, max supp fn ≥ max supp yn . So, we may assume that (fn )n≥1 satisﬁes the requirements of the previous claim. Fix now j such that m2j+1 > 12/(εδ). Claim. There is a (0, j)-dependent sequence (z1 , φ1 , . . . , zn2j+1 , φn2j+1 ) such that for every k ≤ n2j+1 , zk ∈ X, ran φk ⊆ ran T zk and φk T zk > δ. Proof of the claim. Choose j1 even such that m2j1 > n22j+1 , and choose F1 ⊆ N of size n2j1 such that (yk )k∈F1 is a (3, 1/n22j1 )-RIS (going to a subsequence of (yk )k if necessary). Set φ1 = m12j

1

i∈F1

fi ∈ Kω1 and z1 =

m2j1 n2j1

k∈F1

yk .

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

89

fk T yk > δ and by (a) from the above claim, we Note that φ1 T z1 = (1/n2j1 ) k∈F1 have that φ1 z1 = (1/n2j1 ) k∈F1 l∈F1 fk (yl ) = 0. Pick p1 ≥ max{p (supp z1 ∪ supp T z1 ∪ supp φ1 ), # supp z1 ·n22j+1 } and set 2j2 = σ (Φ1 , m2j1 , p1 ). Now choose ﬁnite of length n2j2 such that (xk )k∈F2is a (3, 1/n22j2 )-RIS. Set φ2 = F 2 > F1 (1/m2j2 ) k∈F2 fk ∈ Kω1 and z2 = (m2j2 /n2j2 ) k∈F2 yk . Notice that φ2 > φ1 , φ2 T z2 > δ and φ2 z2 = 0. Pick p2 ≥ max{p1 , p (supp z1 ∪ supp z2 ∪ supp T z1 ∪ supp T z2 ∪supp Φ1 ∪supp Φ2 ), # supp z2 ·n22j+1 }} and set 2j3 = σ (φ1 , m2j1 , p1 , φ2 , m2j2 , p2 ), and so on. Let us check that (z1 , φ1 , . . . , zn2j+1 , φn2j+1 ) is a (0, j)-dependent sequence: Conditions (DS0.1) and (DS0.2) are rather easy to check from the deﬁnition of this sequence. Let us check (DS0.3). There are two cases: (a) Suppose that max supp zk ≤ max supp φk for every 1 ≤ k ≤ n2j+1 . Then supp zk ⊆ pλ −1 supp φλΦ,H −1 Φ,H for every κΦ,H < k < λΦ,H . Then part 2 of (TP.3) gives the desired result. (b) Suppose that max supp φk ≤ max supp zk for every 1 ≤ k ≤ n2j+1 , then supp φk ⊆ supp zλΦ,H −1 pλΦ,H −1 for every κΦ,H < k < λΦ,H , and we are done by part 1 of (TP.3). Fix a (0, j)-dependent sequence (z1 , φ1 , . . . , zn , φn2j+1 ) as in the claim, and set z=

1 n2j+1

n2j+1 k=1

(−1)k+1 zk and φ =

1 m2j+1

n2j+1 k=1

φk .

n2j+1 (−1)k+1 φT zk ≥ δ/m2j+1 and z ≤ 12/m22j+1 . So, Then φT z = 1/n2j+1 k=1 T (z) ≥ δ/m2j+1 ≥ δm2j+1 z/12 > εz as desired. Corollary VIII.4. Let (yk )k be a (C, ε)-RIS, Y its closed linear span and T : Y → Xω1 be a bounded operator. Then limn→∞ d(T yk , Ryk ) = 0. Proof. If not, by previous Proposition VIII.3, we can ﬁnd a vector z ∈ yk k such that z < (1/T )T z which is impossible if T is bounded. Lemma VIII.5. Let (xn )n be a (C, ε)-RIS, X its closed span and T : X → Xω1 be a bounded operator. Then λT : N → R deﬁned by d(T xn , Rxn ) = T xn − λT (n)xn is a convergent sequence. Proof. Fix any two strictly increasing sequences (αn )n and (βn )n with supn αn = supn βn , and suppose that λT (αn ) →n λ1 , λT (βn ) →n λ2 . By going to a subsequences, we can assume that xαn < xβn for every n. Since the closed linear span of {xαn }n ∪ {xβn }n is an HI space, we can ﬁnd for every ε two normalized vectors w1 ∈ xαn n and w2 ∈ xβn n such that T w1 −λ1 w1 ≤ ε/3, T w2 −λ2 w2 ≤ ε/3 and w1 − w2 ≤ ε/3T . Then we have that λ1 w1 − λ2 w2 ≤ T w1 − λ1 w1 + T w1 − T w2 + T w2 − λ2 w2 ≤ ε, (VIII.2) and hence, ε ≥ λ1 w1 − λ2 w2 ≥ |λ1 − λ2 |w1 − |λ2 |w1 − w2 ≥ |λ1 − λ2 | − |λ2 |ε. (VIII.3) So, |λ1 − λ2 | ≤ ε(1 + |λ2 |) for every ε. This implies that λ1 = λ2 .

90

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

Deﬁnition VIII.6. Recall that for a set A of ordinals, A(0) is the set of isolated points of A. Fix a transﬁnite block sequence (xα )α<γ , let X be the closed linear span of it and let T : X → Xω1 be a bounded operator. We deﬁne the step function ξT of T , ξT : Λ(γ +1)(0) → R as follows: Let γ be a successor limit ordinal less than γ. Let ξT (γ) = ξ ∈ R be such that limn→∞ T yn − ξyn = 0 for every (3, ε)-RIS (yn )n satisfying that supn max supp yn = γ. Lemma VIII.5 shows that ξ exists and is unique, and that ξT can be extended to a continuous ξT : Λ(γ + 1) → R. Given a mapping ξ : Λ(γ + 1)(0) → R we deﬁne the diagonal, not necessarily bounded, operator Dξ : X → X in the natural way by Dξ (xα ) = ξ(α+ω)xα . Given a bounded T : X → Xω1 we deﬁne the diagonal step operator DT : xα α<γ → Xω1 of T as DT = DξT . Remark VIII.7. ξT has only countable many values, since it can be extended to a continuous mapping ξT deﬁned in Λ(γ + 1). Proposition VIII.8. The sequence ((T − DT )(yn ))α<γ belongs to c0 (N) for every RIS (yn )n in X. Proof. This is just a consequence of the deﬁnition of DT .

Proposition VIII.9. A bounded operator T : X → Xω1 is strictly singular iﬀ ξT = 0. Proof. Suppose that T is not strictly singular. Then there is a block sequence (yn )n such that T is an isomorphism restricted to the closed linear span Y of (yn )n . Going to a block subsequence if necessary we assume that (yn )n is a RIS. Since T |Y is an isomorphism, limn→∞ T yn > 0. This implies that ξT |Λ(α + 1)(0) = 0, since otherwise ξT (α) = 0, contradicting the above inequality. Suppose now that ξT = 0. Choose some successor limit γ such that ξT (γ) = 0. Then we can ﬁnd a block sequence (yn )n ⊆ Xγ such that T is close enough to ξT (γ)iY,Xω1 , where Y is the closed linear span of (yn )n . Hence, T is not strictly singular. Proposition VIII.10. Let (xα )α<γ be a transﬁnite block sequence, X its closed linear span of (xα )α<γ and a bounded operator T : X → Xω1 . Then DT ≤ CT and hence DT ∈ D(X). Proof. Fix a normalized y ∈ xα α<γ , let y = y1 + · · · + yn be its decomposition in X, yi ∈ xβ β∈[α− ,αi ) for i = 1, . . . , n. Choose φ ∈ Kω1 such that φ(D(y)) = i D(y). Then, n n n n D(y) = i=1 ξT (αi )φ(yi ) = ( i=1 ξT (αi )vi∗ )( i=1 φ(yi )vi ) ≤ i=1 ξT (αi )vi , (VIII.4) the last inequality because ni=1 φ(yi )vαi JT0 ≤ yXω1 ≤ 1. We ﬁnish with the next claim. n Claim. i=1 ξT (αi )vi∗ JT∗ ≤ CT . 0

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

91

Proof of the claim. Fix ε > 0. By the ﬁnite block representability of JT0 in Xω1 and Proposition VIII.8 we can produce inductively w1 , . . . , wn such that (1) wi ∈ xβ β∈[α− ,αi ) , i (2) the natural isomorphism F : wi ni=1 → vi ni=1 is such that F ≤ 1 and F −1 ≤ C, and (3) ni=1 ξT (αi )wi − T wi < ε. n n ∗ ∗ Choose x = i=1 ri vi ∈ JT0 of norm 1 such that i=1 ξT (αi )vi JT0 = n n i=1 ξT (αi )ri . Then i=1 ri wi Xω1 ≤ C and hence DT (

n

i=1 ri wi ) ≥

≥ Hence ε.

n

∗ ∗ i=1 ξT (αi )vi JT0

≤ T (

n

ri ξT (αi )vi JT0

i=1 n

n

i=1

i=1

ξT (αi )ri =

n

(VIII.5) ξT (αi )vi∗ JT∗ . 0

i=1 ri wi )+(T −DT )(

n

i=1 ri wi )

≤ CT +

Theorem VIII.11. Let (xα )α<γ be a normalized block sequence of Xω1 , X its closed linear span. Then, for every bounded operator T : X → Xω1 , DT : X → Xω1 is bounded and T − DT is strictly singular. Proof. It follows from Proposition VIII.9 and Proposition VIII.10.

Corollary VIII.12. Any bounded operator from the closed linear span X of a transﬁnite block sequence into the space Xω1 is the sum of the restriction of a unique diagonal operator D ∈ DX (Xω1 ) and an strictly singular operator. Proof. This follows from previous theorem and Proposition VII.21.

Corollary VIII.13. (a) For T : X → Xω1 bounded TFAE: (i) T is strictly singular, (ii) ξT = 0, and (iii) DT = 0. (b) The transformation T → DT is a projection in the operator algebra L(X) of norm ≤ C. Proposition VIII.14. Let X → Xω1 , I ⊆ ω1 an interval such that PI |X is not strictly singular. Then for every ε > 0 there exist a normalized sequence (xn )n in X and a normalized block sequence (zn )n in XI such that n yz − zn < ε. Proof. Set I = [α, β] and suppose that PI |X is not strictly singular. Let γ0 = {γ ∈ (α, β] : Pγ |X is not strictly singular}. We can ﬁnd for every ε > 0, (yn )n ⊆ X and a block sequence (wn )n ⊆ Xγ0 such that Pγ0 is an isomorphism when restricted to the closed linear span of (yn )n ,

92

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

supn max supp wn = γ0 and n wn − Pγ0 yn ≤ ε/2. Consider U : wn n → X[γ0 ,ω1 ) deﬁned by U wn = P[γ0 ,ω1 ) yn . Notice that U is bounded. Since ξU = 0, U is strictly singular. Hence we can ﬁnd a block sequence (zn )n of (wn )n such that n+1 for all n, U zn ≤ ε/2 and hence the corresponding block sequence (xn )n of (yn )n satisﬁes that n zn − xn ≤ ε. Finally, notice that for n0 large enough, (zn )n≥n0 ⊆ XI . Corollary VIII.15. Xω1 is arbitrarily distortable. Proof. For j ∈ N, and x ∈ Xω1 , let x2j = sup{φ(x) : w(φ) = m2j }. Let X → Xω1 . Since for every ε > 0 we can ﬁnd a subspace of X generated by a Schauder basis (yn )n and a normalized block sequence (zn )n of Xω1 such that n yn −zn ≤ ε, without loss of generality we can assume that X is generated by a block sequence (zn )n . Now, we can ﬁnd an (6, j)-exact pair (x, φ), with x ∈ zn n and hence 1 ≤ x2j ≤ x ≤ 6. And for any other j > j, a (6, 2j )-exact pair (x , φ ) with x ∈ zn n and hence 1 ≤ x ≤ 6 and x 2j ≤ 12/m2j . So, x/x2j x /x 2j

≥

1/6 12/m2j+1

=

m2j+1 72 .

(VIII.6)

Deﬁnition VIII.16. Two Banach spaces X and Y are called totally incomparable if and only if no inﬁnite dimensional closed X1 → X is isomorphic to Y1 → Y . Corollary VIII.17. XI and XJ are totally incomparable for disjoint inﬁnite intervals I and J. Proof. Suppose not, and let X → XI , and Y → XJ such that T : X → Y is an isomorphism onto. By previous Proposition VIII.14, we can assume that X is generated by a block sequence. But since ξT = 0, T cannot be an isomorphism, a contradiction. Another consequence of the representability of JT0 on each transﬁnite block sequence is that we can identify the space D(X) of diagonal step operators on X and hence L(X)/S(X) for every closed span X of a transﬁnite block sequence. (0) Corollary VIII.18. L(X)/S(X) ∼ = L(X, Xω1 )/S(X, Xω1 ) ∼ = JT∗0 (ΓX ) for every X → Xω1 generated by a transﬁnite block sequence. (0)

Proof. This follows from Lemma VII.18, since Λ(γ + 1)(0) and ΓX are orderisomorphic. Remark VIII.19. Note that L(X)/S(X) ∼ = JT∗0 (ΓX ) if ΓX is inﬁnite. To see this, ﬁx a transﬁnite block sequence (xα )α<γ generating X such that γ ≥ ω 2 . Then ΓX \ {max ΓX } and Λ(γ + 1)(0) \ {ω} are order-isomorphic. Theorem VIII.20. Every projection P of Xω1 is of the form P = PI1 +· · ·+PIn +S, where Ii are intervals of ordinals, Ii < Ii+1 and S is strictly singular.

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

93

Proof. Suppose that P : Xω1 → Xω1 is a projection, P = DP + S. Since P 2 = P , we obtain that DP2 − DP is also strictly singular and therefore (ξP (α)2 − ξP (α))iX[α− ,α) ,Xω1 is strictly singular for every successor limit α. This implies that ξP : Λ(ω1 + 1)(0) → {0, 1}. And since ξP has the continuous extension property, there is no strictly increasing sequence {αn }n ⊆ Λ(ω1 +1)(0) such that ξP (α2n ) = 1 and ξP (α2n+1 ) = 0 for every n. Corollary VIII.21. For every n ∈ N there is some m ∈ N such that for every projection P of Xω1 with P ≤ n, P can be written as P = PI1 + · · · + PIk + S such that k ≤ m and I1 << I2 << · · · << Ik , where A < 2nC. We claim that m = n2j works. For suppose that P = PI1 + · · · + PIk + S with I1 << · · · << Ik and k > n2j . Fix ε > 0. Find a normalized block sequence (x1 , y1 , . . . , xn2j /2 , yn2j /2 ) such that (a) xi ∈ XIi , yi ∈ X(sup Ii ,min Ii+1 ) for 1 ≤ i ≤ n2j /2 − 1, and yn2j /2 > xn2j /2 , n2j (b) (x1 , y1 , . . . , xn2j /2 , yn2j /2 ) is C-equivalent to (vi )i=1 , and (c) S|F ≤ ε where F = (x1 , , y1 , . . . , xn2j /2 , yn2j /2 ). Set x = x1 − y1 + · · · + xn2j /2 − yn2j /2 . Then, x ≤ C

n2j

i=1 (−1)

i+1

vi JT0 ≤ C

n2j

i=1 ti T0

= Cn2j /m2j ,

(VIII.7)

vi JT0 − ε = n2j /2 − ε.

(VIII.8)

and P (x) ≥

n2j /2 i=1

xi − ε ≥

n2j /2 i=1

(VIII.7) and (VIII.8) imply that P ≥ (m2j /2 − εm2j /n2j )/C. Hence, P > n, a contradiction. Asymptotically equivalent subspaces and L(X, Xω1 ) Our aim here is to extend the results about operators on subspaces generated by a transﬁnite block sequence to arbitrary subspaces. Deﬁnition VIII.22. Let X be a subspace of Xω1 . A subset Γ of ω1 + 1 is said to be a critical set of X if the following hold: (CS1) Γ is closed of limit ordinals, and 0 ∈ Γ. (CS2) For all γ ∈ Γ, γ < Ω, P(γ,γ + ) |X is not strictly singular and for all α ∈ (γ, γ + ), P(γ,α) |X is strictly singular, where γ + is the successor of γ in Γ and Ω = max Γ. (CS3) P[Ω,ω1 ) |X is strictly singular (we use P∅ = 0).

94

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

Notice that from the above deﬁnition, it follows easily that if Γ is a critical set of X, then max Γ = min{γ ≤ ω1 : P[γ,ω1 ) |X is strictly singular}. Proposition VIII.23. For every X → Xω1 a critical set Γ is uniquely deﬁned, denoted by ΓX . Proof. Fix X → Xω1 . We show ﬁrst that a critical set X exists. We proceed by induction deﬁning an increasing sequence (γα )α<ω1 as follows: We set γ0 = 0. Suppose deﬁned (γβ )β<α satisfying conditions (CS1) and (CS2). If α is limit, then we set γα = supβ<α γβ . Suppose now that α is successor. If P[γα− ,ω1 ) |X is strictly singular, then we set γα = γα− . If not, let γα = min{γ ∈ (γα− , ω1 ) : P[γα− ,γ) |X is not strictly singular}. Let us observe that if X is separable, then the sequence (γα )α<ω1 is eventually constant and we set ΓX = {γα }α<ω1 . If X is non separable, then the sequence (γα )α<ω1 is strictly increasing and ΓX = {γα }α<ω1 ∪ {ω1 }. Next we prove the uniqueness of ΓX . Suppose the opposite, and ﬁx Γ = Γ two diﬀerent critical sets. Set γ = max(Γ ∩ Γ ). First notice that max Γ = max Γ . So, either γΓ+ < γΓ+ or γΓ+ < γΓ+ . This leads to a contradiction using the fact that both Γ and Γ satisfy (CS2). Remark VIII.24. 1. The critical set ΓX provides information concerning the structure of the space X. For example the space X is HI if and only if ΓX = {0, ΩX }. Also, two subspaces X, Y → Xω1 are totally incomparable if and only if ΓX ∩ΓY = {0}. 2. For a transﬁnite block sequence (xα )α<γ its critical set is nothing else but the set introduced from Deﬁnition VII.20 (2). Proposition VIII.25. For every Y → X, the corresponding critical set ΓY is a subset of ΓX . Proof. It follows by an easy inductive argument.

Proposition VIII.26. For every separable X → Xω1 and for every ε > 0 there exist an ordinal γ < ω1 , a normalized sequence (y α )α<γ in X and a normalized transﬁnite block sequence (zα )α<γ such that (a) α<γ zα − xα < ε and (b) ΓX = ΓZ where Z is the closed linear span of (zα )α<γ . Proof. Use Proposition VIII.14, and a standard gliding hump argument.

Deﬁnition VIII.27. Let X, Y → Xω1 . (i) We say that X is asymptotically ﬁner to Y , X ≤a Y , if and only if ΓX ⊆ ΓY . (ii) We say that X is asymptotically equivalent to Y , X ≡a Y , if and only if Γ X = ΓY .

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

95

It follows easily from the above deﬁnition that the relation ≤a is a quasiordering in the class of the subspaces of Xω1 which from Proposition VIII.25 extends the natural inclusion. Notice also that ≡a is an equivalence relation. We now give two alternative formulation of these notions. Proposition VIII.28. For X, Y → Xω1 the following are equivalent: (1) X ≤a Y , (2) if PI |X is not strictly singular, then PI |Y is not strictly singular, for every interval I ⊆ ω1 , and (3) d(SX , SY ) = 0 for every X → X. Proof. Let us observe that for a closed inﬁnite interval I, PI |X is not strictly singular iﬀ there is some γΓ+X ∈ ΓX with min ΓX < γΓ+X ≤ max I. The inverse direction follows immediately from the deﬁnition of the critical sets. So assume now that PI |X is not strictly singular. Set γ0 = max{γ ∈ ΓX : γ ≤ min I}. Observe that γ0 ≤ min I < ΩX , hence min ΓX < γΓ+X ≤ max I by minimality of γΓ+X (Property (CS2)). It is easy to see that the above observation implies easily the equivalence (1) ⇔ (2). (1) ⇒ (3): Suppose that X → X. Then by Proposition VIII.25 and our assumption, ΓX ⊆ ΓY . By Proposition VIII.26, we can ﬁnd two block sequences (zn )n and (wn )n in X0+ such that X

Γ

(a) supn max supp zn = supn max supp wn = 0+ ΓX , and (b) d(SZ , SX ) = d(SW , SY ) = 0 where Z and W are the closed linear span of (zn )n and (wn )n respectively. By Corollary VI.12, d(Z, W ) = 0 and we are done. (3) ⇒ (2): Since for every X → X, d(SX , SY ) = 0, we obtain that for every ε > 0, and every X → X there exists two basic sequences (zn )n and (wn )n such that zn ∈ SX and wn ∈ SY for all n and n zn − wn < ε. Assume now that PI |X is not strictly singular. Choose X → X such that PI |X is isomorphism. Let (zn )n ⊆ X and (wn )n ⊆ Y as above. Then PI |W is isomorphism and hence PI |Y is not strictly singular. Proposition VIII.29. For X, Y → Xω1 the following are equivalent: (1) X ≡a Y , (2) PI |X is not strictly singular if and only if PI |Y is not strictly singular, for every interval I ⊆ ω1 , and (3) d(SX , SY ) = d(SY , SX ) = 0 for every X → X, Y → Y . Corollary VIII.30. 1. For every X → Xω1 and every A ⊆ ΓX there is XA → X such that ΓXA = A. 2. For every non separable X, Y → Xω1 there are non separable X1 → X, Y1 → Y such that X1 ≡a Y1 . We also need the following well known result. Lemma VIII.31. For every Z → X → Xω1 with Z separable there exist W → Z and γ < ω1 such that Pγ |X is a projection onto W . Remark VIII.32. Notice that for W and X as in the lemma, ΓW is an initial part of ΓX .

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

96

Proposition VIII.33. Let X be a subspace of Xω1 and T : X → Xω1 a bounded operator. Then there exists a unique DT ∈ DΓX (Xω1 ) such that (a) DT ≤ 2C 2 T and (b) T − DT |X is strictly singular. Proof. Fix X → Xω1 and a bounded operator T : X → Xω1 . First suppose that X is separable. Then we can ﬁnd a transﬁnite basic sequence (yα )α<γ ⊆ X and a transﬁnite block sequence (zα )α<γ of Xω1 such that α<γ yα − zα < 1 and X ≡a Z, where Z denotes the closed linear span of (zα )α<γ . Consider now T : U

T |Y

Z → Y → Xω1 where Y is the closed linear span of (yα )α<γ and U : Y → Z is the isomorphism deﬁned by U ( α<γ aα zα ) = α<γ aα yα . Notice that U ≤ 2. Then there is a unique D ∈ D(Y ) such that T − D is strictly singular, or equivalently there is unique DT ∈ DΓZ (Xω1 ) such that T − DT |Z is strictly singular. Notice that DT ≤ CD ≤ C 2 T ≤ C 2 U T ≤ 2C 2 T . Let us show that T − DT is strictly singular. Let X → X and ε > 0. Choose Z → Z such that (ΓZ \ {0}) ∩ (ΓX \ {0}) = ∅, U |Z − iZ ,Xω1 ≤ ε/(4T ) and (T −DT )|Z ≤ ε/4. Pick z ∈ Z and x ∈ X such that z −x ≤ ε/(2(DT + T )). Then (T − DT )x ≤(T − DT )x − (T − DT )z + (T − DT )z ε ≤T x − U z + DT x − z + 4 ε (VIII.9) ≤(T + DT )x − z + ≤ ε. 2 Now suppose that X is non separable. By Lemma VIII.31, we can ﬁnd a sequence (Xγ )γ<ω1 of separable complemented subspaces of X such that ΓXγ is an initial part of ΓX for every γ < ω1 . Now the result for X easy follows from the result for the corresponding Tγ = T |Xγ and the fact that DT ∈ DΓX (Xω1 ) and DTγ ∈ DΓXγ (Xω1 ) are unique. The uniqueness of DT ∈ DΓX (Xω1 ) is clear from the analogous result for transﬁnite block sequences. (0) Theorem VIII.34. L(X, Xω1 ) ∼ = DΓX (Xω1 ) ⊕ S(X, Xω1 ) ∼ = JT∗0 (ΓX ) ⊕ S(X, Xω1 ) for every X → Xω1 . If in addition ΓX is inﬁnite, then L(X, Xω1 ) ∼ = JT∗0 (ΓX ) ⊕ S(X, Xω1 ).

Proof. Let H : DΓX → L(X, Xω1 ) be deﬁned by D → D|X. Assume ﬁrst that X is separable. It is clear that D|X ≤ D. For an appropriate ε > 0, we can ﬁnd normalized (yα )α<γ and a normalized block sequence (zα )α such that ΓX = ΓZ and α zα − yα ≤ ε where Z the closed linear span of (zα )α<γ . Since by Proposition VII.21 D|Z ≥ D/C, for C the ﬁnite block representability constant of JT0 in Xω1 , we get that D/C ≤ D|Z ≤ (1 + ε)D|Y ≤ (1 + ε)D|X = (1 + ε)H(D). (VIII.10) Hence, H deﬁnes an isomorphism. To show that H is an isomorphism when X is non separable we use a family (Xα )α<ω1 of separable complemented subspaces

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

97

of X deﬁned as in previous proof. Proposition VIII.33 shows that L(X, Xω1 ) ∼ = DΓX (Xω1 ) ⊕ S(X, Xω1 ). For the latter isomorphism see Remark VIII.19. Notes and Remarks. Chapters VI, VII, VIII and the following appendices originate from [13]. It follows easily from the results of Chapter VIII that every T ∈ L(Xω1 ) is of the form λI +S with S an operator with separable range. The ﬁrst example of a non separable Banach space with few operators, in the above sense, was provided by S. Shelah ([57]) in the late 1970s assuming V = L. Ten years later S. Shelah and J. Steprans, using Todorcevic’ “square-bracket” coloring [60], were able to provide a ZFC example with few operators [58], and in 2001 H.M. Wark using interpolation methods and the previous space, gave a reﬂexive space with few operators [64]. None of the above examples controls the separable subspaces as it happens in Xω1 . There are variants of the ρ function which impose additional properties in Xω1 . Of particular interest is the universal ρ function, introduced in [13] which permits the local structure of Xω1 to be repeated almost everywhere. Using the σρ coding with the universal ρ function one can obtain an HI X with a Schauder basis (en )n which belongs to the asymptotic structure of itself in the sense of [47]. A consequence of the representability of D(Xω2 ) as JT∗ yields that JT∗ with the pointwise multiplication is a Banach algebra. Notice that a similar result for the classical James space [36] is given in [2]. The unconditional counterpart of Xω1 is also presented in [13]. This is a reﬂexive space with an unconditional basis not isomorphic to any of its proper subspaces. In particular for every inﬁnite A ⊂ ω1 denoted by XA = < {ea : a ∈ A} > the shift operator is not continuous. As we have mentioned in the introduction it remains open if there exist reﬂexive Banach spaces not containing an unconditional basic sequence and with dimension greater than ω1 . It is clear that the problem of deﬁning such a space is reduced to deﬁning an eﬃcient coding function σ. S. Todorcevic has extended the notion of a ρ function for cardinals higher than ω1 . However it is not yet clear how we can pass from these ρ functions to an eﬃcient σρ coding. To make more transparent the meaning of Theorem VIII.34, we state the following: Theorem. There exists a separable reﬂexive Banach space X admitting an inﬁnite dimensional Schauder decomposition X = n ⊕Xn such that, denoting by D(X) the class of bounded operators D : X → X with the property D|Xn = λn IXn for all n, the following hold: (i) L(X) ∼ = D(X) ⊕ S(X) ∼ = JT∗0 ⊕ S(X). (ii) For every subspace X of X there exists A ⊆ N which is either an initial ﬁnite interval or is equal to N such that L(X, X) ∼ = JT∗0 (A) ⊕ S(X, X). For example, the space X = Xω2 has all these properties. It is worth pointing out that D(X) is a natural class of operators which behaves similar to the class of

98

Chapter VIII. The Spaces of Operators L(Xγ ), L(X, Xω1 )

operators of the form λI + K with K compact diagonal. For example if xn ∈ Xn with xn = 1 and X = (xn )n then for every D ∈ D(X) we have that D|X = λI + K. Let us mention that there exists a reﬂexive HI space X with a Schauder basis (en )n admitting strictly singular diagonal operators, which are not compact [7]. Moreover the space of strictly singular diagonal operators in this example is non separable. The problem of the existence of a separable reﬂexive Banach space such that the diagonal operators are compact perturbations of the multiples of the identity seems to be also open.

Appendix A

Transﬁnite Schauder Basic Sequences The ﬁrst appendix concerns the presentation of some results related to transﬁnite (Schauder) bases. We recall one of the equivalent formulations of their deﬁnition. Deﬁnition A.1. Let X be a Banach space, and γ be an ordinal number. 1. A total family (xα )α<γ of elements of X (i.e., a family such that X = xα α<γ ) is said to be a transﬁnite basis if there exists a constant C ≥ 1 such that for every interval I of γ the naturally deﬁned map on the linear span of (xα )α<γ α<γ

λα xα →

α∈I

λ α xα

is extended to a bounded projection PI : X → XI = xα α∈I of norm at most C. 2. A transﬁnite basis (xα )α<γ of X is said to be bimonotone if for each interval I of γ, the corresponding projection PI has norm 1. 3. A transﬁnite basis (xα )α<γ of X is said to be unconditional if there exists a constant C ≥ 1 such that for all subset A of γ, the corresponding PA has norm at most C. 4. A transﬁnite basis (xα )α<γ of X is said to be 1-subsymmetricif for every n ∈ N, every α1 < α2 < · · · < αn < γ and every (λi )ni=1 ∈ Rn , ni=1 λi xi = ni=1 λi xαi . Remark A.2. 1. As in the case of the usual Schauder basis (i.e., γ = ω) the above deﬁnition is equivalent to the fact that each x ∈ X admits a unique representation as α<γ λα xα , where the convergence of these series is recursively deﬁned. 2. α<γ λα xα easily yields that for each convergent series The deﬁnition of λ x with (x ) a bounded family, the sequence of coeﬃcients (λα )α<γ α α α α<γ α<γ belongs to c 0 (γ). Furthermore, for every ε > 0 there exists a ﬁnite subset F of γ such that α∈F / λα xα < ε.

100

Appendix A. Transﬁnite Schauder Basic Sequences

3. For every transﬁnite basis (xα )α<γ the dual basis (x∗α )α<γ is also well deﬁned. ∗ As for the usual Schauder bases, (x∗α )α<γ is a w -total subset of X ∗ and each x∗ ∗ in X has a unique representation of the form α<γ x∗ (xα )x∗α where the series is w∗ -convergent. For a detailed study of transﬁnite Schauder bases we refer the reader to [59]. 4. If (xα )α<γ is a transﬁnite basis for the space (X, · ), then there exists an equivalent norm | · | on X such that (xα )α<γ is a bimonotone basis for the space (X, | · |). This norm is deﬁned by |x| = sup{PI (x) : I interval of γ}. In the sequel, for every ordinal γ we shall denote by c00 (γ) the vector space of all sequences (λα )α∈γ of real numbers such that the set {α < γ : λα = 0} is ﬁnite. We also denote by (eα )α<γ the natural Hamel basis of c00 (γ). It is an easy observation that every space X with a transﬁnite basis (xα )α<γ is isometric to the completion of c00 (γ) endowed with an appropriate norm. Moreover if K is a subset of c00 (γ) with the properties (a) {e∗α }α<γ ⊆ K and (b) for every φ ∈ K, φ∞ ≤ 1 and for every interval I of γ, the restriction φI = φ · χI of φ to I is also a member of K, then the norm deﬁned on c00 (γ) by xK = sup{|φ(x)| = φ, x : φ ∈ K} satisﬁes that the completion of (c00 (γ), ·K ) has (eα )α<γ as a transﬁnite bimonotone basis. Fix X with a transﬁnite basis (xα )α<γ . The support supp x of x ∈ X is the set {α < γ : x∗α (x) = 0}. For a given interval I ⊆ γ, let XI = PI X, and for α < γ, let Xα = X[0,α) . For x, y ∈ X ﬁnitely supported, we write x < y to denote that max supp x < min supp y. A sequence (yα )α<ξ is called a transﬁnite block subsequence of (xα )α<γ if and only if for all α < ξ, yα is ﬁnitely supported and for all α < β < ξ, yα < yβ . Notice that a transﬁnite block subsequence of a transﬁnite basis is always a transﬁnite basis of its closed linear span. Fix two Banach spaces X and Y . A bounded operator T : X → Y is an isomorphism iﬀ T X is closed and T is one-to-one. T is called strictly singular if it is not an isomorphism when restricted to any inﬁnite dimensional closed subspace of X (i.e., for all inﬁnite dimensional closed X → X, either T X is not closed or T |X is not 1 − 1). This is equivalent to saying that for all Y → X and ε > 0, there is an inﬁnite dimensional closed subspace Y of Y such that T |Y ≤ ε. It is well known that most of the structure of the inﬁnite dimensional closed subspaces of a separable Banach space X with a basis (xn )n is described by its block sequences. Namely that for every inﬁnite dimensional closed subspace Y of X and every ε > 0 there exists a normalized sequence in Y and a block sequence (wn )n of (xn )n which are 1 + ε-equivalent. The method used for the proof of this result is called the gliding hump argument. This result is not extendable in the case of the transﬁnite block sequences. For example, consider a biorthogonal basis (xα )α<ω·2 of a Hilbert space and set Y the subspace generated by the sequence (xn + xω+n )n .

Appendix A. Transﬁnite Schauder Basic Sequences

101

We now describe how block sequences are connected to the subspaces in the transﬁnite case. Proposition A.3. Let (xα )α<γ be a transﬁnite basis of X and Y an inﬁnite dimensional closed subspace X. Then there exists a λ ≤ γ and a closed subspace Z of Y such that 1. Pλ : Z → Xλ is an isomorphism. 2. For every ε > 0 there exists a semi-normalized block sequence (wn )n in Xλ and a normalized sequence (zn )n in Z such that n Pλ zn − wn < ε. 3. There exists a subspace Z of Z isomorphic to a block subspace of X. 4. If we additionally assume that Y has a Schauder basis (yn )n , then the sequence (zn )n in 2. can be chosen to be a block sequence of (yn )n . Proof. We assume that (xα )α<γ is a bimonotone basis. Let β0 = min{β : Pβ : Y → Xβ is not strictly singular}.

(A.1)

Let us show that λ = β0 is the required ordinal. Notice that β0 has to be necessarily a limit ordinal. Since Pβ0 is not strictly singular on Y , there exists a subspace Z of Y such that Pβ0 : Z → Xβ0 is an isomorphism. On the other hand for every γ < β0 , Pγ : Y → Xγ is strictly singular hence for every ε > 0 and every subspace Z of Z there exists W → Z such that Pγ |W < ε. Now we are ready to apply a modiﬁed gliding hump argument to obtain (zn )n , (wn )n as they are required in 2. Indeed for a given ε we choose (εn )n such that εn > 0, εn < ε/4. We choose a normalized z1 ∈ Z . Since β0 is a limit ordinal, there must exist γ1 < β0 such that P[γ1 ,β0 ) z1 < ε1 . Hence setting w1 = Pγ1 z1 we have that w1 − Pβ0 z1 < ε1 . Since Pγ1 : Z → Xγ1 is strictly singular there exists a normalized z2 ∈ Z and Pγ1 z2 < ε2 . Choose γ2 > γ1 such that P[γ2 ,β0 ) z2 < ε2 and set w2 = P[γ1 ,γ2 ) z2 . Observe that Pβ0 z2 − w2 < 2ε2 and w1 < w2 . Continuing in this manner we obtain (zn )n and (wn )n such that for all n, Pβ0 zn − wn ≤ 2εn , hence n

Pβ0 zn − wn ≤ ε/2.

(A.2)

Since we assume that the transﬁnite basis (xα )α<γ is bimonotone, (A.2) implies that (Pβ0 zn )n and (wn )n are equivalent. The desired property 3. follows from 2, while 4. results of a careful choice of (zn )n in 2. As we have mentioned in the introduction the manner that block subspaces saturate the subspaces of X is weaker than the corresponding result for spaces X with a basis (xn )n . In the next proposition we provide a suﬃcient condition which ensures the complete extension of the result from the Schauder bases to transﬁnite Schauder bases fulﬁlling the additional condition.

102

Appendix A. Transﬁnite Schauder Basic Sequences

Proposition A.4. Let (xα )α<γ be a transﬁnite basis of X. Assume that for every I, J disjoint intervals of γ the spaces XI and XJ are totally incomparable. Then for every Y closed inﬁnite dimensional subspace of X and every ε > 0 there exist (yn )n , (zn )n normalized sequences such that (yn )n ⊂ Y , (zn )n block sequence of (zα )α<γ and n yn − zn < ε. Proof. From Proposition A.3 there exists a subspace Z of Y and λ ≤ γ such that Pλ : Z → Xλ is an isomorphism. Assume that λ < γ and set I = [1, λ) and J = [λ, γ). Then PJ : Z → XJ is a strictly singular operator. Hence we may ﬁnd (w ), (z ) as in Proposition A.3 (2) such that P n J (zn ) < ε which yields that n n z − w < 2ε. n n n Deﬁnition A.5. A transﬁnite basis (xα )α<γ is called shrinking iﬀ for all (αn )n ↑, (xαn )n is shrinking in the usual sense (i.e., (x∗αn ) generates in norm the dual of the closed span of (xαn )n ). It is called boundedly complete iﬀ for all (αn )n ↑, (xαn )n is boundedly complete in the usual sense (i.e., (λn )n , if there is some nfor all sequence of scalars C > 0 such that for all n, i=1 λi xαi ≤ C, then i λi xαi converges in norm). The following result is the extension of the well-known James’ characterization of reﬂexivity [35] in the general setting of a Banach space with a transﬁnite basis. Proposition A.6. Let (xα )α<γ be a transﬁnite basis of X. Then X is reﬂexive iﬀ (xα )α<γ is shrinking and boundedly complete. Proof. The direct implication is consequence of the James’ characterization. The opposite requires the following two claims. Claim 1. If (xα )α<γ is shrinking then the biorthogonal basis (x∗α )α<γ generates in the norm topology the dual space X ∗ . Proof of the claim. Assume the contrary. Then there exists x∗ ∈ X ∗ not in the closed linear span Y of (x∗α )α<γ . Set β0 = min{β ≤ γ : Pβ∗ x∗ ∈ / Y }. Then Pβ∗0 x∗ ∈ / ∗ ∗ Y but for all γ < β0 , Pγ x ∈ Y . Therefore there exists an increasing sequence of successive disjoint intervals I1 < I2 < · · · < In < · · · < β0 and ε > 0 such ∗ ∗ ∗ ∗ that for each that if x∗ ∈ X ∗ , n ∈ N, ∗PIn x ∈ Y and PIn x ≥ ε. Observe ∗ ∗ ∗ x = w − α<γ µα xα , where for each α < γ, µα = x (xα ). Moreover if I is an interval of γ such that PI∗ x∗ ∈ Y and ε > 0, then there is a ﬁnite subset Fε of I such that yε∗ − x∗ < ε, where yε∗ = w∗ − α∈γ\I µα x∗α + α∈Fε µα x∗α . Using this observation we inductively select ﬁnite sets F1 ⊆ I1 ,. . . , Fn ⊆ In such that setting yn∗ = ni=1 α∈Fi µα x∗α + Pβ0 \ ni=1 In x∗ , (A.3) we have that

Pβ∗0 x∗ − yn∗ < εn < 4ε .

(A.4)

Appendix A. Transﬁnite Schauder Basic Sequences

103

Set y ∗ = w∗ − limn yn∗ and (A.3) and (A.4) yield that supp yn∗ ⊆ n Fn and also PF∗n y ∗ > ε/2. Since each Fn is a ﬁnite set we can enumerate n Fn as (αn )n ↑ and clearly y ∗ yields that the sequence (xαn )n is not a shrinking Schauder basis, a contradiction. Claim 2. If (xα )α<γ is boundedly complete then for every x∗∗ ∈ X ∗∗ , the series ∗∗ ∗ α<γ x (xα )xα converges in norm. Proof of the claim. Suppose the contrary and ﬁx x∗∗ ∈ X ∗∗ but not in X. The proof is similar to the previous one. For each α < γ, let λα = x∗∗ x∗α and let β0 = min{β < γ : Pβ∗∗ x∗∗ ∈ / X}. Using a similar argumentwe can choose an increasing sequence (F n )n of ﬁnite subsets of γ such that w∗ − α∈ Fn λα x∗α exists and for every n, α∈Fn λα xα > n ε > 0. This yields that the sequence (xα )α∈ n Fn is not boundedly complete, a contradiction.

Appendix B

The Proof of the Finite Representability of JT0 The goal of this part is to prove the basic inequality and show the ﬁnite interval representability of the James-like space JT0 in Xω1 (Theorem VII.9). Reaching these two goals involves a similar sort of problems and for this reason we introduce a general theory applicable to both cases and hopefully to many other cases to come. General theory The theory deals with a block sequence of vectors (xk )nk=1 , a sequence of scalars (bk )nk=1 , and a functional f ∈ Kω1 , and tries to estimate |f ( nk=1 bk xk )| in terms of n |g( k=1 bk ek )| for an appropriately chosen functional g of an auxiliary Tsirelsonlike space X with basis (ei )i . The natural approach is to start with a tree-analysis (ft )t∈T of f , and to try to replace the functional ft at each node t ∈ T by a functional gt in the norming set of the auxiliary space, and in doing this to try to copy, as much as possible, the given tree-analysis (ft )t∈T . Not all nodes t ∈ T have the same importance in this process. It turns out that the crucial replacements ft → gt are made for t belonging to some sets A ⊆ T such that (ft )t∈A is in some sense responsible for the estimation of the action of the whole functional f on each of the vectors xk . These are the maximal antichains of T deﬁned below. Observe that some of the replacements ft → gt are necessary before this procedure has a chance to work. Suppose for example the replacements are made in an auxiliary mixed Tsirelson space X where a particular (Anj0 , m1j )-operation is not allowed. 0 Then, every time we ﬁnd a node t ∈ T such that the corresponding ft has weight w(ft ) = mj0 , the replacement gt has to be something avoiding this operation, i.e., we cannot put the combination gt = (1/w(ft )) s∈St gs . This sort of nodes are the ones that we call “catchers” below, because their own tree analyses (fs )st cannot be taken into account.

106

Appendix B. The Proof of the Finite Representability of JT0

Antichains and arrays of antichains Recall that every f ∈ Kω1 has a tree-analysis (ft )t∈T such that: For every t ∈ T , (a) if u ) t, then ran fu ⊆ ran ft , and (b) if ft is of type I, then ft = (1/w(ft )) s∈St fs . Recall that A ⊆ T is called an antichain if for every t = t ∈ A, neither t t nor t t. Given t, t ∈ T , we deﬁne t ∧ t = max{v ∈ T : v t, t }. Notice that A ⊆ T is an antichain iﬀ t ∧ t t, t for every t = t ∈ T . Deﬁnition B.1. Fix a tree-analysis (ft )t∈T of f as above. Given a ﬁnitely supported vector x, a set A ⊆ T is called a regular antichain for x and (ft )t∈T if: (a.1) For every t ∈ A, ft is not of type II. (a.2) ft1 ∧t2 is of type II for every t1 = t2 ∈ A, and (a.3) ran ft ∩ ran x = ∅, for every t ∈ A. A is a maximal antichain for x if in addition A satisﬁes (a.4) For every t ∈ T , if supp ft ∩ ran x = ∅, then there is some u ∈ A comparable with t. Let (xk )nk=1 be a block sequence, and let A = (Ak )nk=1 be such that each Ak is a regular antichain for the vector xk and the tree-analysis (ft )t∈T . For a given t ∈ T , we deﬁne DtA = ut {k ∈ [1, n] : u ∈ Ak }, EtA = DtA \ ( s∈St DsA ). Whenever there is no possible confusion we simply write Dt and Et to denote DtA and EtA , respectively. A = (Ak )nk=1 is called a (maximal) regular array for (xk )nk=1 and (ft )t∈T if each Ak is a (maximal) regular antichain for xk and (ft )t∈T , and in addition (a.5) for every t ∈ k Ak such that ft is of type I, either t is a catcher, i.e., Ds = ∅ for every s ∈ St , or for every k ∈ Et , t is splitter of xk , i.e., for every k ∈ Et there are at least s1 = s2 ∈ St such that ran fsi ∩ ran xk = ∅. We denote by S(A) and C(A) the set of splitter nodes and catcher nodes of A, respectively. Notice that if ti ∈ Aki (i = 1, 2) are catcher nodes, then they are incomparable, and that Ak = S(A) ∪ C(A). Note that if no ft (t ∈ T ) is of type II then #Ak ≤ 1 for all k, and so the tree-analysis below becomes much simpler. Deﬁnition B.2. (The functor A(x, C).) Given a block vector x and C ⊆ T consisting of nodes of type I, let A(x, C) be the set of nodes t ∈ T such that (A.1) ft is not of type II. (A.2) ran ft ∩ ran x = ∅.

Appendix B. The Proof of the Finite Representability of JT0

107

(A.3) For every s t if s ∈ Su and fu is of type I, then for every s ∈ Su \ {s} we have that ran fs ∩ ran x = ∅. (A.4) If ft is of type I and t ∈ / C, then t is a splitter of x. (A.5) for every u t, u ∈ / C. Proposition B.3. Let A = A(x, C) be a maximal regular antichain such that {t ∈ A \ C : ft of type I} ⊆ S(A). Moreover, if (xk )nk=1 is a block sequence, then the corresponding A = (A(xk , C))nk=1 is a maximal regular array such that (a) {t ∈ k Ak \ C : ft of type I} ⊆ S(A), and (b) C ⊆ C(A) and for every t ∈ C, Et is an interval of integers. Proof. Fix t = t ∈ Ak . That ft∧t is of type II follows from the fact that if u t, then u ∈ / C, by (A.5), hence if fu is of type I, then (A.3) implies that u is not splitter of x. We show the maximality of A: Fix t ∈ T such that supp ft ∩ran x = ∅. Let t0 ) t be such that ft0 is of type 0 and supp ft0 ⊆ ran wk , and set b = [0, t0 ] = {v ∈ T : v t0 } which is a -well ordered set, and t ∈ b. We distinguish two cases: Suppose ﬁrst that b∩C = ∅. Let u0 = min{u ∈ b : v satisﬁes (A.1), (A.4)}. Notice that u0 exists since t0 satisﬁes (A.1) and (A.4). The minimality of u0 shows that u0 satisﬁes (A.3), hence u0 ∈ A. Suppose now that b ∩ C = ∅, and set v0 = min b ∩ C. It is not diﬃcult to show that u0 = max{u v0 : u satisﬁes (A.1), (A.4)} is in A (notice that v0 satisﬁes (A.1) and (A.4), hence u0 is well deﬁned.) Repeating this procedure for each vector in a given a block sequence (xk )nk=1 , one gets that the array (A(xk , C))nk=1 is maximal and regular. Finally suppose that t ∈ C and suppose that k1 < k2 < k3 with k1 , k3 ∈ Et . It is routine to check that t satisﬁes (A.1)–(A.5) for xk2 , hence it follows that k2 ∈ Et . Proposition B.4. Suppose that A = (Ak )nk=1 is a regular array for a block sequence (xk )nk=1 and (ft )t∈T . Then: (b.0) If t ∈ Ak is splitter or if ft is of type 0, then supp ft ∩ ran xk = ∅. (b.1) If ft is of type I, then {Ds }s∈St ∪ {{k} : k ∈ Et } is a block family, and if t is splitter, then #Et ≤ #St − 1. Suppose that in addition A = (Ak )nk=1 is maximal for (xk )nk=1 . (b.2) Fix t ∈ Ak and ﬁx u s t with fu of type I and s ∈ St . Then for every s ∈ Su \ {s} ran fs ∩ ran xk = ∅. (b.3) Suppose that ft is of type II, k ∈ Dt and s ∈ St . If supp ft ∩ ran xk = ∅, then k ∈ Ds . Proof. (b.0) If ft is of type 0, the conclusion is clear. If t is a splitter, let s1 = s2 ∈ St be such that fs1 < fs2 and ran fs1 ∩ ran xk , ran fs2 ∩ ran xk = ∅. Then max supp fs1 ∈ ran xk . (b.1) For the ﬁrst part, if t is catcher, there is nothing to prove, so we assume t is splitter. First we show that {Ds }s∈St ∪ {{k} : k ∈ Et } is a disjoint family. If

108

Appendix B. The Proof of the Finite Representability of JT0

k ∈ Et ∩Ds for some s ∈ St , then there is some u ) s with u ∈ Ak . But t ∈ Ak and t u, a contradiction. Suppose that k ∈ Ds ∩ Ds with s = s ∈ St . Then there are u, u ∈ Ak such that u ) s, u ) s . Hence u∧u = t but ft is of type I, contradicting (a.2). For the second part, suppose that k1 < k2 < k3 are such that k1 , k3 ∈ Ds for some s ∈ St . This implies that ran xk1 ∩ ran fs , ran xk3 ∩ ran fs = ∅, and hence ran xk2 ⊆ ran fs . This implies that ran xk1 ∩ ran fs = ∅ for every s ∈ St \ {s}. Since t is splitter, k2 ∈ / Et , and, by (a.3), k2 ∈ / Ds for every s ∈ St \ {s}. Let St = {s1 < · · · < sd } be ordered such that fsi < fsj whenever i < j. For k ∈ Et , the set Hk = {i ∈ [1, d] : ran xk ∩ ran fsi = ∅} has at least two elements. We claim that the mapping k → max Hk ∈ {2, . . . , d} is one-to-one. To see this note that for k < k we obtain that Hk ∩ Hk = {max Hk } if max Hk = max Hk , and Hk < Hk otherwise. (b.2) Fix s ∈ St \ {s}, and suppose that ran fs ∩ ran xk = ∅. Since ran fs ∩ ran xk = ∅, we get that supp fs ∩ ran xk = ∅. By maximality of Ak , there is t ∈ Ak comparable with s . Since Ak is an antichain, we get that t ) s , and hence t ∧ t = u. But fu is of type I, a contradiction. (b.3) This follows using (a.4) and (a.1), (a.2). Assignments, ﬁltrations, and their relationships Deﬁnition B.5. Given a block sequence (xk )nk=1 , and a regular array A = (Ak )nk=1 A for (xk )nk=1 , a sequence (gk,t )t∈Ak ,k ⊆ c00 (N) is called an A-assignment provided that supp gk,t ⊆ {k} for every k and t ∈ Ak . The property (b.1) ensures that A every A-assignment (gk,t )t∈Ak ,k naturally ﬁlters down to the whole tree (GA k,t )t∈T A A A as follows: If k ∈ / Dt , then GA = 0, and if t ∈ A , then G = g . Suppose that k k,t k,t k,t A k ∈ DtA \ DsA . If ft is of type I, then we deﬁne recursively GA = (1/w(f ))G t k,t k,s , A where s ∈ St is the unique s = s(k, t) ∈ St such that k ∈ Ds (by (b.1)). If ft is of A type II, ft = s∈St λs fs , then we simply set GA k,t = s∈St λs Gk,s . For t ∈ T , let GA t =

k∈DtA

GA k,t .

A We call (GA t )t∈T the ﬁltration of (gk,t )t∈Ak ,k . Whenever there is no possible conA A fusion, we write gk,t , Gk,t and Gt instead of the respective gk,t , GA k,t and Gt .

Proposition B.6. (c.1) Fix t ∈ T . For every k, supp gk,t ⊆ {k}. Hence supp gt ⊆ Dt . (c.2) If ft is not of type II, then Gt = k∈Et gk,t + (1/w(ft )) s∈St Gs . (c.3) If ft is of type II, ft = s∈St λs fs , then Gt = s∈St λs Gs . Proof. (c.1) is clear. (c.2) If ft is of type 0, this is clear. Suppose that ft is of type I. Then by

Appendix B. The Proof of the Finite Representability of JT0

109

deﬁnition Gk,t + k∈Dt \Et Gk,t = k∈Et gk,t + s∈St k∈Ds Gk,t = 1 1 = k∈Et gk,t + s∈St w(f k∈Ds Gk,s = k∈Et gk,t + w(ft ) s∈St Gs . t) (B.1)

Gt =

k∈Et

(c.3) Suppose that ft is of type II, i.e., ft = s∈St λs fs , and suppose that k ∈ Dt . Then, by (c.1), Gt (ek ) = Gt,k (e ) = λ G (e ) = ( k s k,s k s∈St s∈St λs Gs )(ek ). If k ∈ / Dt , then Gt (ek ) = 0, and s∈St λs Gs (ek ) = 0. Deﬁnition B.7. (Canonical Assignment) Suppose that A = (Ak )k is a regular array for (xk )nk=1 and (ft )t∈T . Let fk,t = ft (xk )e∗k for k ∈ [1, n] and t ∈ Ak . This is the A-canonical assignment. Remark B.8. Note that if the array A is maximal, then ﬁltering down the canonical assignment we get ft (wk ) = Fk,t (ek ), for every t ∈ T , and k ∈ Dt : If k ∈ Et , this is just by deﬁnition. Suppose k ∈ / Et . If ft is of type I, then Fk,t (ek ) = (1/w(ft ))Fk,s (ek ), where s ∈ St is unique such that k ∈ Ds . By the maximality of Ak , we get that supp fs ∩ ran wk = ∅ for every s ∈ St \ {s} (by (b.2)), hence ft (xk ) = (1/w(f t ))fs (xk ) = (1/w(ft ))Fk,s (ek ) = Fk,t (ek ), by inductive hypothesis. If f = λs fs is of type II, then by maximality of Ak , t s∈S t ft (xk ) = s∈St ,k∈Ds λs fs (xk ) = s∈St ,k∈Ds λs Fk,t (ek ) = Fk,t (ek ), the last equality because Fk,u = 0 if k ∈ /D . u n We obtain that ft ( k∈Dt bk xk ) = Ft ( k∈Dt bk ek ) = Ft ( k=1 bk ek ) for every sequence scalars (bk )nk=1 of . nThe last equality follows from supp Gt ⊆ Dt . In n particular, f ( k=1 bk xk ) = G∅ ( k=1 bk ek ), since D∅ = {k : supp f ∩ ran xk = ∅}, by maximality of A. Proposition B.9. Suppose that A = (Ak )nk=1 is a regular array (not necessarily maximal) for (xk )nk=1 and (ft )t∈T . Fix scalars (bk )nk=1 , (ck )nk=1 and suppose that (gk,t )t∈Ak ,k , (hk,t )t∈Ak ,k are A-assignments. (1) If for every t ∈ Ak gk,t (bk ek ) ≤ hk,t (ck ek ), then for every t ∈ T Gk,t (bk ek ) ≤ Hk,t (ck ek ). (2) Gk,u (ek )∞ ≤ max{gk,t ∞ : t ∈ Ak }, for every u ∈ T . n (3) If for every t ∈ k=1 Ak k∈Et gk,t (bk ek ) ≤ k∈Et hk,t (ck ek ), then for every t∈T, Gt ( bk ek ) ≤ Ht ( ck ek ). k∈Dt

(4) Gu ∞ ≤

t∈Ak ,k

k∈Dt

gk,t ∞ for every u ∈ T .

Proof. This follows from Proposition B.6.

110

Appendix B. The Proof of the Finite Representability of JT0

Two successive ﬁltrations In some applications of the theory one needs to do the process of assignment and ﬁltration twice starting with diﬀerent arrays of antichains. To see this, suppose that C and D are regular arrays for (xk )nk=1 and (ft )t∈T . Then we can naturally D D deﬁne a D-assignment (gk,t )t∈Dk ,k by taking the ﬁltration gk,t = GCk,t . For this to work, one needs the following special relationship between C and D. Deﬁnition B.10. We write C ⊀ D if for every k, every c ∈ Ck and every d ∈ Dk , C we have that c ⊀ d. A C-assignment (gk,t )k∈Ck ,k is called coherent provided that C gk,t = 0 whenever ft (wk ) = 0. Proposition B.11. Suppose that C ⊀ D are two regular arrays for (xk )nk=1 and (ft )t∈T , and suppose that D is in addition maximal. Fix a coherent C-assignment C (gk,t )t∈Ck ,k . Then: (a) For every k ∈ DtC ∩ DtD , GCk,t = GD k,t . (b) g∅C = g∅D . Proof. (a) If k ∈ EtD , this is just by deﬁnition. Suppose ft is of type I, and suppose D that k ∈ DsD , for some s ∈ St . Then GD k,t = (1/w(ft ))Gk,s . Since D is a maximal regular array, by Proposition B.4 (b.2), ran fs ∩ ran wk = ∅ for every s ∈ St \ {s}. If k ∈ DsC , then we are done by the inductive hypothesis. So, suppose k ∈ EtC , i.e., t ∈ Ck . Hence, t u for some u ∈ Dk (because k ∈ DsD ), contradicting our assumption that C ⊀ D. If ft = combination, s∈St λs fs is a sub-convex D D D then GD = G = G = D D C k,t k,s k,s s∈St ,k∈Ds s∈St ,k∈Ds ∩Ds s∈St ,k∈DsD ∩DsC Gk,s = GCk,t . To see the last equality note that if k ∈ / DsD , then, by maximality of D, supp fu ∩ ran wk = ∅ for every u ) s, so, by coherence of the assignment, GCk,t = 0; C if k ∈ / DsC , then k ∈ / DuC for all u ) s, and so gk,u = 0 for all u ) s u ∈ Ck , giving D us Gk,s = 0. D D C (b) Note now that g∅D = k∈D C gk,∅ = k∈D C ∩D C gk,∅ = g∅ . For if k ∈ ∅

∅

∅

D∅C \ D∅D , then by maximality of D, for all u ∈ T , supp fu ∩ ran wk = ∅, hence, by C D = 0 for all u, and hence gk,∅ = 0; if k ∈ D∅D \D∅C , coherence of the C-assignment gk,u C then every Ck = ∅, and so gk,∅ = 0.

Let us now give the two main applications of this general theory of treeanalyses. The proof of the basic inequality Let W be the minimal subset of c00 (N) containing {±e∗k }k , and closed under ((A4nj , m1j )) operations. Fix a (C, ε)-RIS (xk )nk=1 , and ﬁx (jk )nk=1 witnessing that (xk )nk=1 is a (C, ε)-RIS, i.e., a) xk ≤ C, b) | supp xk | ≤ mjk+1 ε and c) for all type I functionals φ of K with w(φ) < mjk , |φ(xk )| ≤ C/w(φ). Fix a sequence

Appendix B. The Proof of the Finite Representability of JT0

111

(bk )nk=1 of scalars, maxk |bk | ≤ 1, and f ∈ Kω1 . Let (ft )t∈T be a tree-analysis of f . Consider the maximal regular array A = (A(xk , C))nk=1 , where C is the set of nodes t such that ft is of type I and w(ft ) = mj0 . We introduce the following two A-assignments (gk,t )t∈Ak ,k , and (rk,t )t∈Ak ,k . Fix k and t ∈ Ak . If tt is of type 0, then we set gk,t = e∗k and rk,t = 0. Suppose that t is of type I, and w(ft ) = mj0 . Let lt = min{k ∈ Et : w(ft ) ≤ mjk } if it exists, and lt = ∞ otherwise. If k > lt , then we set gk,t = (1/w(ft ))e∗k and rk,t = 0. If k < kt , then we set gk,t = 0 and rk,t = εe∗k . If k = lt , then we set gk,t = e∗k and rk,t = 0. Suppose now that w(ft ) = mj0 . Notice that Et is an interval. Set kt = max{l ∈ Dt : |bl | = (bi )i∈Et ∞ }. If k = kt then we set gk,t = e∗k , and rk,t = εek ; if not, then we set gk,t = 0, and rk,t = εe∗k . Let (Gt )t∈T and (Rt )t∈T be the corresponding ﬁltrations. Claim (D). Fix t ∈ T . Then: (d.1) Rt ∞ ≤ ε. (d.2) |ft ( k∈Dt bk xk )| ≤ C(Gt + Rt )( k∈Dt |bk |ek ). (d.3) For every t for which ft is of type I, either Gt ∈ conv{h ∈ W : w(h) = w(ft )} or Gt = e∗k + ht for some k ∈ / supp ht and ht ∈ conv{h ∈ W : w(h) = w(ft )}. Proof of the claim. (d.1) follows from Proposition B.9, and (d.2) follows also from the same proposition applied to the canonical A-assignment, the assignment (C(gk,t + hk,t ))t∈Ak ,k , and the sequences of scalars (bk )k and (|bk |)k . (d.3) If w(ft ) = mj0 , then t is a catcher and Gt = k∈Et gk,t = e∗kt ∈ W . Suppose that t is of type I, w(ft ) = m j0 . By (c.2) and the particular A-assignment, ∗ ∗ we know that either Gt = (1/w(ft ))( k∈E e + s∈St Gs ) or Gt = elt + ht , t ,k>lt k ∗ where ht = (1/w(ft ))( k∈Et ,k>lt ek + s∈St Gs ). Assume this last case holds. Subcase 1a. For every s ∈ St the functional fs is not of type II. From the inductive hypothesis, we have that for every s ∈ St , Gs = hs or Gs = e∗ls + hs , hs ∈ W . For s ∈ St such that Gs = e∗ls + hs , set Is1 = {n ∈ N : n < ls } and Is2 = {n ∈ N : n > ls }. We set h1s = Is1 hs , h2s = Is2 hs . Then, for every s ∈ St the functionals h1s , e∗ls , and h2s are successive and belong to W . By (b.1), for s = s ∈ St the corresponding functionals together with {e∗k }k∈Et ,k>lt form a block family, and we obtain that #{e∗k }k∈Et ,k>lt + #{e∗ls : s ∈ St } + #{h1s : s ∈ St } + #{h2s : s ∈ St } ≤ 4#St . (B.2) Therefore, (1/w(ft ))( k∈Et ,k>lt e∗k + s∈St Gs ) ∈ W . Subcase 1b. There are s ∈ St for which fs is of type II. Let B1 be the set of immediate successors s of t such that fs isof type II, and B2 = St \ S1 . Observe that every sub-convex combination fs = u∈Ss ru fu satisﬁes that fu is of type I. We may assume, allowing repetitions if needed, that for every s ∈ St such that fs is of type II, fs = (1/k) kq=1 fs,q , where each fs,q ∈ {fu : u ∈ Ss }. For each q = 1, 2, . . . , k we set hqt = (1/mj )( l∈Et ,l>lt e∗l + s∈B1 Gs + s∈B2 Gs,q ), where Gs,q = Gu for u ∈ Ss such that fs,q = fu . A similar argument as in the previous subcase shows that hqt ∈ W with w(hqt ) = mj for q = 1, 2, . . . , k and ht = (1/k) kq=1 hqt , as required.

112

Appendix B. The Proof of the Finite Representability of JT0

The particular case t = ∅, the root of T , gives us the conclusion of the basic inequality. Remark B.12. Note that a ﬁner assignment using the same array of antichains will actually give us the conclusion of the basic inequality for a bit smaller auxiliary space T [(A2nj , m1j )j ]. The proof of the ﬁnite interval representability of JT0 The general scheme of the proof is quite similar to the proof of the basic inequality though the input block sequence of vectors is slightly diﬀerently chosen. Notice however that the ﬁnite interval representability involve two inequalities needed for showing that the representing operator as well as its inverse are uniformly bounded. Thus, while in the proof of the Basic Inequality we could aﬀord to go the auxiliary space T [(A4nj , m1j )j ] this is no longer possible in this case. In other words, we need to improve on the counting (B.2). It is exactly for this reason that we introduce below two arrays of antichains and use two successive ﬁltrations as explained above in Subsection B. Fix a transﬁnite block sequence (xα )α<γ , n ∈ N, a sequence I1 ≤ I2 ≤ · · · ≤ In of successive, not necessarily distinct, inﬁnite intervals of γ, and ε > 0. Let j0 be such that m2j0 +1 > 100n/ε and set l = n2j0 +1 /m2j0 +1 . Find a (1, j0 )dependent sequence (z1 , ψ1 , . . . , zn2j0 +1 , ψn2j0 +1 ) such that (a) ran ψi ⊆ ran zi for every i = 1, . . . , n2j0 +1 and (b) (zk )il k=(i−1)l+1 ⊆ xα α∈Ii for every i = 1, . . . , n. Let n2j0 +1 φ = m2j1 +1 i=1 ψi , 0

and for each k = 1, . . . , n we set m wk = 2jl0 +1 kl i=(k−1)l+1 zi and φk =

1 m2j0 +1

kl

i=(k−1)l+1 ψi

∈ Kω 1 .

Proposition B.13. Fix k = 1, . . . , n. Then (1) ran φk ⊆ ran wk , φk wk = 1 and 1 ≤ wk ≤ 24. (2) For every f ∈ Kω1 of type I with w(f ) > m2j0 +1 , |f (wk )| ≤ 1/m22j0 +1 . d (3) Let f ∈ Kω1 be of type I, f = (1/w(f )) i=1 fi with w(f ) = m2j+1 for j < j0 and d ≤ n2j+1 . Let d0 = max{i ≤ d : w(fi ) < m2j0 +1 }, and set d0 −1 d fL = 1/m2j+1 i=1 fi and fR = 1/m2j+1 i=d0 +1 fi . Then |fL (wk )| ≤ 1/m22j0 +1 and |fR (wk )| ≤ 1/m2j0 +1 . d (4) Let f = (1/w(f )) i=1 fi with w(f ) = m2j0 +1 and d ≤ n2j0 +1 be such that #{i ∈ [1, d] : w(fi ) = w(ψi ) and supp zi ∩ supp fi = ∅} ≤ 2. Then, |f (wk )| ≤ 1/m22j0 +1 .

Appendix B. The Proof of the Finite Representability of JT0

113

Proof. First of all, note that (zi )kl i=(k−1)l+1 is a (12, 1/n2j0 +1 )-RIS. Note also that (1) and (2) follow from Proposition II.19. (3) By the properties of special sequences, #

d0 −1 i=1

supp fi ≤ w(fd0 ) < m2j0 +1 .

(B.3)

So, |fL (wk )| ≤ f0 1 wk ∞ ≤ m32j0 +1 /n2j0 +1 ≤ 1/m22j0 +1 . Let us now estimate |fR (wk )|. To save on notation we only estimate for k = 1. Set F0 = {r ∈ [1, l] : #({i ∈ [d0 + 1, d] : ran zr ∩ supp fi = ∅}) ≥ 2}, F1 = [1, l] \ F0 . Notice that |F0 | ≤ n2j+1 − 1. For i = 0, 1 let wi = (mj1 /l) k∈Fi zk . Since fR ∈ Kω1 and since (zk )k is a (12, 1/n2j0 +1 )-RIS, we have that m m |fR (w0 )| ≤ w0 ≤ 2jl0 +1 k∈F0 zk ≤ 2jl0 +1 6n2j+1 . (B.4) To estimate |fR (w1 )| we use the basic inequality. For each i = d0 + 1, . . . , d, let Hi ={k ∈ F1 : ran zk ∩ supp fi = ∅}. Note that {Hi }i is apartition of F1 and is a block family. For i = d0 + 1, . . . , d, we set w1,i = mj1 /l k∈Hi zk . Clearly w1 = w1,d0 +1 + · · · + w1,d and hence |fR (w1 )| ≤

d i=d0 +1

|fR (w1,i )| =

1 m2j+1

d i=d0 +1

|fi (w1,i )|.

(B.5)

Let us estimate now |fi (w1,i )|, for i = d0 +1, . . . , d. For a ﬁxed such i, applying again the basic inequality, we obtain |fi (w1,i )| ≤ 12(g1i + g2i )(m2j0 +1 /l k∈Hiek ), where in the worst case, g1i = hi + e∗ki , with hi ∈ W , and hi ∈ convQ {h ∈ W : w(h) = w(fi )}. Since the auxiliary space is 1-unconditional, by Proposition II.9, |hi ((m2j0 +1 /l) k∈Hi ek )| ≤ m2j0 +1 /w(fi ). Note that g2i ∞ ≤ 1/n2j0 +1 . Putting all these inequalities together we get ' ( m2j0 +1 n2j+1 m2j0 +1 m2j0 +1 d 12 |fR (w1 )| ≤ m2j+1 + n2j i=d0 +1 w(fi ) + l +1 0 2 m n m m 2j+1 d 2j +1 2j 2j +1 12 0 0 +1 0 (B.6) + n2j +1 . ≤ m2j+1 i=d0 +1 w(fi ) + n2j +1 0

Using (B.4) and (B.6) we obtain ' 12m2j0 +1 2n2j+1 m2j0 +1 |fR (w1 )| ≤ m2j+1 + n2j +1 0

1 n2j0 +1

+

d

1 i=d0 +1 w(fi )

0

(

≤

1 m2j0 +1 .

(B.7)

(4) Let E = {i ∈ [1, d] : w(fi ) = w(ψi ) and supp zi ∩ supp fi = ∅}. By our assumptions, #E ≤ 2. For i ∈ [(k − 1)l, kl] \ E the properties of the dependent sequences yield that |f (zi )| ≤ 1/n2j0 +1 . Hence, |f (wk )| ≤ 2 · 24m2j0 +1 /l + m2j0 +1 /n2j0 +1 ≤ 1/m22j0 +1 .

114 Lemma B.14. (bk )nk=1 .

Appendix B. The Proof of the Finite Representability of JT0 n

k=1 bk wk

≤ 121

n

k=1 bk vk JT0

for every choice of scalars

Proof. Fix a sequence (bk )nk=1 of scalars with maxk |bk | ≤ 1, an f ∈ Kω1 , and its tree (ft )t∈T . Antichains. A node t ∈ T is called relevant if (1) w(ft ) ≤ m2j0 +1 , and (2) if u t is its immediate predecessor, if fu is of type I, and if w(fu ) = m2j+1 < m2j0 +1 , then t = s(u) = max{s ∈ Su : w(fs ) < m2j0 +1 }, where the maximum is taken according to the block ordering Su = {s1 < · · · < sd }. Let C be the set of nodes t which are either non-relevant, or such that ft is of type I and w(ft ) = m2j0 +1 . Let B = (Bk )nk=1 where Bk = A(wk , C) for k = 1, . . . , n (see (A.1)–(A.5) in Proposition B.3 above). For each k, let Bkunc = S(Bk ) \ C be the set of splitters that are not in C, Bkcnd = Bk ∩ C, and Bkat = Bk \ (Bkunc ∪ Bkcnd ). Fix u ∈ Bkunc , and observe that u is splitter of xk for every k ∈ Eu . List all s ∈ Su such that ran fs ∩ ran wk = ∅ ,{sk,1 , . . . , sk,d } ordered according to the block ordering fsk,1 < · · · < fsk,d . We set now in =wk |[min supp wk , max supp fsk,1 ] wk,u ﬁn in wk,u =wk − wk,t . = A(wk,u , Cnr ), where Cnr is the set of non-relevant nodes For ∈ {in, ﬁn}, let Bk,u of T . Set Bk = u∈Bunc Bk,u . Observe that B = (Bk )nk=1 is a regular (not necesk sarily maximal) arrow for (wk )nk=1 and (ft )t∈T , whenever ∈ {in, ﬁn, cnd, at}. Assignments and ﬁltrations. Consider the following B -assignments (gk,t )k∈Bk ,k

where ∈ {in, ﬁn, cnd}, and (r ) where ∈ {in, ﬁn, cnd, at}: Fix k, and k,t k∈Bk ,k t ∈ ∈{in,ﬁn,cnd,at} Bk . at (a) Suppose that ft is of type 0. Then we set rk,t = (1/m2j0 +1 )e∗k if t ∈ Bkat , and ∗ we set gk,t = 0 and rk,t = (1/m2j0 +1 )ek , if t ∈ Bk for some ∈ {in, ﬁn}. (b) Suppose that t is non-relevant. Then clearly t ∈ / Bkat . Fix ∈ {in, ﬁn, cnd}. We set gk,t = 0 in all cases. Suppose that w(ft ) > m2j0 +1 . Then we set rk,t = ∗ cnd ∗ (1/m2j0 +1 )ek for ∈ {in, ﬁn}, and rk,t = (sgn(bk )/m2j0 +1 )ek . Finally, if t = s(u), where u is the immediate predecessor of t (see the deﬁnition of relevant node), cnd then we set rk,t = ft (wk )e∗k for ∈ {in, ﬁn} and rk,t = sgn(bk )ft (wk )e∗k . (c) Suppose now that t is relevant. (c.1) w(ft ) = m2j0 +1 . If t ∈ Bk , for ∈ {in, ﬁn}, then we set gk,t = (1/w(ft ))e∗k n2j0 +1 and rk,t = 0. Suppose that t ∈ Bkcnd . Suppose that ft = ±I(1/m2j0 +1 ) i=1 gi , where I ⊆ ω1 is an interval, and Φ = (g1 , . . . , gn2j0 +1 ) is a 2j0 + 1-special sequence. Set Ψ = (ψ1 , . . . , ψn2j0 +1 ). Consider It = {i ∈ [1, κΦ,Ψ − 1] : Igi = 0} = k(t,2)−1 [k(t, 1), k(t, 2)], and let εt = sgn( k=k(t,1)+1 bk ). If k = k(t, i) for i = 1, 2, then we cnd,Bcnd

cnd,Bcnd

cnd cnd = sgn (bk(t,i) )e∗k(t,i) and rk,t = 0. We set gk,t k = εt e∗k and rk,t k = 0 set gk,t cnd cnd if k ∈ (k(t, 1), k(t, 2)). We set gk,t = 0, rk,t = sgn(bk )(1/m2j0 +1 )e∗k otherwise. (c.2) Suppose that w(ft ) = m2j0 +1 . Then t ∈ Bk , for some ∈ {in, ﬁn}. Set gk,t = (1/w(ft ))e∗k and rk,t = 0 for all cases, except for w(ft ) = m2j+1 < m2j0 +1 .

Appendix B. The Proof of the Finite Representability of JT0

115

In this case, we observe that since t is splitting there are at least two immediate successor s1 = s2 ∈ St such that ran wk,u ∩ ran fsi = ∅ (i = 1, 2) for some u ∈ Bk .

This implies that there is at most one k ∈ EtB such that ran fs(t) ∩ ran wk,v = ∅ unc ∗ for v ∈ Bk , and t ∈ Bk,v . Then we set gk,t = (1/m2j0 +1 )ek and rk,t = 0 if k is this one, and gk,t = 0 and rk,t = (1/m2j0 +1 )e∗k otherwise. Let (Gt )t∈T , (Rt )t∈T be the corresponding ﬁltrations. Recall that given a regular array A = (Ak )k for (xk )k and (ft )t∈T the canonical A-assignment A A (fk,t )t∈Ak ,k is deﬁned by fk,t = f (xk )e∗k . It was shown in Remark B.7 that if in ad dition A is maximal, then for every (ak )nk=1 and every t ∈ T , F A ( k∈DtA ak ek ) = ft ( k∈DtA ak wk ).

Claim. Fix t ∈ T , and for ∈ {in, ﬁn, cnd, at} let Dt = DtB . Then: (e.1) |Ft ( k∈Dt bk ek )| ≤ 24(Gt + Rt )( k∈Dt |bk |ek ) for ∈ {in, ﬁn}. + Rtcnd )( k∈Dcnd bk ek ). (e.2) |Ftcnd ( k∈Dcnd bk ek )| ≤ 24(Gcnd t t t (e.3) |Ftat ( k∈Dat )bk ek | ≤ 24|Rtat ( k∈Dat )bk ek |. t t

∈ 3W (JT0 ). (e.4) Gt ∈ W (T0 ) for ∈ {in, ﬁn}, and Gcnd t . For ∈ {in, ﬁn, cnd}, either t is non-relevant, w(ft ) < (e.5) Rtat ∞ ≤ 1/m 2j0 +1 m2j0 +1 and Gk = k∈E B ft (wk )e∗k or Rt ∞ ≤ 1/m2j0 +1 . t

Proof of the claim. (e.1)–(e.3) are immediate applications of Proposition B.9. (e.4) Most of the cases follows immediately by deﬁnition of the corresponding assignments. We hint the non-trivial ones: Suppose that t is relevant. If w(ft ) = m2j0 +1 , then Dtcnd = Etcnd ,and the corresponding assignment gives that Gcnd = λ1 e∗k(t,1) + λ2 e∗k(t,2) + εt k∈Etcnd ∩(k(t,1),k(t,2)) e∗k ∈ 3W (JT0), where t λi = sgn(bk(t,i) )χEtcnd (k(t, i)), for i = 1, 2, and where χE denotes the characteristic function of E. Fix ∈ {in, ﬁn}. We claim that #Dt ≤ 1: Suppose not, and say that k1 < k2 ∈ Dt . Then since w(ft ) = m2j0 +1 there are ui ∈ Bkunc i and si ∈ Bki ,ui (i = 1, 2) such that u1 , u2 t s1 , s2 . If = in, then since ran ft ⊆ ran fs(k1 ,u1 ) , it follows that ran ft < ran wk2 , and since ran fs2 ⊆ ran ft , we obtain that ran fs2 ∩ ran wk2 = ∅, contradicting the fact that s2 ∈ Bkin2 ,u2 . If = ﬁn, in a similar manner we obtain that ran wk1 ∩ ran fs1 = ∅, a contradiction. Hence, either Gt = 0, or Gt = (1/m2j0 +1 )e∗k , certainly in W (T0 ) considered as sub-convex combinations. Suppose now that w(ft ) = m2j0 +1 . There are three subcases to consider: If w(f t ) > m2j0 +1 , then t is non-relevant, hence a catcher node, and Gt = k∈Et gk,t = 0. If w(ft ) = m2j with j ≤ j0 , then the inductive hypothesis gives that Gcnd ∈ 3W (JT0 ) (since Etcnd = ∅). Fix ∈ {in, ﬁn}. Observe t that for every k ∈ Et there is s ∈ St such that ran fs ⊆ ran wk , in which case Ds =∅, and so #Et + #{s ∈ St : Gcnd = 0} ≤ #St , and then, Gt = s ∗ (1/w(ft ))( k∈E ek + s∈St Gs ) ∈ W (T0 ). t If w(ft ) = m2j+1 < m2j0 +1 , then using that there is at most one immediate successor s(t) of t which is relevant we obtain that either Gcnd = 0, t

116

Appendix B. The Proof of the Finite Representability of JT0

∗ or Gcnd = (1/m2j )Gcnd t s(t) , and for ∈ {in, ﬁn}, either Gt = (1/m2j+1 )ek , or Gt = (1/m2j+1 )Gs(t) . (e.5) Rtat ∞ ≤ 1/m2j0 +1 follows from Proposition B.9, since this is so for the corresponding assignment of which Rtat is a ﬁltration. Suppose that ∈ {in, ﬁn, cnd}. The proof is by backwards induction over t. Again we concentrate on non-trivial cases. Suppose that ft is of type I and t is relevant. Then if w(ft ) = m2j with j ≤ j0 , the desired result follows from the deﬁnition of the corresponding assignments, and hypothesis. inductive Suppose that w(ft ) = m2j+1 with j < j0 . Then Rt = k∈E rk,t +(1/w(ft )) s∈St Rs . By the deﬁnition of the assignments, t rk,t ∞ ≤ 1/m2j0 +1 for every k ∈ Et . Observe that all s ∈ St , except possibly one, s(t), are non-relevant and that rk,s = fs (wk )e∗k for every k ∈ Es = Ds . Hence, for every s ∈ St \ {s(t)}, (1/w(ft ))Rs ∞ = max{(1/w(ft ))fs (wk ) : k ∈ Es } ≤ 1/m2j+ 1 ; the last inequality follows from Proposition B.13. By inductive hypoth esis Rs(t) ∞ ≤ 1/m2j0 +1 , so we are done. Suppose that t is non-relevant. The case w(ft ) > m2j0 +1 is immediate. Suppose that w(ft ) = m2j and t = s(u), where u is the immediate predecessor of t (see the node). Notice that t is a catcher, so Et = Dt , and deﬁnition of relevant ∗ Rt = k∈E ft (wk )ek , as desired. t

We are now ready to ﬁnish the proof using part B of the general theory above. Notice that for each ∈ {in, ﬁn, cnd, at}, B ⊀ B, and that the canonical assignments of B are coherent. Let (hk,t )t∈Bk ,k be the assignments induced by the canonical B -assignments (fk,t )t∈Bk ,k , for ∈ {in, ﬁn, cnd, at}. Claim. For every t ∈ T , Htin + Htﬁn + Ftcnd + Ftat = FtB , the canonical assignment of B. ﬁn cnd at Proof of the claim. We show that for every t ∈ Bk , hin k,t + hk,t + hk,t + hk,t = ft (wk )e∗k . The only non trivial case is if t ∈ Bkunc . Notice that since Bk,t is a maximal antichain for wk,t and (fs )st , we obtain that hk,t = Fk,t = ft (wk,t )e∗k . in ﬁn in ﬁn ∗ ∗ Hence, fk,t + fk,t = (ft (wk,t ) + ft (wk,t ))ek = ft (wk )ek , and hk,t = 0 for ∈ {cnd, at}.

Finally, by Proposition B.11, H∅ = F∅ , for ∈ {in, ﬁn, cnd, at}. Hence, n n |f ( k=1 bk wk )| =|F∅B ( k=1 bk ek )| ≤ ∈{in,ﬁn,cnd,at} |F∅ ( k∈D bk ek )| ≤ ∅ ≤24(5 nk=1 bk ek JT0 +4 nk=1 bk ek ∞ ) ≤ 120 nk=1 ek JT0 +ε. (B.8) Corollary B.15. The natural isomorphism F : w1 , . . . , wn → v1 , . . . , vn deﬁned by F (wi ) = vi satisﬁes that F ≤ 1 and F −1 ≤ 120 + ε. Consequently, JT0 is ﬁnite interval representable on the basis (eα )α<ω1 of Xω1 with a constant C < 121. Proof. Proposition VII.11 shows that F ≤ 1; the other inequality follows from Lemma B.14.

Bibliography [1] D. E. Alspach, S. A. Argyros, Complexity of weakly null sequences, Dissertationes Mathematicae, 321, 1992. [2] A. D. Andrew, W. L. Green, On James’ quasireﬂexive Banach space as a Banach algebra, Canad. J. Math. 32 (1980), no. 5, 1080–1101. [3] S. A. Argyros, A universal property of Hereditarily Indecomposable Banach spaces, Proc. Amer. Math. Soc., 129 (2001), 3231–3239. [4] S.A. Argyros, A. Arvanitakis, A.Tolias, Strictly singular extensions and HI spaces with no reﬂexive subspaces, in preparation, [5] S. A. Argyros, I. Deliyanni, Banach spaces of the type of Tsirelson, preprint, 1992. [6] S. A. Argyros, I. Deliyanni, Examples of asymptotic l1 Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. [7] S. A. Argyros, I. Deliyanni, On the structure of the Diagonal strictly singular operators in HI spaces, in preparation, [8] S. A. Argyros, I. Deliyanni, D. N. Kutzarova, A. Manoussakis, Modiﬁed mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), no. 1, 43–109. [9] S. A. Argyros, I. Deliyanni, A. Manoussakis, Distortion and spreading models in mixed Tsirelson spaces, Studia Math. 157 (3) 2003, 199–236. [10] S. A. Argyros, V. Felouzis, Interpolating Hereditarily Indecomposable Banach spaces, Journal Amer. Math. Soc., 13 (2000), 243–294. [11] S.A. Argyros, G. Godefroy, H. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, 1007–1069, NorthHolland, Amsterdam, 2003. [12] S. A. Argyros, J. Lopez-Abad, S. Todorcevic, A class of Banach spaces with no unconditional basic sequence, C. R. Acad. Sci. Paris. Ser. I 337 (2003), 43–48. [13] S. A. Argyros, J. Lopez-Abad, S. Todorcevic, A class of Banach spaces with few non strictly singular operators, J. of Func. Anal. (to appear).

118

Bibliography

[14] S. A. Argyros, A. Manoussakis, An Indecomposable and Unconditionally saturated Banach space, Studia Math. 159 (1), (2003), 1–32. [15] S. A. Argyros, A. Manoussakis, A sequentially unconditional Banach space with few operators, Proc. LMS (to appear). [16] S. A. Argyros, A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc. 170 (2004), no. 806, [17] S. A. Argyros, A. Tolias, Indecomposability and unconditionality in duality, Geom. Funct. Anal. 14 (2004), no. 2, 247–282. [18] S. F. Bellenot, Tsirelson superspaces and p , Journal of Funct. Anal., 69, No 2, 1986, 207–228. [19] S. Bellenot, R. Haydon, E. Odell, Quasi-Reﬂexive and tree spaces, Contemporary Mathematics, 85 (1989), 19–43. [20] J. Bernu´es, I. Deliyanni, Families of ﬁnite subsets of N of low complexity and Tsirelson type spaces, Math. Nachr. 222 (2001), 15–29. [21] P. G. Casazza, T. Shura, Tsirelson’s space, Lecture Notes in Math. 1363, Springer Verlag, 1989. [22] I. Deliyanni, A. Manoussakis, Asymptotic p HI-spaces, in preparation. [23] G.A. Edgar, A long James space, Proc. Conf. on Measure Theory, Lecture Notes in Math. 794, (1980) 31–37. [24] M. Fabian, P. Habala, P. Hajek,V. Montesinos Santaluca, J. Pelant, V. Zizler, Functional analysis and inﬁnite-dimensional geometry., CMS Books in Mathematics 8, Springer-Verlag, New York, 2001. [25] V. Ferenczi, Operators on subspaces of Hereditarily Indecomposable Banach spaces, Bull. London Math. Soc. 29 (1997), no. 3, 338–344. [26] V. Ferenczi, Hereditarily ﬁnitely decomposable Banach spaces, Studia Math. 123 (1997), no. 2, 135–149. [27] V. Ferenczi, A uniformly convex Hereditarily Indecomposable Banach space, Israel J. Math., 102 (1997), 199–225. [28] T. Figiel, W. B. Johnson, A uniformly convex Banach space which contains no p , Compositio Math. 29, 1974, 179–190. [29] W. T. Gowers, A Banach space not containing c0 , 1 or a reﬂexive subspace, Transactions of the AMS 344, No 1, 1994, 407–420. [30] W. T. Gowers, An Inﬁnite Ramsey Theorem and Some Banach-Space Dichotomies, Ann. of Math. (2) 156, (2002), 797–833. [31] W. T. Gowers, A solution to Banach’s hyperplane problem, Bull. London. Math. Soc., 28, (1996), 297–304.

Bibliography

119

[32] W. T. Gowers, Ramsey methods in Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, 1071–1097, North-Holland, Amsterdam, 2003 [33] W. T. Gowers, B. Maurey, The unconditional basic sequence problem, Journal of AMS 6, 1993, 851–874. [34] W. T. Gowers, B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), no. 4, 543–568. [35] R. C. James, Bases and reﬂexivity of Banach spaces. Ann. of Math. (2) 52, (1950). 518–527. [36] R. C. James, A non-reﬂexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177. [37] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80, 1964, 542–550. [38] R. C. James, A separable somewhat reﬂexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. [39] W. B. Johnson,A reﬂexive Banach space which is not suﬃciently Euclidean Studia Math. 55 (1976), no. 2, 201–205. [40] J. Lindenstrauss On nonseparable reﬂexive Banach spaces. Bull. Amer. Math. Soc. 72 (1966) 967–970. [41] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I , Springer-Verlag Vol. 92, (1977) [42] J. Lopez-Abad, A. Manoussakis, On Tsirelson type spaces (in preparation). [43] A. Manoussakis, On the structure of a certain class of mixed Tsirelson spaces, Positivity 5 (2001), no. 3, 193–238. [44] A. Manoussakis, Some remarks on spreading models and mixed Tsirelson spaces, Proc. Amer. Math. Soc. 131 (2003), 2515–2525. [45] B. Maurey, A remark about distortion, Operator Theory: Advances and Applications 77, 1995, 131–142. [46] B. Maurey, Banach spaces with few operators, Handbook of the geometry of Banach spaces, Vol. 2, 1247–1297, North-Holland, Amsterdam, 2003 [47] B. Maurey, V. D. Milman, N. Tomczak-Jaegermann, Asymptotic inﬁnite dimensional theory of Banach spaces, Operator Theory: Advances and Applications 77, 1995, 149–175. [48] B. Maurey, H. Rosenthal, Normalized weakly null sequences with no unconditional subsequence, Studia Math. 61, 1977, 77–98. [49] V. D. Milman, The geometric theory of Banach spaces, part II: Geometry of the unit sphere, Math. Surveys 26, 1971, 79–163.

120

Bibliography

[50] V. D. Milman, N. Tomczak-Jaegermann, Asymptotic p spaces and bounded distortions, Banach spaces, Contemp. Math. 144, 1993, 173–196. [51] E. Odell, On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic, Analysis and logic (Mons, 1997), 189–267, London Math. Soc. Lecture Note Ser., 262, Cambridge Univ. Press, Cambridge, 2002. [52] E. Odell, Th. Schlumprecht, The distortion problem, Acta Math. 173, 1994, 259–281. [53] E. Odell, Th. Schlumprecht, Distortion and Stabilized Structure in Banach spaces; New Geometric Phenomena for Banach and Hilbert Spaces, Proc. International Congress of mathematicians, vol. 1,2,(Z¨ urich, Switzerland 1994), Birkh¨ auser Verlag, Basel, 1995, 955–965. [54] J. Rainwater, Weak convergence of bounded sequences, Proc. Amer. Math. Soc. 14 (1963), 999. [55] Th. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76, 1991, 81–95. [56] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2, 1930, 58–62. [57] S. Shelah, A Banach space with few operators, Israel J. Math. 30 (1978), no. 1-2, 181-191. [58] S. Shelah, J. Stepr¯ ans, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 101–105. [59] I. Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste Romˆania, Bucharest, 1981. [60] S. Todorcevic, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261–294. [61] S. Todorcevic, Coherent sequences, Handbook of set theory, to appear. [62] N. Tomczak-Jaegermann, Banach spaces of type p have arbitrarily distortable subspaces, GAFA, 6 (1996), 1074–1082. [63] B. S. Tsirelson, Not every Banach space contains p or c0 , Funct. Anal. Appl. 8 1974, 138–141. [64] H. M. Wark, A non-separable reﬂexive Banach space on which there are few operators, J. London Math. Soc. (2) 64 (2001), no. 3, 675–689. [65] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. of Math. 4 (1966), p. 265–272

Part B

High-Dimensional Ramsey Theory and Banach Space Geometry Stevo Todorcevic

Introduction One of the ﬁrst remarkable applications of Ramsey theory to Banach space theory dates back to the mid 1970s and is due to Brunel and Sucheston. They were able to isolate the notion of a spreading model (en ) of a basic sequence (xn ) in a given Banach space X. The spreading model (en ) is a basic sequence in some other Banach space Y . It is ﬁnitely block equivalent to (xn ) but it typically has considerably more properties especially if one is willing to pass to its inﬁnite block subsequences. For example, (en ) has a block subsequence which is unconditional. Working from an unconditional spreading model (en ) one can go further and ﬁnd a basic sequence (fn ) in some Banach space Z such that (fn ) is ﬁnitely block representable in (en ) and such that (fn ) is equivalent to the standard basis of some p (1 ≤ p < ∞) or c0 , a well-known result of J. L. Krivine. It follows that for any basic sequence (xn ) in some Banach space X one of the classical spaces p (1 ≤ p < ∞) or c0 is ﬁnitely block representable in (xn ). This in particular gives precise connection between the local theory of general Banach spaces and the local theory of classical Banach spaces, as well as a new proof of Dvoretzky’s theorem that 2 is ﬁnitely represented in every inﬁnite dimensional Banach space. Starting from this set of results we have split the rest of the lectures into two parts, those dealing with problems of ﬁnding subsequences of a given sequence with some desired properties (like being unconditional, for example) and those of ﬁnding block subsequences with desired properties. In Ramsey theory this corresponds to concentrating on diﬀerent “Ramsey spaces” so we tried to explain this in some details. The discovery of the ﬁrst Ramsey space N[∞] of all inﬁnite increasing sequences of natural numbers has in fact been started in the early 1960s from purely utilitarian reasons of developing a theory of well-quasi-orderings, a theory that culminated in Laver’s proof of Fra¨ıss´e’s conjecture stating that the class of countable linear orderings is well-quasi-ordered under embeddability. We have presented parts of this beautiful theory of Nash–Williams not only because of its own right but also because of the importance of his notions of ‘fronts’ and ‘barriers’ in the theory of Tsirelson norms. This is an area of functional analysis that has seen a considerable growth in the last 10 to 15 years. Some of this deep theory is presented in the parallel set of lecture notes of Spiros A. Argyros which form the ﬁrst part of this volume and which we warmly recommend to the reader.

124

Introduction

In the early 1970s Nash–Williams theory has been reformulated and considerably strengthened by the work of Galvin, Prikry, Silver and specially Ellentuck by introducing the topological Ramsey theory. In hindsight the ﬁrst application of this new topological Ramsey theory in the Banach space theory is Rosenthal’s 1 -theorem stating that a bounded inﬁnite sequence (xn ) of elements of some Banach space contains either an inﬁnite subsequence (xnk ) which is weakly Cauchy or an inﬁnite subsequence (xnk ) such that for some positive constants δ < ε, δ

m k=1

|λk | ≤

m k=1

λk xnk ≤ ε

m

|λk |

k=1

for all choices of integers m ≥ 1 and scalars λ1 , . . . , λk . The proof of Rosenthal’s 1 -theorem which used the topological Ramsey theory has actually been found by Farahat [27] shortly after Rosenthal’s original proof [78]. In fact, it was Farahat’s proof which has introduced topological Ramsey theory to this area of Banach space theory. Out of the numerous results proved in the wake of Farahat’s proof of Rosenthal’s 1 -theorem one could not go without mentioning Elton–Odell’s beautiful result that every inﬁnite dimensional normed space X contains an inﬁnite normalized sequence (xn )∞ n=0 of vectors such that for some ε > 0 and all m = n, xm − xn > 1 + ε. Diestel’s book [20] contains an exposition of several other results about sequences in Banach spaces obtained using the topological Ramsey space N[∞] of inﬁnite subsets of N. The Ramsey theory of N[∞] is particularly well suited for the study of sequence — convergence and summability of series in Banach spaces, or more generally, in topological abelian groups. We have presented this in Sections II.9 and II.10. For example, we show how Nash–Williams’ version of this theory enters naturally into the proof of Rosenthal’s theorem stating every weakly null sequence (xn ) in some Banach space X contains either a subsequence (xnk ) all of whose further subsequences are Ces`aro summable, or a subsequence (xnk ) whose spreading model is isomorphic to 1 . In Section II.10 we show that a simple application of the topological Ramsey theorem for N[∞] gives a simple and uniform way for deducing several basic principles of functional analysis such as principles of uniform boundedness and automatic continuity, Schur’s 1 -theorem, Orlicz–Pettis theorem, Vitali–Hahn–Saks theorem, etc. While the Ramsey space N[∞] has numerous applications to Banach space theory the block Ramsey spaces such as the space FIN[∞] of all inﬁnite block sequences of ﬁnite sets seem to be more relevant to deeper problems of this theory. We have chosen to exemplify this on the problem of ﬁnding an unconditional sequence of a given weakly null sequence (xn ) of some Banach space X. The Ramsey theory of N[∞] via an unpublished result of Rosenthal shows that every weakly null sequence in ∞ (Γ) consisting of characteristic functions of subsets of Γ contains an unconditional subsequence. In these notes (see II.5), we shall extend this result to all weakly null sequences in l∞ (Γ) that are in some sense separated away from 0. There is however a bound on how far one can go in this direction.

Introduction

125

The famous example of Maurey–Rosenthal shows that this is no longer true even 2 in the case of weakly null sequences of the function space C(ω ω + 1). However, it still makes sense asking for an unconditional block-subsequence of a given weakly null sequence. The search for a block Ramsey theorem applicable to this problem from the Banach space theory lasted for almost twenty years. The culmination of this search is the famous Gowers’ dichotomy which although it does not depend on any previous result about known Ramsey spaces, such as N[∞] or FIN[∞] , it uses the basic idea of the ‘combinatorial forcing’ present in essentially all results establishing the existence of various Ramsey spaces. We have taken some eﬀort in pointing explicitly any place where combinatorial forcing is used not only to point out the similarities in various arguments but to also indicate the existence of some sort of method that might be applicable to many other cases. We have also taken some eﬀort to point out the diﬀerence between the classical notion of a Ramsey subset of a given Ramsey space and Gowers’ notion of strategically and approximately Ramsey sets of inﬁnite normalized block sequences of vectors of a given Banach space E. For example, in the last sections of these notes we show that while classical Ramsey property is always closed under the operation of complementation, Gowers’ notion of strategic or approximate Ramsey property in general is not preserved by taking complements. This is done by transferring the strategic or approximate Ramsey property of a given class of sets of normalized block sequences to the class of their continuous images. So, for example, if the Banach space E contains no isomorphic copy of c0 then there is a rather automatic way to transfer the strategic Ramsey property of analytic sets of inﬁnite block sequences of E to the strategic Ramsey property of Gδ sets of inﬁnite block sequences of E. This explains why the full strength of Gowers’ analytic Ramsey theorem for Banach spaces (Theorem 4.1 of [35]) is really contained in its Gδ -case and why all known applications of this theorem involve only simple sets of block sequences. There is an aspect of Ramsey theory that calls for applications. This is a part of Ramsey theory that tries to classify ‘canonical’ equivalence relations on a given Ramsey space. For example in the case of the Ramsey space N[∞] one is interested in the behavior of equivalence relations (equivalently, maps from N[∞] into some other space) on N[∞] , not globally but relative to the possibility of going to an arbitrary chosen sub-cube M [∞] over an inﬁnite subset M of N. We have chosen to present a result of Pudlak and R¨ odl that essentially deals with the classiﬁcation of all Borel equivalence relations on N[∞] with at most countably many classes. We have also included some block Ramsey analogues such as the beautiful result of A. D. Taylor which identiﬁes the list of ﬁve equivalence relations on FIN and the results of J. Lopez-Abad which does the same for the generalizations FINk of FIN. Finally, we mention that this set of notes is a reworked version of the material that was originally prepared for the Advanced Course on Ramsey Methods in Analysis at the Centre de Recerca Matem`atica during January 2004. We have tried to present some of the aspects of Ramsey theory that have shown to be

126

Introduction

useful in some areas of functional analysis but undoubtedly have missed many others. Even the small selection when fully presented would ﬁll an amount of lecturing of a much greater length than that given given to us. So it was necessary to compromise and give only the key concepts and ideas of the theory. The length restriction is also responsible for our compromise of being rather brief and at times superﬁcial regarding the credits and other historical discussions. We choose this opportunity to apologize to the numerous authors that have built this subject and hope that the rich source of literature that we include at the end of these notes will indirectly remedy this ﬂaw since any reader interested in more details will have no problems of ﬁnding a more complete picture there. During the preparation of this set of notes I have proﬁted from remarks of many participants of the Advanced Course. Special thanks go to Neus Castells and Jordi Lopez Abad who helped not only with mathematical contents of these notes but also with the technical aspects of preparing these notes. Naturally, all the remaining omissions and errors are my sole responsibility.

Chapter I

Finite-Dimensional Ramsey Theory: Finite Representability of Banach Spaces I.1

Finite-Dimensional Ramsey Theorem

Consider a set S with possibly some structure such as for example a partial semigroup structure, where the associative relation is deﬁned only for x and y respecting some order. Then given a positive integer k, Ramsey-theoretic results at this level are typically not about colorings of the full k-power S k but rather some restricted version S [k] ⊆ S k that respects the partial semigroup structure on S. In the case S = N with the usual order, S [k] is the set of all k-element subsets of N, and similarly for any other ordered set (S, <). Theorem I.1.1 (Ramsey). For every positive integer k and for every ﬁnite coloring of the symmetric power N[k] there is an inﬁnite set M ⊆ N such that M [k] is monochromatic. Proof. Choose a nonprincipal ultraﬁlter U on N. It is very convenient to think about U as a quantiﬁer over N in the sense that a formula of the form (Un)ϕ(n) is interpreted as saying that the set {n : ϕ(n)} belongs to U. The corresponding quantiﬁer respects all propositional connectives in the sense that for every choices of formulas ϕ(x) and ψ(x) we have: (a) ¬(Un)ϕ(n) ←→ (Un) ¬ϕ(n), (b) (Un)ϕ(n) & (Un)ψ(n) ←→ (Un) (ϕ(n) & ψ(n)), (c) (Un)ϕ(n) ∨ (Un)ψ(n) ←→ (Un) (ϕ(n) ∨ ψ(n)).

128

Chapter I. Finite-Dimensional Ramsey Theory

Given a positive integer k and an ultraﬁlter U we can deﬁne its k-power U k by deciding when a subset X of N[k] belongs to U k as follows: X ∈ U k iﬀ (Un0 )(Un1 ) · · · (Unk−1 )X(n0 , n1 , . . . , nk−1 ) where, of course, X(n0 , n1 , . . . , nk−1 ) has the meaning that the k-element set {n0 , n1 , . . . , nk−1 } is written in its <-increasing order and that it belongs to X. Then the properties (a), (b), and (c) of the quantiﬁer (Un) for this particular formula translate to the following facts about U k : (a ) X ∈ U k iﬀ (N[k] \X) ∈ U k . (b ) X, Y ∈ U k iﬀ X ∩ Y ∈ U k . (c ) X ∈ U k or Y ∈ U k iﬀ X ∪ Y ∈ U k . It follows that U k is indeed a nonprincipal ultraﬁlter on N[k] . Hence, given a ﬁnite coloring of N[k] we can ﬁnd a color P such that P ∈ U k . Given a set s ⊆ N of size l < k, let Ps = {t ∈ N[k−l] : max(s) < min(t) & s ∪ t ∈ P }. Note the following properties of these sets: (0) P∅ = P ∈ U k (1) {s ∈ P [l] : Ps ∈ U k−l } ∈ U l for 0 ≤ l < k. Let T = {s ∈ N[≤k] : Ps ∈ U [k−|s|] }, where in case s ∈ N[k] , we choose the convention that “Ps ∈ U 0 ” means “s ∈ P ”. Note that: (2) ∅ ∈ T (3) s ∈ T ∩ N[
I.1. Finite-Dimensional Ramsey Theorem

129

[2] → {0, 1} by Proof. Given a sequence {Mi }∞ i=0 of elements of U deﬁne c : N [2] letting c({i, j}) = 0 iﬀ j ∈ Mi . If M ∈ U is such that c M is constant, then the constant value has to be 0, giving us the conclusion j ∈ Mi whenever i < j ∈ M . For the converse implication, given c : N[2] → {0, 1}, for each i let ε(i) ∈ {0, 1} be such that Mi = {j > i : c(i, j) = ε(i)} ∈ U. Get M ∈ U such that j ∈ Mi whenever i < j ∈ M . Shrinking M we may assume that ε M is constant, so c M [2] is also constant.

Lemma I.1.5. A nonprincipal ultraﬁlter U on N is Ramsey iﬀ for every f : N → N there exists an M ∈ U such that f M is either 1 − 1 or constant. Proof. Given f , deﬁne a coloring c : N[2] → {0, 1} by letting c(i, j) = 0 iﬀ f (i) = f (j). Find M ∈ U with monochromatic M [2] . Then f is either constant or oneto-one on M. We prove the converse implication using the characterisation given in the previous Lemma. Let {Mi }∞ i=0 ⊆ U be a given sequence which we may assume to be decreasing and with empty intersection. Deﬁne f : N → N by f (i) = min{j : i ∈ Mj }. Then there is M ∈ U on which f is one-to-one. Choose a strictly increasing inﬁnite sequence (nk ) in N such that for all i, j, k ∈ N, if f (i) = j, then i ≤ nk implies j < nk+1 and j ≤ nk implies i < nk+1 . Applying the selection property to the function that collapses each interval [nk , nk+1 ) to a point, we obtain an N ∈ U which takes at most one point from each of the intervals. Going to a subset of N , we may assume that N does not intersect two consecutive intervals [nk , nk+1 ). Then P = M ∩ N is a member of U such that j ∈ Mi whenever i < j ∈ P. Exercise I.1.6. Show that if U is a Ramsey ultraﬁlter on N then for every (xn ) ∈ c0 there is M ∈ U such that n∈M |xn | < ∞. Note that for a given ultraﬁlter U on some set S and a sequence (xs )s∈S of elements of some compact space X there is a unique member x of X such that {s ∈ S : xs ∈ U} ∈ U for every open neighborhood U of x. We shall denote this fact by the formula lim xs = x.

s→U

Note also that the set βS of all ultraﬁlters on S has a natural compact Hausdorﬀ topology generated by basis {U ∈ βS : A ∈ U}

(A ⊆ S).

ˇ Exercise I.1.7. Show that βS is the Stone-Cech compactiﬁcation of the discrete space S, or in other words, that for every f : S → X where X is a compact Hausdorﬀ space there is a continuous map βf : βS → X such that βf S = f . Exercise I.1.8. Use Ramsey’s Theorem to show that every bounded inﬁnite se∞ quence (xn )∞ n=0 of real numbers has an inﬁnite subsequence (xnk )k=0 such that ∞ ∞ both (xnk )k=0 and (xnk+1 −nk )k=0 are convergent.

130

I.2

Chapter I. Finite-Dimensional Ramsey Theory

Spreading Models of Banach Spaces

Recall the notion of a Schauder basis of an inﬁnite-dimensional Banach space X, a sequence (ei ) ⊆ X with the property that for every x ∈ X there is a unique sequence (ai ) of scalars such that x=

∞

ai ei .

i=0

Given a Schauder basis (ei ) of X the sequence (Pn )∞ n=1 of canonical projections is n−1 ∞ deﬁned by Pn (x) = i=0 ai ei where x = i=0 ai ei . The sequence (Pn )∞ n=1 of norms is uniformly bounded and the corresponding least upper bound is the basis constant of (ei ). By going to an equivalent norm of X this constant can be made to be equal 1. One also implicitly assumes that ei = 1 for all i. Exercise I.2.1. Show that if(ei ) is a Schauder basis of a Banach space X with ∞ constant C > 0, and if x = i=0 ai ei is a vector from X of norm at most 1 then |ai | ≤ 2C for all i. Deﬁnition I.2.2. A sequence (ei ) ⊆ X is an unconditional basic sequence if there n−1 n is C < ∞ such that for all ﬁnite sequence (ai )n−1 i=0 of scalars and (εi )i=0 ∈ {±1} ,

n−1

εi ai ei ≤ C

i=0

n−1

ai ei .

i=0

The smallest C satisfying this is called the unconditional basis constant of (ei ). Exercise I.2.3. Show that (ei ) ⊆ X is an unconditional basic sequence if for every x in the closed linear span of (ei ) there exists a sequence (ai ) of scalars such that x=

∞

aπ(i) eπ(i)

i=0

for every permutation π of N. Exercise I.2.4. Show that the standard bases of p (1 ≤ p < ∞) and c0 are 1-unconditional. One of the ﬁrst uses of Ramsey Theory in the study of Banach spaces is based on the following concept. Deﬁnition I.2.5. Given a normalized basic sequence (xn ) in some Banach space X, its spreading model is a normalized basic sequence (yn ) in possibly another Banach space Y such that for every integer k ≥ 0 there is an integer l ≥ 0 such that k k 1 | ani xni − ani yni | < k i=0 i=0

I.2. Spreading Models of Banach Spaces

131

for every sequence of integers l ≤ n0 < · · · < nk and for every sequence (ani )i≤k of scalars of modulus at most 1. Note that while a spreading model for a given normalized basic sequence (xn ) needs not to exist, it is unique whenever it does. Recall that two basic sequences (xn ) and (yn ) in possibly diﬀerent Banach spaces X and Y are C-equivalent for some constant C > 0 whenever (1/C)

∞ n=0

a n xn ≤

∞

an yn ≤ C

n=0

∞

a n xn

n=0

for every choice of scalars (an ) of modulus at most 1. Deﬁnition I.2.6. A basic sequence (xn ) in some Banach space X is C-spreading for some C > 0 if it is C-equivalent to all of its inﬁnite subsequences. Example I.2.7. The canonical basic sequences of c0 and p (p ≥ 1) are 1-spreading. Note the following reformulation of the existence of a 1-spreading model. Deﬁnition I.2.8. A basic sequence (xn ) in some Banach space X is asymptotically spreading if there is a 1-spreading basic sequence (yn ) in a possibly diﬀerent Banach space Y such that for all k, ε > 0 there is n ∈ N such that (xni )ki=0 ∼1+ε (yi )ki=0 1 for every choice of integers n < n0 < · · · < nk . The following Lemma gives the intended meaning to this deﬁnition in regard to Deﬁnition I.2.5. Lemma I.2.9. A basic sequence (xn ) is asymptotically spreading iﬀ it admits a 1-spreading model. Theorem I.2.10 (Brunel–Sucheston). Every normalized basic sequence (xn ) in some Banach space X has a subsequence which is asymptotically spreading. Proof. Let C > 0 be the basic constant of (xn ). For every integer k > 0 we ﬁx a 1 ﬁnite ( 2k )-net Dk in [−2C, 2C]k in the metric determined by the supremum norm. We assume moreover that Dk is closed under scalar multiplication by ±/k (0 ≤ ≤ k). Then for every k > 0 and d = (d0 , . . . , dk−1 ) ∈ Dk we have a map fd : N[k] −→ R+ deﬁned by fd ({n0 , n1 , . . . , nk−1 }< ) =

k−1

di xni .

i=0 1 (u

i)

∼C (vi ) is our notation for C-equivalence deﬁned above.

132

Chapter I. Finite-Dimensional Ramsey Theory

k−1 Note that fd ({n0 , n1 , . . . , nk−1 }) ≤ i=0 |di | for every {n0 , n1 , . . . , nk−1 } ∈ N[k] . In other words, the range of fd is bounded in R for all d ∈ Dk and k > 0. It follows that lim fd (s) = rd s→U k

is ﬁnite for all > 0 and d ∈ Dk . To simplify the notation we assume that U is, in fact, a Ramsey ultraﬁlter, so by Corollary 1.3 we can ﬁnd Mk ∈ U (k > 0) such that 1 |fd (s) − rd | < . 2k [k]

for all k > 0, d ∈ Dk and s ∈ Mk . We assume that the sets Mk are decreasing. Pick inﬁnite M = (ni )i∈N (in fact in U) such that nj ∈ Mni whenever i < j. Deﬁne a norm · on c00 by letting

k−1

di ei = rd

(k > 0, d = (d0 , . . . , dk−1 ) ∈ Dk ).

i=0

Note that indeed this set of equalities deﬁnes a norm since for example for a scalar λ for which λd = (λd0 , . . . , λdk−1 ) belongs to Dk , we have

k−1

λdi ei = λrd .

i=0

k−1 Note that in this norm the basis (ei ) of c00 is 1-spreading, i.e., i=0 λai eni = k−1 i=0 λai ei for every choice of n0 < · · · < nk−1 , and k > 0. By the choice of M = (nj ), k−1 k−1 1 di ei − di xnji | < | k i=0 i=0 for every k > 0 and every choice of j0 < j1 < · · · < jk−1 as long as j0 ≥ k. Since (Dk ) is a sequence of ﬁner and ﬁner nets, this sequence of estimates are suﬃcient to conclude that ((ei ), · ) is a spreading model of the subsequence (xnj ) of our given sequence (xn ). The following fact shows the going to a block-subsequence of the spreading model (en ) one can improve its behavior. ∞ Lemma I.2.11. If (en )∞ n=0 is spreading then (e2n − e2n+1 )n=0 is 1-unconditional.

Proof. Let fi = e2i − e2i+1 , i = 0, 1, . . . . We need to verify that for every pair of ﬁnite sets of integers I ⊆ J and every choice aj (j ∈ J) of scalars of modulus at most 1, ai fi ≤ aj fj . i∈I

j∈J

I.2. Spreading Models of Banach Spaces

133

Pick an integer n > 0. Choose a block sequence sj (j ∈ J) of ﬁnite subsets of N (i.e., si < sj whenever i < j in J) such that |sj | = 2 for j ∈ I and |sj | = n + 1 for j ∈ J \ I. For j ∈ J and k < |sj | let sj (k) denote the kth member of sj according to the increasing enumeration of sj . Then by the 1-spreading property of (ei ) and for all k < n, aj fj = ai (esi (1) − esi (0) ) + aj (esj (k+1) − esj (k) ). j∈J

i∈I

j∈J\I

Summing up these n inequalities and dividing by n, we get

aj fj ≥

j∈J

n−1 1 n ai (esi (1) − esi (0) ) + aj (esj (k+1) − esj (k) ) n i∈I

≥

k=0 j∈J\I

1 |aj | (esj (n) − esj (0) ) n

ai (esi (1) − esi (0) ) −

i∈I

≥

j∈J\I

2|J \ I| . n

ai fi +

i∈I

Since

2|J\I| n

→ 0 as n → ∞ this gives us the desired inequality.

The following result gives us a suﬃcient condition for the spreading model (en ) to be unconditional. Theorem I.2.12 (Brunel–Sucheston). Every normalized weakly null basic sequence (xn ) in some Banach space X has a subsequence with a 1-spreading unconditional model. Proof. On the basis of the previous proof, it suﬃces to show that the spreading model (ei ) of the subsequence (xni ) of (xn ) is 1-unconditional under the additional assumption that (xn ) is weakly null. In other words we have to prove that

k

ai ei ≤

k

ai ei

i=0

i=i0

for all 0 ≤ i0 ≤ k and all choices of scalars ai (i ≤ k) with ai0 = 0. Fix > 0. Find n such that k k | ai ei − ai xni | < i=i0

|

k i=0

i=i0

ai ei −

k

ai xni | <

i=0

for all choices of integers nk > nk−1 > . . . > n0 ≥ n. Since (xni ) is weakly null by

134

Chapter I. Finite-Dimensional Ramsey Theory

Mazur’s theorem (see II.3.35 below) there is a sequence of integers kl > kl−1 > · · · > k0 ≥ nk . and a sequence λi (i ≤ l) of positive scalars which add to 1 such l that the convex-combination y = i=i0 λi xni has norm at most |ai | . Then for 0 every j ∈ [k0 , kl ],

k

ai ei ≥

i 0 −1

i=0

ai xni + ai0 xj +

i=0

k

ai xni −

i=i0 +1

Note that this inequality remains valid if we replace xj by any convex combination of vectors satisfying this inequality, and so in particular if we replace xj by y. Hence, i k k 0 −1 ai ei ≥ ai xni + ai0 y + ai xni − i=0

≥

i=0 k

ai xni + |ai0 |y − ≥

i=i0

i=i0 +1 k

ai xni − 2.

i=i0

Since > 0 was arbitrary this ﬁnishes the proof.

We shall see below that going to block subsequences of the spreading model one can always achieve the unconditionality. This will be the subject matter of our next section but let us ﬁnish the present section by pointing out some standard examples that can be used to test these new notions. Example I.2.13. The Schreier space S is the completion of (c00 , · S ), where the norm is given by xS = sup |x(n)| : E ⊆ N and |E| ≤ min(E) + 1 n∈E

Note that while 1 does not embed into S the spreading model of its Schauder basis is isomorphic to 1 . This is essentially the ﬁrst nontrivial example of what is usually called asymptotic 1 space. The second example of an asymptotic 1 space is the following well-known space. Example I.2.14. The Tsirelson space is the completion of (c00 , · T ), where · T is implicitly given by a system of equations n−1 1 xT = x∞ ∨ sup E x : n ≤ E0 < E1 < · · · < En−1 , 2 =0

where for a subset E of N and x ∈ c00 by Ex we denote the member of c00 which is equal to x on E and 0 otherwise. Notice that the standard basis (en ) of c00 is a Schauder basis of T .

I.3. Finite Representability of Banach Spaces

135

Exercise I.2.15. (1) Prove that for every ε > 0 there is a vector x = n an en ∈ T such that εx1 > xT . Give explicitly such a vector x. (2) Prove the statement (1) where (en ) is replaced by an arbitrary normalized sequence (xn ) of ﬁnitely supported vectors of T such that for every n max supp xn < min supp xn+1 2 . Use this to show that 1 does not embed into T . (3) Verify that Schreier’s as well as Tsirelson’s space have spreading models isomorphic to 1 .

I.3

Finite Representability of Banach Spaces

The construction of the spreading model presented in the previous section suggests considering ultrapowers X I /U of Banach spaces X. Thus, for a Banach space X, an (inﬁnite) index set I and a (nonprincipal) ultraﬁlter U on I we let X I /U = {(xi )i∈I ∈ X I : lim xi ≤ ∞}/ ∼, i→U

where we put (xi )i∈I ∼ (yi )i∈I if limi→U xi − yi = 0, and where we equip X I /U with the norm [(xi )i∈I ] = lim xi . i→U

Deﬁnition I.3.1. A Banach space X is ﬁnitely representable in a Banach space Y if for every > 0 and every ﬁnite-dimensional subspace X0 of X there is a ﬁnite-dimensional subspace Y0 of Y and an isomorphism T : X0 → Y0 such that T T −1 < 1 + . Note that if (en ) is a spreading model of some sequence (xn ) in some Banach space X then any space which is ﬁnitely representable in (en ) is also ﬁnitely representable in X. Theorem I.3.2. The ultrapower X I /U is a Banach space which contains a natural isometric copy of X and which is ﬁnitely representable in X. Proof. To see that X I /U is complete, consider a Cauchy sequence (v n )∞ n=0 of elements of X I /U. Going to a subsequence of (v n )∞ we can choose a decreasing n=0 sequence I = I0 ⊇ · · · ⊇ In ⊇ . . . of elements of U and representatives (xni )i∈I ∈ v n (n = 0, 1 . . . ) such that n m for all i ∈ In (1) xm i − xi ≤ 2 m+1 m (2) xi = xi for i ∈ Im \ Im+1 . Then for every i ∈ I the sequence (xni )∞ n=0 is Cauchy in X so we can assign to it I a limit xi . Then [(xi )i∈I ] is the limit of the sequence (v n )∞ n=0 in X /U. Identifying x ∈ X with the constant sequence (x)i∈I gives us an isometric embedding of X into X I /U. 2 such

a sequence (xn ) is usually called a block subsequence of (en )

136

Chapter I. Finite-Dimensional Ramsey Theory

To show that X I /U is ﬁnitely representable in X, choose a ﬁnite-dimensional subspace V0 of X I /U spanned by vectors v 0 , . . . , v n of norm 1. Choose C > 0 such that n n n C |ak | ≤ ak v k ≤ |ak | k=0

k=0

k=0

for every choice of scalars Let v = for k ≤ n. and for d ∈ D Fix > 0. Choose a ﬁnite (C · )-net D in the unit ball of n+1 1 and k ≤ n let dk be the kth coordinate of d relative to the basis of n+1 . Then 1 (ai )ni=0 .

(∀ d ∈ D)(Ui )C ·

n

k

[(xki )i∈I ]

|dk | ≤

k=0

n

dk xki ≤

k=0

n

|ak |.

k=0

Since D is ﬁnite we can ﬁnd a single set J ∈ U such that for all i ∈ J and d ∈ Dn , C·

n k=0

|dk | ≤

n k=0

dk xki ≤

n

|ak |.

k=0

Fix i ∈ J and let X0 be the linear span of (xki )nk=0 , and let T : V0 → X0 be determined by T (v k ) = xki . Then one straightforwardly checks that T · T −1 ≤ 1 + 3

The following result shows that the notion of ultrapower captures the notion of ﬁnite representability in Banach spaces in a very precise sense. Theorem I.3.3. If a Banach space X is ﬁnitely representable in some other Banach space Y then there is an ultrapower Y I /U of Y which contains an isometric copy of X. Proof. Let I be the collection of all pairs (V, ) where V is a ﬁnite-dimensional subspace of X and > 0. We order I by letting (U, ) ≤ (V, δ) if U ⊆ V and ≥ δ. Note that (I, ≤) is upward directed, so we can ﬁnd an ultraﬁlter U on I such that {(V, δ) ∈ I : (V, δ) ≥ (U, )} ∈ U for all (V, ) ∈ I. By the hypothesis of the theorem, for every ﬁnite-dimensional subspace V ⊆ X and > 0 we can choose an isomorphism T(V,) : V → U where U is some ﬁnite-dimensional subspace of Y . This determines T : X → Y I /U by T (x) = (T(V,) (x))(V,)∈I) , / V . Then T is a norm-preserving linear operator where we let T(V,) (x) = 0 if x ∈ from X into Y I /U.

I.3. Finite Representability of Banach Spaces

137

The ultrapower technique leads to a natural realization of the spreading model of a given basic sequence (xn ) in some Banach space X. To see this take a nonprincipal ultraﬁlter U on N and consider its ﬁnite Fubini powers U k (k ≥ 1). Note that there is a natural identiﬁcation X N /U 1 ⊆ X N /U 2 ⊆ · · · ⊆ X N /U k ⊆ . . . [1]

[2]

[k]

via the sequence of linear isometric embeddings ϕ : X N /U k → X N /U l (k ≤ l) [k]

[l]

deﬁned by ϕ([(ys )s∈N[k] ]) = [(zt )t∈N[l] ] where zt = ytk . Each ultrapower X N /U k has a distinguished vector ek determined by our starting sequence (xn )∞ n=0 ⊆ X as [k]

ek = [(ys )s∈N[k] ], where ys = xmax(s) . Then for every x ∈ X, every integer n > 0 and every sequence ak (k ≤ n) of scalars we have the equality x +

l−1 k=o

ak ek l = lim x + s→U l

a|s∩n| xn

n∈s

where for s ∈ N[l ] and n ∈ s, we let s ∩ n = {m ∈ s : m < n} and where on the [l] left-hand side x ∈ X is identiﬁed with the constant sequence [(x)s∈N[l] ] ∈ X N /U l . It follows that (ek )∞ k=0 is a spreading model of a subsequence of (xn ) found via the simple diagonal procedure presented above during the course of the proof of Theorem I.2.10. The spreading sequence (ek ) has some interesting block-sequences, i.e., sequences (e2n − e2n+1 )∞ n=0 considered above in Lemma I.2.11. In fact one can ﬁnd block-subsequences of (en ) with stronger properties. Lemma I.3.4. Every spreading normalized basic sequence (en )∞ n=0 has a block∞ subsequence whose spreading model (fk )∞ is 1-equivalent to ( n fn )n=0 for any k=0 ∞ choice of (n )n=0 of signs. Proof. By Lemma I.2.11 we may assume that (en )∞ n=0 is 1-unconditional. We distinguish two cases Case 1. li=0 ei → ∞ as l → ∞. Choose ﬁnite sets Fk (k = 0, 1, . . . ) such that for every k |Fk |−1 (a) i=0 ei ≥ k2 , and (b) max Fk + 1 < min Fk+1 . For k = 0, 1, . . . , let i∈Fk σk (i)ei fk = , |Fk |−1 i=0 ei

138

Chapter I. Finite-Dimensional Ramsey Theory

where σk : Fk → {−1, 1} is the alternating sign-assignment starting with 1. We want to show that the spreading model of (fk ) has the desired property. To see this, it suﬃces to show that for every n ≤ k0 < · · · < kn−1 , every sequence of signs (k )nk=0 and every choice of scalars (ai )n−1 i=0 , |

n−1

ai fki −

i=0

n−1

i ai fki | ≤

k=0

2n maxn−1 i=0 |ai | . k02

(I.1)

Fix n ≤ k0 < · · · < kn−1 , a sequence of signs (k )nk=0 and a sequence of scalars (ai )n−1 i=0 . For i ≤ n − 1, let i∈Gk τk (i)ei gki = , |Gk |−1 i=0 ei where Gi = (Fi \ {min Fi }) ∪ {max Fi + 1}, and τi : Gi → {−1, 1} is the alternating sign-assignment starting with 1. Notice that (c) for every i ≤ n − 1 |Fi | = |Gi |, (d) for every i < n − 1, Gi < Fi+1 , and |Fk |−1 ei )(eli + (−1)|Fi |+1 eri +1 ), where (e) for every i ≤ n − 1 fki + gki = (1/ i=0i li = min Fi and ri = max Fi . Let I + = {i < n : k = 1} and I − = {i < n : k = −1}. Note that since (ei ) is spreading we have that

n−1

i ai fki =

i=0

ai fki −

ai gki .

(I.2)

i∈I −

i∈I +

So, using that (en ) is 1-unconditional, and equation (I.2) |

n−1 i=0

ai fki −

n−1

i ai fki | =|

i=0

n−1 i=0

≤

ai fki −

ai fki −

i∈I +

ai gki | ≤

i∈I −

ai (fki + gki ) ≤

i∈I −

≤

maxi≤n−1 |ak | (eli + (−1)|Fi |+1 eri +1 ) ≤ k02 − i∈I

maxi≤n−1 |ak | ≤ 2n, k02

(I.3)

as desired. Case 2. supl li=0 ei < ∞. Then (ei ) is equivalent to the standard basis of c0 , so by James’ theorem for each l > 0 we can ﬁnd a block subsequence (fkl )∞ k=0 of (ei ) which is (1 + 1/l)-equivalent to the standard basis of c0 . So the spreading k ∞ model (fi )∞ i=0 of (fk )k=0 is 1-equivalent to the standard basis of c0 and is therefore ∞ equivalent to (i fi )∞ i=0 for every choice (i )i=0 of signs.

I.3. Finite Representability of Banach Spaces

139

To summarize what has been proved upto this point we need a deﬁnition quite analogous to Deﬁnition I.3.1 above. Deﬁnition I.3.5. Suppose X and Y are Banach spaces with Schauder bases (en ) and (fn ), respectively. We say that X is ﬁnitely block representable in Y if for every > 0 and every ﬁnite-dimensional block subspace X0 of X there is a ﬁnitedimensional block subspace Y0 of Y and an isomorphism T : X0 → Y0 such that T T −1 < 1 + . Clearly, if (ei ) is a spreading model of a subsequence of some basic sequence (xn ) in some Banach space X, then the space generated by (ei ) is ﬁnitely block representable in (xn ). So the following result summarizes the results obtained so far. Theorem I.3.6. Suppose (xn ) is a normalized basic sequence in some Banach space X. Then there is a Banach space Y with a Schauder basis (yn ) which is ﬁnitely block representable in (xn ) and has the property

n i=0

ai yi =

n

σi ai yki

i=0

for every n, every sequence k0 < k1 < · · · < kn of integers, every sequence a0 < a1 < · · · < an of scalars, and every sequence σ0 , σ1 , . . . , σn of signs. Corollary I.3.7. For every positive integer n and > 0 there is an integer m such that every basic sequence (ei )m i=0 in some Banach space X contains a block subsequence (fi )ni=0 such that

n

σi ai fi ≤ (1 + )

i=0

n

ai fi

i=0

for every choice a0 , a1 , . . . , an of scalars and σ0 , σ1 , . . . , σn of signs.

Starting from the basic sequence (yn ) of Theorem I.3.6 one can ﬁnd its blocksubsequence whose spreading model is equivalent to the basis of one of the standard sequence space p (1 ≤ p < ∞) or c0 . This is the content of the following result. Theorem I.3.8 (Krivine). Suppose (ei ) is a basic sequence in some Banach space such that n n ai ei = σi ai eki i=0

i=0

for every n ≥ 0, sequence of integers 0 ≤ k0 < k1 < · · · < kn , sequence a0 , a1 , . . . , an of scalars and sequence σ0 , σ1 , . . . , σn of signs. Then the unit vector basis of either p for some 1 ≤ p < ∞ or of c0 is ﬁnitely block representable in (ei ).

140

Chapter I. Finite-Dimensional Ramsey Theory

Proof. We need a convenient way to “double” a given ﬁnitely supported vector of (ei ) as well as to “triple” it. In other words to have bounded operators x → D(x) and x → T (x) which commute and which perform these two operations, respectively. A natural way to formalize this is to re-enumerate (ei ) as (ei )i∈Q and deﬁne D and T on the span X1 of (ei )i∈Q∩[0,1) by letting D(eq ) = eq/2 + e(q+1)/2 and T (eq ) = eq/3 + e(q+1)/3 + e(q+2)/3 . Note that this indeed deﬁnes bounded linear operators on X1 which commute. In fact D ≤ 2 and T ≤ 3. Taking an ultrapower over a block subsequence (xn ) of (ei ) such that for some 1 ≤ λ ≤ 2, the sequence D(xn ) − λxn converges to 0 one ﬁnds a sequence (y˜n ) in the ultrapower and 1 ≤ µ ≤ 3 such that T (xn ) − µy˜n converges to 0. So, for some normalized block subsequence (zn ) of (xn ) we have that D(zn ) − λzn → 0 and D(zn ) − µzn → 0. For n, k ≥ 0 let zn k be the copy of the vector zn when Q ∩ [0, 1) is shifted to Q ∩ [k, k + 1) in the natural way. Using (fk ) to denote the standard basis of c00 we use (zn k ) to deﬁne a norm on c00 by

l

ak fk = lim n→U

k=0

l

ak zn k .

k=0

Let Y be the completion of c00 relative to this norm. Then for every pair x and y of ﬁnitely supported vectors of Y and index k such that x < k, k + 1, k + 2 < y, we have x + fk + fk+1 + y = x + λfk + y and x + fk + fk+1 + fk+2 + y = x + µfk + y. It follows that for every pair of integers m and n,

m n 2 3

fk = λm µn

(I.4)

k=1

More generally,

l k=1

ek =

j

λmi µni ei

(I.5)

i=1

m whenever l = ji=1 2mi 3ni . So if λ = 1 or µ = 1 we would get that 2k=1 fk = 1 3n or k=1 fk = 1, respectively, so the basis (fk ) of Y is 1-equivalent to the basis of c0 . So we may concentrate on the case when λ, µ > 1. Note that by (I.4) the function ϕ(2m 3−n ) = λm µ−n is a nondecreasing multiplicative function, and since {2n 3−n : m, n ∈ N} is dense in R+ , it extends to a nondecreasing multiplicative function ϕ : R+ → R+ . So there is 1 ≤ p < ∞ such that ϕ(v) = v 1/p . In particular, λ = 21/p and µ = 31/p . l Consider now the function ψ(l) = k=1 fk . Then taking mi = m and ni = n 1 (1 ≤ i ≤ j) in (I.5), we obtain that ψ(j · 2m · 3n ) = λm · 3n · ψ(j) = (2m · 3n ) p · ψ(j).

I.3. Finite Representability of Banach Spaces

141

Since ψ is nondecreasing and subadditive and since {2n 3−n : m, n ∈ N} is dense in R+ one easily concludes that, in fact, ψ(j) = j 1/p for all j ∈ N. Using this and (I.5) again we see that if for each i ≤ j we have scalars ai of the form λmi µni , then for l = ji=1 2mi 3ni we have that

j i=1

ai fi =

l k=1

1 p

fk = l = (

j

1

api ) p .

i=1

Since the sequence normalizations of sequences (a1 , . . . , aj ) of this form are dense in the unit sphere of pj we have this equality for every choice of scalars. Corollary I.3.9. For every basic sequence (xn ) in some Banach space X the standard basis of some p (1 ≤ p < ∞) or of c0 is ﬁnitely block representable in the closed linear span of (xn ). It is worth stating also a ﬁnite form of Corollary I.3.9 proved via the usual compactness argument. Corollary I.3.10. For every integer n and constants 0 < , C < ∞ there is an integer m = m(n, , C) such that if X is an m-dimensional Banach space with normalized basis (xj )m j=1 and constant C there is a p ∈ [1, ∞] and a block subsequence (fi )ni=1 of (xj )m j=1 which is (1 + )-equivalent to the standard basis of pn . We ﬁnish this section by mentioning another ﬁnite-dimensional result. Theorem I.3.11 (Odell–Rosenthal–Schlumprecht). Suppose h is a uniformly continuous real-valued function deﬁned on the unit sphere of some Banach space X with Schauder basis (xn ). Then there is λ ∈ R such that for every > 0 and every integer n > 0 there is a block subsequence (fi )ni=0 such that |h(y) − λ| < for all y ∈ (fi )ni=0 of norm 1. It turns out that an inﬁnite-dimensional analogue of this result is false even for Banach spaces of the form p (1 ≤ p < ∞), though it is true for the Banach space c0 . We return to this point in another section of these notes.

Chapter II

Ramsey Theory of Finite and Inﬁnite Sequences II.1

The Theory of Well-Quasi-Ordered Sets

Inﬁnite-dimensional Ramsey theory is a branch of Ramsey theory initiated in the early 1960s by Nash–Williams in his attempts to extend a well-known theorem of Kruskal that the ﬁnite trees are well-quasi-ordered (w.q.o.) under the inf-preserving embeddings (see Theorem II.1.9 below). Recall that a quasi-order (q.o.) is a set Q with a binary transitive and reﬂexive relation ≤. We write x < y if x ≤ y and y x, we write x|y if x y and y x, and we write x ≡ y if x ≤ y and y ≤ x. A quasi-ordered set Q is well-quasi-ordered (w.q.o.) if there are no inﬁnite descending sequences x0 > x1 > · · · > xn > · · · and no inﬁnite antichains xm | xn

(m = n).

Lemma II.1.1. A quasi-ordered set Q is w.q.o. iﬀ for every inﬁnite sequence (xn ) ⊆ Q there exists m < n such that xm ≤ xn iﬀ for every inﬁnite sequence (xn ) ⊆ Q there exists an inﬁnite sequence n0 < n1 < · · · < nk < · · · of integers such that xni ≤ xnj whenever i < j. Proof. Use the 2-dimensional Ramsey theorem.

Corollary II.1.2. If Q0 and Q1 are w.q.o., then so is their cartesian product Q0 × Q1 . Lemma II.1.3. Suppose (xn ) is a sequence of elements of a q.o. set Q. Then there is an inﬁnite subsequence (xnk ) such that either

144 (1) xnk | xn

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences whenever k = , or

(2) xnk ≤ xn

whenever k < , or

(3) xnk > xn

whenever k < .

Proof. Apply the 2-dimensional Ramsey theorem to the 3-coloring given by (1), (2) and (3). A quasiordered set Q is well-founded if it contains no inﬁnite sequence (xn ) such that xm > xn whenever m < n. Corollary II.1.4. If a well-founded q.o. set Q is not w.q.o., then it contains an inﬁnite sequence (xn ) such that xm xn whenever m < n. An inﬁnite sequence (xn ) of elements of some q.o. set Q is a bad array if xm xn holds for all m < n. Deﬁnition II.1.5 (Nash–Williams). A bad array (xn ) of a q.o. set Q is minimal bad if there is no bad array (yn ) of Q such that: (1) ∀m∃n ym ≤ xn , (2) ∃m∃n ym < xn . Lemma II.1.6. Suppose Q is a well-founded but not well-quasi-ordered set. Then there is a minimal bad array (xn ) in Q. Proof. By Corollary II.1.4 we can start with a bad array (x0n ) of Q and assume that its ﬁrst term x00 is minimal ﬁrst term of all possible bad arrays. Choose now a bad array (x1n ) such that x11 is minimal second term among all bad arrays (yn ) such that yo = x00 . Then choose a bad array (x2n ) ≤ (x1n ) such that x22 is minimal third term among all bad arrays (yn ) such that y0 = x00 and y1 = x11 , and so on. We claim that the diagonal sequence (xnn ) is a minimal bad array of Q. For suppose there is a bad array (yi ) such that ∀i∃ n yi ≤ xnn and ∃ i∃ n yi < xnn . Let m be the minimal n such that for some i, yi < xnn . Going to a subsequence of (yi ) we may assume that in fact y0 < xm m and that for all i there is n ≥ m such that yi < xnn . Then the sequence x00 , . . . , xn−1 n−1 , y0 , y1 , y2 , . . . is a bad array, contradicting the choice of xnn . Corollary II.1.7. If (xn ) is a minimal bad array of a quasi-ordered set Q then {q ∈ Q : (∃ i)q < xi } is w.q.o. Given a q.o. set Q we let the set Q<∞ of all ﬁnite sequences of elements of Q be ordered by: (x0 , . . . , xm ) ≤ (y0 , . . . , ym ) if there exists k0 < k1 < · · · < km ≤ n such that xi ≤ yki for all i ≤ m. Theorem II.1.8 (Higman). If Q is w.q.o. then so is Q<∞ .

II.1. The Theory of Well-Quasi-Ordered Sets

145

Proof. First of all note that Q<∞ is well-founded. So if Q<∞ is not w.q.o., then <∞ by Lemma II.1.6 we have a minimal bad array (tn )∞ . Note n=0 of members of Q n ∞ <∞ that a subsequence of (t )n=0 is also a bad array of Q , so by Lemma II.1.1 we may assume that if tn = (tn0 , . . . , tnkn ) (n = 0, 1, 2, . . . ), then n tm 0 ≤ t0 whenever m ≤ n.

Note that in particular kn ≥ 1 for all n. So if we let sn = (tn1 , . . . , tnkn ), we get an <∞ inﬁnite sequence (sn )∞ . Note that (1) and (2) n=0 of nonempty members of Q n of Deﬁnition II.1.5 are satisﬁed for (xn ) = (t ) and (yn ) = (sn ), so we conclude that (sn ) is not a bad array of Q<∞ . Thus there exists m < n such that sm ≤ sn . n m n Combining this with the fact that tm 0 ≤ t0 we conclude that t ≤ t , contradicting n the assumption that (t ) is a bad array. This ﬁnishes the proof. Recall that a tree is a partially ordered set (T, ≤T ) with the property that {x ∈ T : x ≤T y} is totally ordered for all y ∈ T . A tree T is rooted if it has a minimal element, root(T ). Given a quasi-ordered set Q, a Q-tree or a tree labeled by Q is a pair (T, f ) where T is a rooted tree and f : T → Q. Given two Q-trees (S, f ) and (T, g) we put (S, f ) ≤ (T, g) if there is a Φ : S → T such that for all x, y ∈ S: (1) x ≤S y iﬀ Φ(x) ≤T Φ(y) (2) Φ(x ∧ y) = Φ(x) ∧ Φ(y) (3) f (x) ≤Q g(Φ(x)). Theorem II.1.9 (Kruskal). If Q is w.q.o., then so is the class of all ﬁnite Q-trees. Proof. If the class of Q-trees is not w.q.o., then by Lemma II.1.6 we could choose a minimal bad array (Ti , fi ) (i = 0, 1, 2, . . . ) of Q-trees. Going to a subsequence we may assume that fn (root(Tn )) ≤Q fm (root(Tm )) where n < m. Let J be the class of all Q-trees of the form (T, f ) for which we can ﬁnd n and an immediate successor x of root(Tn ) such that T = {y ∈ Tn : x ≤T y} and f = fn T. By II.1.7 the class J is w.q.o. and by Higman’s theorem so is J <∞ . Thus we can ﬁnd n < m such that the element of J <∞ corresponding to the immediate successors of root(Tn ) is ≤ the element of J <∞ corresponding to the immediate successors of root(Tm ). It follows that (Tn , fn ) ≤ (Tm , fm ) contradicting our original assumption that the sequence {(Ti , fi )} is bad. Unfortunately Higman’s Theorem is not true in the inﬁnite dimension as the following example shows:

146

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Example II.1.10 (R. Rado). Let Q1 = {(i, j) : i < j ∈ N} quasiordered by (i, j) ≤ (k, l) iﬀ either i = k and j ≤ l, or j < k. Then

Q∞ 1

is not w.q.o.

In fact Rado’s example is a minimal quasi-ordered set with this property. Theorem II.1.11 (R. Rado). Suppose Q is w.q.o. while its power Qω is not w.q.o. Then Q contains an isomorphic copy of the quasi-ordered set Q1 . Proof. Suppose xn : ω → Q (n = 0, 1, 2, . . . ) is a bad array of members of Qω . By our assumptions Q is w.q.o., so for each n one can ﬁnd an integer kn such that for each k ≥ kn there are inﬁnitely many l with the property that xn (k) ≤ xn (l). Using Higman’s theorem and then Ramsey’s theorem and going to a subsequence we may assume that xn kn ≤ xm km whenever n ≤ m. It follows that we can cut oﬀ the initial segment xn kn from each of the xn and still obtain a bad array. So to save on notation we assume that kn = 0 for all n. Then for each n ≤ m there is an integer knm such that xn (knm ) xm (k) for all k ∈ ω. So in particular we have that xn (knm ) xm (kmp ) whenever n < m < p. Color a triple {n, m, p}< red or blue depending whether or not xn (knm ) ≤ xn (kmp ). Also color a quadruple {n, m, p, q}< green or yellow depending whether or not xn (knm ) ≤ xp (kpq ). Let M be an inﬁnite subset of ω with M [3] and M [4] monochromatic. Then {xn (knm ) : n, m ∈ M, n < m} is an isomorphic copy of Q1 in Q. Exercise II.1.12. Show that if Q is w.q.o., then its power Qω is well-founded. It was this diﬃculty with the notion of w.q.o. that has motivated the following remarkable deﬁnition. Deﬁnition II.1.13 (Nash–Williams). A quasi-ordered set Q is better-quasi-ordered (b.q.o.) if for every continuous map f : N∞ −→ Q1 there exists X ∈ N∞ such that f (X) ≤ f (∗ X), where ∗ X denotes the shift of X, i.e., ∗ X(n) = X(n + 1). This is in fact a modern reformulation of Nash–Williams’ original deﬁnition. In practice it is more convenient to work with the set N[∞] of all inﬁnite strictly increasing sequences of integers rather than with N∞ since in this case we can in particular identify sequences with the corresponding subsets of N. Latter advances of this area of Ramsey theory have shown that this deﬁnition does not change if we quantify over Borel functions rather than continuous ones. In the following section we shall examine this remarkable deﬁnition moving towards its original form as given by Nash–Williams. Exercise II.1.14. Show that every w.q.o. is b.q.o. Exercise II.1.15. Show that the product of two b.q.o.’s is a b.q.o. 1Q

is taken here with its discrete topology.

II.2. Nash–Williams’ Theory of Fronts and Barriers

II.2

147

Nash–Williams’ Theory of Fronts and Barriers

The purpose of this section is to study properties of families F that can always be achieved by taking restrictions of the form F M = {s ∈ F : s ⊆ M } = F ∩ P(M ), where M is an inﬁnite subset of N. Note that there is another form of restriction that one can take: F[M ] = {s ∩ M : s ∈ F}. One can also take two possible kind of closures of F, the downwards closure F = {t : t ⊆ s for some s ∈ F}, as well as topological closure F in the Cantor space 2N when sets are identiﬁed with their characteristic functions. One of the point behind the notion of barrier that we study in this section is that we have the equality F M = F M = F[M ] whenever F is a barrier on some superset of M, so this way one avoids the possible traps caused by mixing up between these two kinds of restrictions and the two kinds of closures. So let us introduce the key notions of this section. Deﬁnition II.2.1. A family B of ﬁnite subsets of N is called a front if (1) s t2 whenever s = t ∈ B. (2) B is inﬁnite and for every inﬁnite M ⊆ B there exists s ∈ B such that s M. If B instead of (1) has the following stronger condition: (1 ) s ⊆ t whenever s = t ∈ B. then B is called a barrier on B. To connect this notion with the Deﬁnition II.1.13 note that whenever we have a front B such that B = N and a mapping f : B −→ Q, we can extend f to a continuous map fˆ: N[∞] −→ Q by letting fˆ(M ) = f (s) where s ∈ B is unique with the property that s M . Clearly every continuous map g : N[∞] −→ Q (with discrete topology) has the form fˆ for some f : B −→ Q. 2s

t means that s is an initial segment of t.

148

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Exercise II.2.2. Prove this last assertion. Deﬁnition II.2.3. (i) A family F of ﬁnite subsets of N is thin if s t for every pair s, t of distinct members of F. (ii) A family F of ﬁnite subsets of N is Sperner if s t for every pair s = t ∈ F. (iii) A family F of ﬁnite subsets of N is Ramsey if for every ﬁnite partition F = F0 ∪ · · · ∪ Fk there is an inﬁnite set M ⊆ N such that at most one of the restrictions F0 M, . . . , Fk M 3 is non-empty. Exercise II.2.4. Show that the family N[k] of all k-element subsets of N is Ramsey (and Sperner). Lemma II.2.5. For every Ramsey family F there is an inﬁnite set M such that F M is Sperner. Proof. Let F = F0 ∪ F1 , where F0 is the family of all ⊆-minimal elements of F and apply the Ramsey property of F. Lemma II.2.6. For every Ramsey family F there is an inﬁnite set M such that F M is thin. Proof. Let F = F0 ∪ F1 , where F0 is the family of all -minimal members of F and apply the Ramsey property of F. Lemma II.2.7 (Nash–Williams). Every thin family is Ramsey. Proof. We present the original 1965 proof which involves the ﬁrst instance of what can be called ‘combinatorial forcing’. We let the variable M, N, P, . . . run over inﬁnite subsets of N, and s, t, m, . . . to run over ﬁnite subsets of N. A set D of inﬁnite subsets on N is open if N ⊆ M ∈ D implies N ∈ D. We say that such a D is dense below M if for every N ⊆ M there is a P ⊆ N in D. A dense-open-set assignment on M is a family Ds (s ∈ M [<∞] ) such that for all s ∈ M [<∞] ,4 Ds is dense-open below M/s Note the following property of this notion: (1) For every dense-open-set assignment Ds (s ∈ M [<∞] ) on M there is an N ⊆ M such that N/s ∈ Ds for all s ∈ N [<∞] . A function E : N[<∞] −→ P(N[<∞] ) is an extensor if s t for all s and t in E(s). Fixing an extensor E we give a deﬁnition: 3 Recall 4 M/s

that Fi M = Fi ∩ P(M ). = {m ∈ M : m > max(s)}.

II.2. Nash–Williams’ Theory of Fronts and Barriers

149

Deﬁnition II.2.8 (Combinatorial forcing). We say that s is inextensible in M if E(s) ∩ P(M ) = ∅. We say that s is strongly extensible in M if E(s) ∩ P(N ) = ∅ for every N ∈ [s, M ]5 . Note the following property of these notions: (2) For every M there is N ⊆ M such that every ﬁnite subset of N is either inextensible in N or is strongly extensible in N . Back to the proof of Nash–Williams’ Lemma. It suﬃces to show that for every pair F0 ⊆ F of thin families of ﬁnite subsets of N and every M there is N ⊆ M such that either F0 N = ∅ or F N ⊆ F0 Given such F0 ⊆ F deﬁne an extensor E by t ∈ E(s) iﬀ t s and t has an initial segment in F0 . By (2) shrinking the given M we may assume that every ﬁnite subset of M is either inextensible in M , or is strongly extensible. For s ∈ M [<∞] let Ds be the collection of all P ⊆ M such that either (a) s ∪ {n} is inextensible in M for all n ∈ P , or (b) s ∪ {n} is strongly extensible in M for all n ∈ P . Clearly, Ds (s ∈ M [<∞] ) is a dense-open-set assignment on M , so by (1), there is N ⊆ M such that (c) N/s ∈ Ds for all s ∈ M [<∞] . Case 1. ∅ is inextensible in M (and therefore in N ). In this case we have that F0 N = ∅. Case 2. ∅ is strongly extensible in M (and therefore in N ). Note that by the deﬁnition of extension E, an s is extensible in M iﬀ there is n ∈ M/s such that s ∪ {n} is extensible in M . So by the choice of N ⊆ M an s ⊆ N is extensible in N iﬀ for some (all) n ∈ N/s, s ∪ {n} is extensible in N . It follows that: (3) Every ﬁnite subset s of N is (strongly) extensible in N . Using (3) and the fact that F is thin, one concludes that N cannot contain a member of F \ F0 (see the deﬁnition of extension E). Corollary II.2.9. For every front F there is an inﬁnite set M ⊆ F such that the restriction F M is a barrier on M . Proof. Given a front B, let B0 be the set of all ⊆-minimal elements of B. By Nash– Williams’ lemma there is an inﬁnite M ⊆ B such that either B0 M = ∅ or (B \ B0 ) M = ∅. Note that the second alternative holds. It follows that in the deﬁnition of b.q.o., we may restrict ourselves to continuous maps of the form fˆ: N[∞] −→ Q where f : B −→ Q has its domain B to be 5 [s, M ]

= {P : s P ⊆ s ∪ M }

150

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

a barrier rather than a front. This gives us a way to ﬁnitize the outcome of the Deﬁnition II.1.13 . For a ﬁnite set ∅ = s ⊆ N, let ∗s

= s \ {min(s)}.

For ﬁnite nonempty subsets s and t of N, put s t iﬀ ∗ s t where is the strict version of the initial segment ordering . Thus, {2} {3}, {2, 3, 4} {3, 4, 6}, . . . The relation captures the shift-operator on N[∞] as shown in the following reformulation of Deﬁnition 3.3. Lemma II.2.10. A quasi-ordered set Q is b.q.o. iﬀ for every f : B −→ Q deﬁned on some barrier B there exists s t in B such that f (s) ≤ f (t). Deﬁnition II.2.11. For an integer k 1 and a barrier B, let B [k] := {s0 ∪ s1 ∪ · · · ∪ sk−1 : s0 s1 · · · sk−1 in B} Lemma II.2.12. If B is a barrier then so is any of its ﬁnite stretches B [k]. Proof. Given an inﬁnite set M , let M = M0 , . . . , Mi+1 = ∗ Mi , . . . , Mk−1 = ∗ Mk−2 be the sequence of ﬁrst k shifts of M . For each i < k there is unique si ∈ B such that si Mi . Then s0 s1 · · · si · · · sk−1 (II.1) and s = s0 ∪ s1 ∪ · · · ∪ sk−1 is a member of B[k] such that s M . Note also since B satisﬁes II.2.1(1) every member s of B[k] has a unique decomposition s = s0 ∪ · · · ∪ sk−1 into members of B. From this one easily concludes that B[k] also satisﬁes II.2.1(1 ), and so B[k] is a barrier. Lemma II.2.13. If Q is b.q.o., then for every barrier B and f : B −→ Q there exists an inﬁnite M ⊆ N such that f (s) ≤ f (t) for all s, t ∈ B M such that s t. Proof. Apply Nash–Williams’ generalization of Ramsey’s theorem (Lemma II.2.7) to the coloring B [2] = C ∪ D deﬁned by letting s ∈ C iﬀ f (s0 ) ≤ f (s1 ) where s0 , s1 ∈ B are unique such that s0 s1 and s = s0 ∪ s1 . Corollary II.2.14. The product of two b.q.o. is a b.q.o. Proof. Consider a map f : B → Q0 × Q1 from a barrier B into the cartesian product of the quasi-orderings. Let fi : B → Qi (i < 2) be the composition of f with the corresponding projection. Applying Lemma II.2.13 successively to f and then to f1 we get an inﬁnite set M ⊆ N such that fi (s) ≤ fi (t) for all s t from B M and i < 2. It follows that f (s) ≤ f (t) for all s t from B M , as required.

II.2. Nash–Williams’ Theory of Fronts and Barriers

151

The following lemma gives us an ordinal index on which one can carry inductive arguments about fronts and barriers. Lemma II.2.15. Every front is lexicographically well-ordered. Proof. The lexicographical ordering on ﬁnite subsets of N to which one refers is deﬁned as: s
(II.2)

for all i < j < k in N. Applying Ramsey’s theorem we get an inﬁnite set X ⊆ N such that either (a) min(si sj ) < min(sj sk ) for all i < j < k in X, or (b) min(si sj ) > min(sj sk ) for all i < j < k in X. Clearly the case (b) is impossible se we are left with (a). Applying Ramsey’s theorem once again we ﬁnd an inﬁnite subset Y of X such that for some ρ ∈ { , =, }, (c) si ∩ min(si sj )ρsj ∩ min(sj sk ) for all i < j < k in Y . Clearly the case when ρ is equal to is impossible. The case when ρ is equal to gives (d) si ∩ min(si sj ) sj ∩ min(sj sk ) for all i < j < k in Y and this contradicts property (2) of B. Se we are left with the case ρ == which together with (a) gives us that si

Exercise II.2.19. Show that the lexicographical rank of B1 ⊕ B0 is equal to the sum of lexicographical ranks of B0 and B1 . Deﬁnition II.2.20. Given two families B, C ⊆ FIN, let B ⊗ C = {s1 ∪ · · · ∪ sn : s1 < · · · < sn , si ∈ B, {min si }ni=1 ∈ C}. Exercise II.2.21. Prove that if B, C are fronts on M , then so is B ⊗ C. The following immediate fact gives us a way to construct fronts of arbitrary high lexicographical ranks.

152

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Lemma II.2.22. For an inﬁnite sequence (Bi )i∈N of fronts on N, the family B=

∞

B0 ⊕ B1 ⊕ · · · ⊕ Bi ⊕ {i}.

i=0

is also a front on N.

In particular, the construction of Lemma II.2.22 gives us the following example of a barrier of lexicographical rank ω ω . ∞ Example II.2.23. S = i=0 N[1] ⊕ N[1] ⊕ · · · ⊕ N[1] ⊕ {i} where at the level i we [1] [1] take N ⊕ · · · ⊕ N i times is an example of what one can call Schreier barrier. In general we have the following fact. Lemma II.2.24. For every countable indecomposable6 ordinal δ there is a barrier on N of lexicographical rank equal to δ. Proof. The lexicographical rank of N[1] is clearly equal to ω. To see the inductive step, suppose ∞ ωα = ω αi i=0

where α0 ≤ α1 ≤ · · · < α. For each i ∈ ω pick a barrier Bi on N of rank ω αi and let ∞ B= B0 ⊕ B1 ⊕ · · · ⊕ Bi ⊕ {i}. i=0

Then B is a front of rank ω α . Note that B is not necessarily a barrier. Note, however, that B is ‘uniform’ in the sense that it looks the same (and so in particular it has the same lexicographical rank) when restricted to an arbitrary inﬁnite subset of N. Applying II.2.9 and rescaling back to N we get the conclusion of the Lemma. Exercise II.2.25. Compute the lexicographical rank of the Schreier barrier. Banach space theory oﬀers many applications of fronts and barriers (see for example [2], [3], [7], [28], [68]) though the experts in this area prefer to think about them as compact (under the topology of pointwise convergence) families of ﬁnite subsets of N by considering in fact not fronts B themselves but their downwards closures B = {s ⊆ N : (∃t ∈ B)s ⊆ t}. For example, instead of Schreier’s barrier introduced above in Example II.2.23 they consider the corresponding Schreier family S = {s ⊆ N : |s| ≤ min(s) + 1}. 6A

limit ordinal δ is called indecomposable if there is no α, β < δ such that α + β = δ

II.3. Uniform Fronts and Barriers

153

During the last ten years or so (see [3], [2] and [68]) Schreier family has been generalized into the transﬁnite as follows: S0 ={{n} : n ∈ N} ∪ {∅}, n Sα+1 ={ Ek : n ≤ E0 < E1 < · · · < En , Ek ∈ Sα }, k=0

Sα ={E : E ∈ Sδn , n ∈ N, n ≤ E}, for δ a countable limit ordinal and (δn )∞ n=0 an increasing sequence of smaller ordinals converging to δ. The reader is invited to examine the corresponding sequence Sα (α < ω1 ) of fronts and compute their lexicographical ranks. While the sequence Sα (α < ω1 ) leaves out certain lexicographical ranks the main diﬃculty with Sα (α < ω1 ) is however their non uniformity, i.e., the diﬃculty in taking restrictions to inﬁnite subsets of N while preserving their initial structure, a feature that is very important in Ramsey theory. The purpose of the next section is to examine a uniform hierarchy of fronts and barriers that has been quite useful in Ramsey theory over the last three decades and which hopefully will ﬁnd its uses in Banach space theory as well. Exercise II.2.26. Show that for each countable ordinal α the Schreier family Sα is spreading, i.e., it has the property that a ﬁnite set {n0 , . . . , nk } enumerated increasingly belongs to Sα if one can ﬁnd a set {m0 , . . . , mk } ∈ Sα of the same size and enumerated increasingly such that mi ≤ ni for all i ≤ k.

II.3

Uniform Fronts and Barriers

For a family B of inﬁnite subsets of N and n ∈ N set Bn ={s ∈ B : n = min(s)} B{n} ={s ∈ N[<∞] : {n} ∪ s ∈ B, n < min(s)}. Deﬁnition II.3.1. Let α be a countable ordinal and let M be an inﬁnite subset of N. We say that a family B of ﬁnite subsets of N is α-uniform on M provided that: (a) α = 0 implies B = {∅} (b) α = β + 1 implies that ∅ ∈ / B and B{n} is β-uniform on M/n7 for all n ∈ M . (c) α > 0 limit implies that there is an increasing sequence {αn }n∈M of ordinals converging to α such that B{n} is αn -uniform on M/n for all n ∈ M . Note the following three properties of uniform families proved by straightforward induction on α. Lemma II.3.2. Every α-uniform family on M is a front on M . 7 M/n

= {m ∈ M : m > n}.

154

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

The second property is the main reason behind our choice of the word ‘uniform’ in the name for these families. Lemma II.3.3. If B is α-uniform on M and if N is an inﬁnite subset of M , then B N is α-uniform on N . So in particular (see II.2.9), every α-uniform family B has a restriction B M that is not just an α-uniform family but also an α-uniform barrier. Lemma II.3.4. If B is uniform (i.e., α-uniform for some α) on M , then B is a maximal thin family on M. Exercise II.3.5. Show that for every positive integer k, the family N[k] is the only k-uniform family on N. Exercise II.3.6. Find examples of ω-uniform and (ω + 1)-uniform families on N. Are they unique? The next immediate property (proved again by a straightforward induction on α and β) shows that the notion of uniform families behaves well with respect the operations of taking sums and products between two families of ﬁnite subsets of N. Lemma II.3.7. (a) Suppose that B and C are α and β-uniform on M , respectively. Then B ⊕ C is α + β-uniform on M . (b) If B is an α-uniform front on M , and C is a β-uniform front on M , then B ⊗ C is an α · β-uniform front on M . The following is one of the most important results connecting arbitrary families of ﬁnite subsets of N with fronts and barriers. It is also the key result in the development of the topological Ramsey theory of N[∞] (see Section II.6). Lemma II.3.8 (Galvin). For every family B of ﬁnite subsets of N and every inﬁnite M ⊆ N there exists an inﬁnite N ⊆ M such that the restriction B N is either empty or it contains a barrier. Proof. We may assume that the given M is equal to N and use variables M , N , P, . . . for inﬁnite subsets of N and s, t, n, . . . for ﬁnite subsets of N. Deﬁnition II.3.9 (Combinatorial forcing). We say that M accepts s if every X ∈ [s, M ]8 has an initial segment in B. If there is no N ⊆ M which accepts s we say that M rejects s. We say that M decides s if M either accepts or rejects s. Note the following monotonicity properties of this forcing relation: (i) If M accepts (rejects) s then every N ⊆ M accepts (rejects) s. (ii) For every M and s there exists N ⊆ M which decides s. (iii) M accepts s if and only if M accepts s ∪ {n} for all n ∈ M/s9 . 8 [s, M ] 9 M/s

= {X ∈ N[∞] : s X ⊆ s ∪ M }. = {m ∈ M : m > max(s)}.

II.3. Uniform Fronts and Barriers

155

(iv) If M rejects s then M does not accept s ∪ {n} for all but ﬁnitely many n ∈ M/s. To ﬁnish the proof a simple diagonalization will give us an M which decides all of its ﬁnite subsets. If M accepts ∅ we have reached one of the conclusions of the lemma. If M rejects ∅, let m0 ∈ M be minimal such that M rejects {n} for all n ≥ m0 in M (use (iv) and that M decides all of its ﬁnite subsets). Having deﬁned an increasing sequence m0 < m1 < · · · < mk of members of M with the property that M rejects s for all s ⊆ {m0 , . . . , mk } let mk+1 > mk be the minimal m ∈ M such that M rejects s ∪ {n} for all n ≥ m in M (use (iv) and that M decides all of its ﬁnite subsets). Finally let N = {mi : i ∈ N}. Then B N = ∅, ﬁnishing the proof of Lemma II.3.8. Corollary II.3.10. For every family B of ﬁnite subsets of N and for every inﬁnite M ⊆ N there exists an inﬁnite N ⊆ M such that B N = ∅ or B N contains an uniform barrier. Proof. By lemma II.3.8 we may assume to have an inﬁnite M ⊆ N such that B M contains a barrier. Shrinking B we may assume B is a barrier on M . Now the proof that we can shrink M further to have B M uniform is by induction on, say, lexicographical rank of B M . To see that this can be done note that rank(Bn M ) < rank(B M ) for every n ∈ M

Theorem II.3.11. The following are equivalent for a family F of ﬁnite subsets of N: (a) F is Ramsey. (b) There is an inﬁnite M ⊆ N such that F M is Sperner. (c) There is an inﬁnite M ⊆ N such that F M is either empty or uniform on M. (d) There is an inﬁnite M ⊆ N such that F M is thin. (e) There is an inﬁnite M ⊆ N such that for every inﬁnite N ⊆ M the restriction F N cannot contain two disjoint uniform families on N . Proof. (a) → (b) Apply the Ramsey property of F to the partition F = F0 ∪ F1 where F0 is the set of all ⊆-minimal elements of F. (b) → (c) This follows from Corollary II.3.10. (c) → (d) This follows from Lemma II.3.4. (d) → (e) By Lemma II.3.4 the union of two disjoint uniform families on the same set cannot be thin. (e) → (a) Let F = F0 ∪ · · · ∪ Fk be a given ﬁnite partition of F. By Corollary II.3.10 and Lemma II.3.3 we can ﬁnd an inﬁnite M such that for all i ≤ k the restriction Fi M is either empty or uniform. Going to a further subset of M we may assume that F M satisﬁes the conclusion of (5). It follows that at most one Fi M is nonempty.

156

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

The following general comparability property between diﬀerent barriers is quite useful. Lemma II.3.12. Suppose B and C are two uniform barriers on the same inﬁnite set M . Then there is an inﬁnite subset N of M such that either for every s ∈ B N there is t ∈ C N such that s ⊆ t, or vice versa for every t ∈ C N there is s ∈ B N such that t ⊆ s. Proof. Let B0 be the collection of all s ∈ B which are not included in any member of C. Similarly, let C0 be the collection of all t ∈ C that are not included in any member of B. By Theorem II.3.11 we can ﬁnd an inﬁnite N ⊆ M such that : (a) B0 N = ∅ or B N ⊆ B0 , and (b) C0 N = ∅ or C N ⊆ C0 . Now note that since B and C are barriers it is not possible to have at the same time the ﬁrst alternative in (a) and the ﬁrst alternative in (b). Given two barriers B and C we say that C dominates B if every member of B is included in some member of C. Recall the standard identiﬁcation of the power set of N with the Cantor space 2N via characteristic functions. This will give us the natural topology on the power set of N. Referring to this topology note the following property of uniform barriers: Lemma II.3.13. The topological closure B of an uniform barrier B is equal to its downward closure relative to the inclusion, or in other words, B = B, where B = {s ⊆ N : (∃t ∈ B)s ⊆ t}.

Note also the following pleasant properties of barriers relative to the operation B of taking the downwards closure under inclusion. Lemma II.3.14. (a) If B is a barrier on M the F N = F N for every N ⊆ M . (b) If F is a barrier on M , then for every N ⊆ M such that M \ N is inﬁnite, F N = F[N ], and so in particular F[N ] is downwards closed under inclusion. Exercise II.3.15. Show that the Cantor–Bendixson rank of the topological closure B of an α-uniform barrier B is equal to α + 1. Corollary II.3.16. Suppose that A is an α uniform barrier on M and B is a β uniform barrier on M , with α < β. Then there is some N ∈ M [∞] such that AN ⊆B Proof. By Lemma II.3.12, there is some inﬁnite subset N ⊆ M such that either A N ⊆ B or B N ⊆ A. We work to show that the second alternative is impossible. Suppose not; since B N is also a β-uniform barrier on N , Cantor– Bendixon rank of B N is β + 1, hence the Cantor–Bendixon rank of A is at least β + 1, which is impossible since A is α-uniform.

II.3. Uniform Fronts and Barriers

157

Lemma II.3.17. The lexicographical rank of an α-uniform barrier is equal to ω α .

Proof. Induction on α.

Note that Lemma II.3.13 relates uniform barriers on inﬁnite subsets of N and compact hereditary 10 families of ﬁnite subsets of N. In order to examine this relationship even more precisely, it is convenient to have the notation F[M ] = {s ∩ M : s ∈ F} for the trace of F on M , where F is an arbitrary family of subsets of N and M ⊆ N. This must not to be confused with the restriction F M = F ∩ P(M ) already used several times above as they are seldom equal. There is however an exception as the following fact shows. Lemma II.3.18. If F is a hereditary family of sets, then F[M ] = F M for every set M . Deﬁnition II.3.19. A family F of ﬁnite sets is called pre-compact if its topological closure F consists only of ﬁnite sets. Note the following immediate property of this notion. Lemma II.3.20. Suppose that F is a hereditary family of ﬁnite subsets oﬀ N. Then F is compact if and only if it is pre-compact. Lemma II.3.21. Suppose that F is a compact hereditary family of ﬁnite subsets of N. Then there is an inﬁnite set M such that the set F[M ]max of ⊆-maximal elements of F[M ] is a uniform barrier on M . Moreover, F[M ]max is also equal to the set F[M ] − max of -maximal elements of F[M ]. Proof. The proof is by induction on the Cantor–Bendixon rank α of the family F, the minimal countable ordinal α such that the αth -derivative of F is ﬁnite. If α = 0, then F is ﬁnite and in this case the conclusion of the lemma is immediate. Let us consider the case α > 0. Going to a restriction of the form F[N/n0 ] = F (N/n0 ), we may assume that F (α) = {∅}. For n ∈ N, we set F{n} = {t ∈ N[<∞] : t ∪ {n} ∈ F } ⊆ F.

(II.3)

Claim. For every n, the section F{n} is a compact hereditary family of CantorBendixon rank strictly smaller than α. Proof of the claim. It is clear that F{n} is pre-compact since it is a subset of F. To show that it is hereditary, consider t ⊆ u with u ∈ F{n} . Then {n}∪t ⊆ {n}∪u ∈ F 10 A

family F of sets is hereditary if A ⊆ B ∈ F implies A ∈ F .

158

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

and so {n} ∪ t ∈ F, since F is hereditary. This implies that u ∈ F{n} , as desired. It follows that F is compact. An easy induction on γ shows that {t ∪ {n} : t ∈ (F{n} )(γ) } ⊆ (F)(γ) .

(II.4)

This gives that the Cantor–Bendixon rank of F{n} is strictly small than α since otherwise {t ∪ {n} : t ∈ (F{n} )(α) } ⊆ {∅} which leads to a contradiction. Applying the inductive hypothesis, we can build a sequence (Mi ) of inﬁnite subsets of N such that, setting mi = min Mi , then for every i, we have that mi < Mi+1 and that F{mi } [Mi+1 ]max = Bi is a uniform barrier on Mi+1 . Let M∞ = {mi }, and let B= Bi M∞ ⊕ {mi }. (II.5) i

It is clear that B is a uniform family on M∞ but not necessarily a barrier on M∞ . Let M = {mi }i∈I ⊆ M∞ be such that B M is a uniform barrier. Note that BM = Bi M ⊕ {mi }. (II.6) i∈I

Let us show that F[M ]max = B M . Consider ﬁrst an s ∈ F[M ]max , and let i ∈ I be such that mi = min s . Fix t ∈ F such that s = t ∩ M ⊆ t ∩ Mi+1 . Since mi ∈ t and F is hereditary, we obtain that ∗ s ∈ F{mi } [Mi+1 ]. Let u ∈ F{mi } [Mi+1 ]max = Bi be such that ∗ s ⊆ u. Case 1. ∗ s = u. Then ∗ s ∈ Bi , and ∗ s ⊆ M , so ∗ s ∈ Bi M , and therefore s = {mi } ∪ ∗ s ∈ B M . Case 2. ∗ s u. Consider the inﬁnite set ∗ s ∪ (M/s) ⊆ Mi+1 . Since Bi is a barrier on Mi+1 we can ﬁnd v ∈ Bi such that v ∗ s ∪ (M/s). Note that in this case v ⊆ M/mi . Subcase 2.1. ∗ s is a strict initial part of v. Then s = {mi } ∪ ∗ s {mi } ∪ v = ({mi } ∪ v) ∩ M.

(II.7)

Let w ∈ F{mi } be such that v = w ∩ Mi+1 . Then s ({mi } ∪ v) ∩ M = ({mi } ∪ (w ∩ Mi+1 )) ∩ M = ({mi } ∪ w) ∩ M ∈ F[M ], (II.8) since {mi } ∪ w ∈ F. But this contradicts the maximality of s. Subcase 2.2 v ∗ s. Then v ∗ s u, and u, v ∈ Bi , which contradicts the fact that Bi is Sperner. Let us now show the other inclusion B M ⊆ F[M ]max . So, consider an s ∈ B M , and let mi = min s. Then ∗ s ∈ Bi M , hence ∗ s ∈ F{mi } [Mi+1 ]max . So ∗ s = t ∩ Mi+1 , for some t ∈ F{mi } . Since ∗ s ⊆ M/mi we obtain s = ({mi } ∪ ∗ s) ∩ M = ({mi } ∪ (t ∩ Mi+1 )) ∩ M ⊆ ({mi } ∪ t) ∩ M ∈ F[M ]. (II.9) Hence, there is some u ∈ F[M ]max ⊆ B M such that s ⊆ u. But B M is Sperner, so we must have s = u, as desired.

II.3. Uniform Fronts and Barriers

159

Theorem II.3.22. Suppose that F is a pre-compact family. Then there is an inﬁnite set M such that (a) F[M ]max = F[M ] − max is a uniform barrier on M , and ⊆

] = F[M ] = F[M ], and so in particular the trace F[M ] is a (b) F[M ] = F[M hereditary family. Proof. By Lemma II.3.21 applied to the compact hereditary family F there is ] = F[M ], some M such that F[M ]max is a uniform barrier on M . Note that F[M max max max = F [M ] . Let B = F[M ] . Choose N ⊆ M such that M \ N is so F[M ] inﬁnite. By Lemma II.3.14 (b), we know that B[N ] = B N.

(II.10)

Note also that B N = (F[M ]max ) N ⊆ (F[M ]max )[N ] ⊆ F[M ][N ] = F[N ].

(II.11)

It follows that F[N ]max = B N , and therefore F[N ] ⊆ B N = B[N ] ⊆ F[N ].

(II.12)

Corollary II.3.23. Suppose that F0 and F1 are two pre-compact families. Then there is an inﬁnite set M such that either F0 [M ] ⊆ F1 [M ] or F1 [M ] ⊆ F0 [M ]. Proof. Let M1 ⊆ M0 be such that Fi [Mi ] is the closure of a uniform barrier Bi on Mi (i = 0, 1). Let M ⊆ M1 be such that Bi M ⊆ B j M for i = j ∈ {0, 1}. By Lemma II.3.14 (a) and Lemma II.3.18 B i M = Bi M = Bi [M ] = Fi [Mi ][M ] = Fi [M ], so Fi [M ] ⊆ Fj [M ].

(II.13)

Exercise II.3.24. Show that if (Cn ) is an inﬁnite sequence of clopen subsets of α + 1 where α is an ordinal < ω ω then either (a) (Cn ) has an inﬁnite subsequence (Cnk ) with nonempty intersection, or (b) (Cn ) has a subsequence (Cnk ) such that for some positive integer l, every intersection of at most l of Cnk is nonempty but every intersection of at least l + 1 Cnk ’s is empty. (Hint: let F = {{n : γ ∈ Cn } : γ ∈ α + 1}, apply Theorem II.3.22 and argue that the resulting barrier B must be of some ﬁnite rank l). There is another notion of heredity which is enjoyed by many uniform barriers. It rests on another ordering on N[<∞] reminiscent to the one appearing in the w.q.o. theory (see Theorem II.1.8 above): s t iﬀ the only strictly increasing map σ : t → s satisﬁes that n ≥ σ(n) for all n ∈ t. We say that a family F of ﬁnite subsets of N is spreading on M if s t ⊆ M and s ∈ F M implies t ∈ F. This notion can be quite useful but unfortunately is not shared by closures of all uniform barriers as the following example shows.

160

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Example II.3.25. For n ∈ N, let fn : N → N by fn (i) = max{1, i − n + 1}. For n ∈ N, let B=

∞

Cn ⊕ {n}.

n=0

Then B M is not a spreading family for any inﬁnite set M : Fix M = {mk }∞ k=0 increasing enumeration. Notice that B = {s ∈ FIN : |∗ s| = fmin s (min(∗ s))}. Let s = {m0 , m2 , . . . , mk } and t = {m1 , m2 , . . . , mk } where k = fm0 (m2 ) = m2 − m0 + 1. It is clear that s t and s ∈ B, but t ∈ / B since if u t, then |∗ t| = k, but fm1 (m2 ) = m2 − m1 + 1 < k. To get spreading one needs to make a slight variation on Deﬁnition II.3.1. Deﬁnition II.3.26. Let α be a countable ordinal and let M be an inﬁnite subset of N. We say that a family B of ﬁnite subsets of N is α-extrauniform on M provided that: (a) α = 0 implies B = {0}, (b) α = β + 1 implies B = C ⊕ M [1] for some β-extrauniform family C on M . (c) α > 0 limit implies that there is a strictly increasing sequence (αm )m∈M of ordinals converging to α such that B{m} is αm -extrauniform on M/m for all m ∈ M. Proposition II.3.27. Suppose that B is extrauniform on M . Then there is some inﬁnite set N ⊆ M such that B N is spreading in N . Proof. Suppose that B is α-extrauniform on M . The proof goes by induction on α. Suppose α = β + 1. Then B = C ⊕ M [1] for some β-extrauniform family C on M , so there is some N ∈ M [∞] such that C N is spreading. Is it easy to see that B M is also spreading. Now suppose that α is limit. Then ﬁx an increasing sequence of ordinals (αm )m∈M with limit α such that for every m ∈ M , B{m} is αm -extrauniform on M/m. We ﬁnd a strictly decreasing sequence (Mk )∞ k=0 , of inﬁnite subsets of M such that: (a) The sequence nk = min Mk is strictly increasing in k, (b) B{n0 } M0 is spreading, (c) B{nk+1 } Mk+1 is spreading and B{nk } Mk+1 ⊆ B{nk+1 } Mk+1 . Let N = {nk }∞ k=0 . We claim that B N is spreading on N : For suppose that s t ⊆ N , s ∈ B N , and set nk = min s, nl = min t. Since B{nk } Nk is spreading on Nk , and since ∗ s ∗ t and ∗ t ⊆ Nk we have that ∗ t ∈ Bnk . If k = l, then t = {nk } ∪ ∗ t ∈ B; if k < l, then since ∗ t ⊆ Nk+1 and B{nk } Nk+1 ⊆ B{nk+1 } Nk+1 we obtain that ∗ t ∈ B{nk+1 } and so t ∈ B. This ﬁnishes the proof.

II.3. Uniform Fronts and Barriers

161

Given a family B of ﬁnite sets of N, let ∗B

= {∗ s : s ∈ B}.

Note that if B is an uniform family on some set M and of some rank > 0, then ∗ B is also an uniform family on ∗ M . Corollary II.3.28. Let B be an α-uniform family on M for some ordinal α > 0. Then there is an inﬁnite subset N ⊆ M such that the family ∗ (B N ) is spreading on ∗ N . Proof. Choose an α-extrauniform family C on M . By Proposition II.3.27, Lemma II.3.12 and Corollary II.3.16 there is some inﬁnite subset N ⊆ M such that C N is spreading in N and either (a) B N ⊆ C N ⊆ B N ⊕ N [≤1] or (b) C N ⊆ B N ⊆ C N ⊕ N [≤1] ⊆ B N ⊕ N [≤1] . Using this one easily proves that ∗ (B N ) is spreading on ∗ N . Corollary II.3.29. For every pre-compact family F of ﬁnite subsets of N there is an inﬁnite set M such that ∗ F[M ] is spreading.

Proof. Use the previous result and Theorem II.3.22.

We ﬁnish this section by mentioning two applications. The ﬁrst requires the following deﬁnition. Deﬁnition II.3.30. (a) For a compact hereditary family B of ﬁnite subsets of N and a real θ ∈ (0, 1), let T [B, θ] be the corresponding Tsirelson-like space deﬁned to be the completion of c00 (N) under the norm deﬁned implicitly by x = max x∞ , sup θ (Ei )n i=1

n

Ei x

i=1

where in the last sup (Ei )ni=1 is running over all ﬁnite sets E1 < · · · < En such that there is some {k1 < · · · < kn } ∈ B such that k1 ≤ E1 < k2 ≤ E2 < · · · < kn ≤ En . (b) Given two inﬁnite dimensional Banach spaces X and Y we write X → Y iﬀ there is some closed subspace of Y isomorphic to X. Given a compact hereditary family B and θ ∈ (0, 1) one is interested in knowing whether the space T [B, θ] shares some of the properties of the Tsirelson space T = T [S, 1/2] and in determining when two spaces of the form T [B, θ] contain isomorphic copies of a common inﬁnite-dimensional Banach space. It turns out that answer to these questions requires a ﬁne analysis of uniform fronts and barriers that naturally follows the one exposed above. Here is an example of a result whose proof uses this kind of analysis in an essential way.

162

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Theorem II.3.31 ([51]). Suppose that B, B are two compact, hereditary, and spreading families of ﬁnite subsets of N with Cantor–Bendixon ranks

i(B) = ω α0 n0 + · · · + ω αk nk and i(B ) = ω α0 n0 + · · · + ω αk nk , respectively, written in their Cantor normal forms. Suppose that θ, θ ∈ (0, 1). Then the following conditions are equivalent: (a) There is X → T [B, θ], T [B , θ ]. (b) For every X → T [B, θ] and Y → T [B , θ ] there is Z → X, Y . (c) There is some integer l such that either (i) θ l = θ and max{ω α0 , n0 }l = max{ω α0 , n0 }, or l (ii) θ = θ and max{ω α0 , n0 }l = max{ω α0 , n0 }. To state the second application, let MFIN be the collection of all ﬁnite convex means on N, i.e., ﬁnitely supported maps µ : N → [0, 1] such that ∞

µ(n) = 1.

n=0

For F ⊆ N and µ ∈ MFIN , let µ(F ) = u∈F µ(F ). The following is a generalization of a result in [4] and [70] where a similar conclusion is reached for the Schreier hierarchy. Lemma II.3.32. Let α and β be countable ordinals such that β is bigger or equal to the minimal indecomposable ordinal > α. Let A be an α-uniform barrier on some inﬁnite set M and let B be a β-uniform barrier on some set N ⊇ M . Then for every ε > 0 there is µ ∈ MFIN such that supp(µ) ∈ B and µ(s) < ε for all s ∈ A. Proof. The proof is by induction on α. Notice that the requirement on β is equivalent to saying that β ≥ ω α0 +1 , where α = ω α0 n0 + · · · + ω αk nk is the Cantor’s normal form of α. By Lemma II.3.12 it suﬃces to consider only the case of β = ω α0 +1 . There are two cases: Case 1. Suppose ﬁrst that α = ω α0 , β = ω α0 +1 . The proof is by induction on α0 . If α0 = 0, then B is an ω-uniform barrier on M , while A is 1-uniform on M , hence A = M [1] . Fix s ∈ B M such that ε · |s| < 1, and let µ = (1/|s|)χs , where χs is the characteristic function of s. Suppose that α0 > 0. Fix a sequence (γm )m∈M such that γm ↑m ω α0 and such that for every m ∈ M , A{m} is γm -uniform on M/m. Fix also k such that 1/k < ε/2. Let N ∈ M [∞] be such that A N ⊗ N [k] ⊆ B N ,

(II.14)

(this is possible since A ⊗ M [k] is ω α0 · k-uniform on M , while B is ω α0 +1 -uniform on M ). Now let t1 ∈ A N and µ ∈ MFIN be such that supp µ ⊆ t1 . Since for every n ∈ N ∩ [1, max s1 ], (A N ){n} is γn -uniform, and γn < ω α0 , by inductive

II.3. Uniform Fronts and Barriers

163

hypothesis (for the pair γn < ω α0 ) we can ﬁnd µ2 ∈ MFIN such that supp µ2 ⊆ t2 ∈ A (N/n1 ), and such that for every s ∈ A N with min s ≤ max t1 , we obtain that µ2 (s) < 1/2. In general we can ﬁnd µ1 , . . . , µk ∈ MFIN in such a way that: (a) For every i = 2, . . . , k, µi ⊆ ti ∈ A (N/(max ti−1 )), and (b) for every i = 2, . . . , k, and every s ∈ A N such that min s ≤ max ti−1 , µi (s) ≤ 1/2i−1 . Let µ = (1/k) ki=1 µi . It is clear that µ ∈ MFIN , and that supp µ =

k

supp µi ⊆

i=1

k

ti ∈ A N ⊗ N [k] ⊆ B.

(II.15)

i=1

Now ﬁx s ∈ A. Let s ∈ A N be such that s ∩ N ⊆ s. (This is possible: Since A N is a barrier on N , we may ﬁnd s ∈ A N such that either s s ∩ N or s ∩ N ⊆ s; the ﬁrst alternative is not possible since it implies that s s and both are elements of the barrier A.) So we have that 1 µ(s) =µ(s ∩ N ) ≤ µ(s) = (µi0 (s) + · · · + µk (s)) ≤ k 1 2 1 1 1 + i + · · · + k−1 ≤ < ε. ≤ 0 k 2 2 k

(II.16)

Case 2. General case α = ω α0 n0 + · · · + ω αk nk . Let C be an ω α0 -uniform barrier on M . Then C ⊗ M [n0 +1] is ω α0 (n0 + 1)-uniform on M . Since ω α0 (n0 + 1) > α, by Proposition II.3.16 there is some N ∈ M [∞] such that A N ⊆ C N ⊗ N [n0 +1] . Now ﬁx ε > 0. Then there is some µ ∈ MFIN with support in B N and such that for every s ∈ C N , µ(s) < ε/(n0 + 1). We work to show that µ(s) < ε for every s ∈ A. Fix s ∈ A. Since supp µ ⊆ N , we have that µ(s) = µ(s ∩ N ). Again we can ﬁnd s ∈ A N such that s ∩ N ⊆ s, and since µ(s) = µ(s ∩ N ) ≤ µ(s), we may assume that s ∈ A N . Now using that A N ⊆ C N ⊗ N [n0 +1] , we can ﬁnd s1 < · · · < sn0 +1 , si ∈ C N , such that s ⊆ s1 ∪ · · · ∪ sn0 +1 . Hence, µ(s) ≤

n 0 +1 i=1

as desired.

µ(si ) <

ε (n0 + 1) = ε, n0 + 1

(II.17)

Remark II.3.33. One might wonder if the convex mean µ satisfying conclusion of Lemma II.3.32 can have a simpler form in comparison to the one obtained in the previous proof. For example, can one produce such mean µ of the form µt = (1/|t|)χt ? It turns out that such a simple form of µ cannot be found if, for example, A is equal to the Schreier barrier {s ∈ N[<∞] : |s| = min s + 1}. Note that in this case for every t ∈ N[<∞] there is some s ∈ A such that |s ∩ t| ≥ (1/2)|t| or in other words µt (s) ≥ 1/2.

164

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Corollary II.3.34 (Pt´ak). For every pre-compact family F of ﬁnite subsets of N and every ε > 0 there is µ ∈ MFIN such that µ(s) < ε for all s ∈ F. Proof. This follows from Theorem II.3.22 and Lemma II.3.32.

Corollary II.3.35 (Mazur). For every bounded weakly null sequence (xn ) in some Banach space X and every ε > 0 there is a ﬁnite convex combination y = k λ x i=1 i ni such that y ≤ ε. Proof. We may assume that xn ≤ 1 for all n. For x∗ ∈ BX ∗ , let Fx∗ = {n ∈ N : |x∗ (xn )| ≥ ε/2}. Let F = {Fx∗ : x∗ ∈ BX ∗ }. Note that F is a pre-compact family of ﬁnite subsets of N. By Corollary II.3.34 there is µ ∈ MFIN such that µ(F ) < ε/2 for all F ∈ F. Let G = supp(µ) and y= µ(n)xn . n∈G

Then y is the required convex combination. To see this consider an x∗ ∈ BX ∗ . Then |x∗ (y)| = | µ(n)x∗ (xn ) + µ(n)x∗ (xn )| n∈Fx∗ ∩G n∈G\Fx∗ ≤ µ(n)|x∗ (xn )| + µ(n)|x∗ (xn )| n∈Fx∗ ∩G

n∈G\Fx∗

˙ < ε/2 + ε/2 = ε. < µ(Fx ) + µ(G \ Fx∗ )ε/2 It follows that y ≤ ε.

Remark II.3.36. (a) There is also a direct way from Mazur’s theorem to Ptak’s lemma. To see this, ﬁx a pre-compact family F of ﬁnite subsets of N and ε > 0. Deﬁne an inﬁnite sequence (fn )n ⊆ C(F) as follows:

0 if n ∈ /s fn (s) = 1 if n ∈ s. Then (fn ) is a weakly null sequence in the Banach space (C(F), ·∞ ). By Mazur’s theorem there exist a ﬁnite sequence n1 < · · · < nk of integers and a sequence (λi )ki=1 ⊆ [0, 1] such that k λi fni ∞ < ε. (II.18) k

i=1

Deﬁne µ = i=1 λi χ{ni } ∈ MFIN . Then µ(s) < ε for all s ∈ F as required for the conclusion of Ptak’s Lemma. (b) Note that Schreier space S introduced in Example I.2.13 has the property that every average (1/|s| i∈s ei ) of its natural basis (en ) has norm at least 1/4. For Tsirelson space T (see Example I.2.14) something stronger is true: Every

II.4. Canonical Equivalence Relations on Uniform Fronts and Barriers

165

weaklynull sequence (xn ) of T has a subsequence (xnk ) such that all averages (1/|s| i∈s xni ) have norm at least 1/4. (c) If a normalized basic sequence (xn ) has 1 as spreading model, then every average on (xn ) has norm far away from zero. Note that the natural basis of S, and every normalized block sequence (xn ) of T have 1 as a spreading model. Rosenthal’s Theorem II.9.8 below clariﬁes the situation.

II.4

Canonical Equivalence Relations on Uniform Fronts and Barriers

The purpose of this section is to examine what are the essential equivalence relation that one can put on an uniform barrier B modulo the possibility of going to an arbitrary restriction of the form B M for M an inﬁnite subset of N. It will be instructive to ﬁrst understand the canonical equivalence relations on barriers M [k] of ﬁnite rank. The following equivalence relations on N[k] suggest themselves. Example II.4.1. For a subset I ⊆ {0, . . . , k − 1} deﬁne the following equivalence relation EI on the uniform barrier N[k] : {m0 , . . . , mk−1 }< EI {n0 , . . . , nk−1 }< iﬀ mi = ni for all i ∈ I. The following classical result shows that EI for I ⊆ {0, . . . , k − 1} exhaust the list of all canonical equivalence relations on N[k] . Theorem II.4.2 (Erd¨ os–Rado). For every integer k ≥ 1 and every equivalence relation E on N[k] there is an inﬁnite M ⊆ N and I ⊆ {0, . . . , k − 1} such that E M [k] = EI M [k] Proof. The proof is by induction on k. The case k = 1 is clear. So assume k > 1 and we have an equivalence relation E on N[k] . Let E be the collection of all equivalence relations on {0, 1, . . . , 2d + 1}[k] . Deﬁne c : N[2k+2] −→ E by letting c(S) the copy of the equivalence relation E (S [k] ) under the order preserving map from s onto {0, 1, . . . , 2d + 1}. By Ramsey’s theorem there is an inﬁnite M ⊆ N such that c M [2k+2] is constantly equal to some E0 ∈ E. If E M [k] is the equality relation then we are done. So assume that there exist s = t in M [k] such that s E t. Let s = {m0 , m1 , . . . , mk−1 }, t = {n0 , n1 , . . . , nk−1 } and let j < k be minimal i such that mi = ni . It follows that the equation x E y for x, y ∈ M [k] does not depend on the j th member of x and the j th member of y, so we are done by the inductive hypothesis. Corollary II.4.3. For every regressive11 f : N[k] −→ N there is an inﬁnite M ⊆ N such that f (s) = f (t) for every pair s, t ∈ M [k] with min(s) = min(t). 11 i.e.,

f (s) < min(s) for all s ∈ N[k] with min(s) = 0.

166

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Remark II.4.4. The ﬁnite form of this result (which follows from it via a simple compactness argument) reads: For every k and m there is n such that for every regressive f : n[k] −→ n there is a min-homogeneous X ⊆ n of size m. It turns out that this result is not provable in Peano arithmetic. Corollary II.4.5 (Ramsey). For every integer k ≥ 1 and every equivalence relation E with ﬁnitely many classes there exists an inﬁnite M ⊆ N such that s E t for all s, t ∈ M [k] . Exercise II.4.6. Prove the ﬁnite version of the Erd¨ os–Rado canonization theorem by working just in PA. Deﬁnition II.4.7. An equivalence relation E deﬁned on some uniform barrier B is canonical if there is an uniform barrier C (not necessarily on the same domain as B) and a mapping f : B −→ C such that (a) f (s) ⊆ s for all s ∈ B (b) for s, t ∈ B, s E t iﬀ f (s) = f (t). Theorem II.4.8 (Pudlak–R¨ odl). For every equivalence relation E deﬁned on some (uniform) barrier B there is an inﬁnite set M ⊆ N such that E (B M ) is canonical. Typical equivalence relations are deﬁned using a map h : B → N by s Eh t iﬀ h(s) = h(t). So let us examine behaviors of such maps h : B −→ N. Lemma II.4.9. Suppose B is an uniform barrier on some inﬁnite set M ⊆ N and that h : B −→ N is such that h(s)∈ / s for all s ∈ B. Then there is inﬁnite N ⊆ M such that (h [B N ]) ∩ N = ∅. Proof. Let B be α-uniform for some α. The proof is by induction on α.

Lemma II.4.10. Suppose h0 : B0 −→ N and h1 : B1 −→ N are 1-1 mappings deﬁned on two uniform barriers B0 and B1 on the same domain M ⊆ N. Then there is an inﬁnite N ⊆ M such that either (1) B0 N = B1 N and h0 (B0 N ) = h1 (B1 N ), or (2) h0 [B0 N ] ∩ h1 [B1 N ] = ∅. (1)

(2)

Proof. Deﬁne a partition B0 = B0 ∪ B0 (1)

s ∈ B0

by letting

iﬀ s ∈ B1 and h0 (s) = h1 (s).

By the Ramsey property of B0 we can ﬁnd an inﬁnite N ⊆ M such that B0 N ⊆ (1) (1) B0 or (B0 N ) ∩ B0 = ∅. In the ﬁrst case we have B0 N ⊆ B1 N which by maximality of B0 N means B0 N = B1 N and so we have the conclusion (1). So suppose that for every s ∈ B0 N we have that either s ∈ / B1 or s ∈ B1 but h0 (s) = h1 (s). Deﬁne now g0 : B0 N −→ N and g1 : B0 N −→ N

II.4. Canonical Equivalence Relations on Uniform Fronts and Barriers

167

as follows: given s ∈ B0 N , if there is t ∈ B1 such that h(s) = h1 (t) and if the unique t has the property that t \ s = ∅, let g0 (s) = min(t \ s); otherwise g0 (s) = 0. The function g1 is deﬁned symmetrically, i.e., g1 (s) = 0 implies that g1 (s) ∈ s \ t for the unique t ∈ B1 such that h0 (s) = h1 (t) . By the previous Lemma there exists an inﬁnite P ⊆ N such that gi [Bi P ] ∩ P = ∅ for i = 0, 1. Then h0 [B0 P ] ∩ h1 [B1 P ] = ∅.

Corollary II.4.11. Suppose h0 : B0 −→ N and h1 : B1 −→ N are 1-1 mappings deﬁned on uniform barriers acting on the same set M . Suppose B0 is α0 -uniform, B1 is α1 -uniform, and α0 = α1 . Then there is an inﬁnite N ⊆ M such that h0 [B0 N ] ∩ h1 [B1 N ] = ∅. Proof of Theorem II.4.8. Suppose B is α-uniform on N. The proof is by induction on this α. If α = 0, the conclusion is clear, so assume α > 0. For n ∈ N, deﬁne an equivalence relation En on B{n} by letting sEn t iﬀ ({n} ∪ s)E({n} ∪ t). Using the inductive hypothesis and diagonalizing we get uniform barriers Cn (n ∈ N) and mappings 1−1 fn : B{n} −→ Cn such that (a)n fn (s) ⊆ s for all s ∈ B{n} (b)n for s, t ∈ B{n} , s En t iﬀ fn (s) = fn (t). For each n ∈ N let ψn : Cn −→ B{n} /En be the mapping such that for v ∈ Cn , fn (s) = v for all s ∈ ψn (v). A simple diagonalization argument using Lemma II.4.10 will give us an inﬁnite set M ⊆ N such that for all m < n in M, either (1) Cm (M/n) = Cn (M/n) and ψm (Cm (M/n)) = ψn (Cn (M/n)). (2) ψm [Cm (M/n)] ∩ ψn [Cm (M/n)] = ∅. Applying Ramsey’s theorem we have the following two cases to consider: Case 1: For all n < m in M , Cm (M/n) = Cn (M/n) and ψm (Cm (M/n)) = ψn (Cn (M/n).

168

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Let C = Cmin(M ) , N = M \ {min(M )}, and let f : B N −→ C be deﬁned by f (s) = fmin(s) (s \ {min(s)}). Then f canonizes E (B N ). Case 2: ψm [Cm (M/n)] ∩ ψn [Cm (M/n)] = ∅. For m < n in M and u ⊆ M ∩ (m, n], let Cm(u)n = {v : n < min(v) and u ∪ v ∈ Cm } and deﬁne ψm(u)n : Cm(u)n −→ B{n} /En by ψm(u)n (v) = ψn (u ∪ v). Shrinking M , we may assume that Cm(u)n is uniform on M/n for all m < n and u ⊆ (m, n] ∩ M . Performing another diagonalization procedure using Lemma II.4.10 we get an inﬁnite set N ⊆ M such that for all m < n in N and u ⊆ N ∩ (m, n], either (3) Cm(u)n (N/n) = Cn (N/n) & ψm(u)n (Cm(u)n (N/n)) = ψn (Cn (N/n)), or (4) ψm(u)n Cm(u)n (N/n) ∩ ψn [Cn (N/n)] = ∅. Let C= {{n} ∪ t : t ∈ Cn (N/n)} n∈N

and deﬁne f : B N −→ C by f (s) = {min(s)} ∪ fmin(s) (s \ {min(s)}). It is now routine to check that f canonizes B N .

We ﬁnish this section with the following fact which shows that there is essentially only one solution to the canonization problem for equivalence relations deﬁned on uniform barriers. Theorem II.4.12. Suppose E is a canonical equivalence relation deﬁned on some uniform barrier B and witnessing that this is so by two pair (f0 , C0 ) and (f1 , C0 ) as in Deﬁnition II.4.7. Then there is an inﬁnite M ⊆ N such that C0 M = C1 M and f0 (B M ) = f1 (B M ).

II.5. Unconditional Subsequences of Weakly Null Sequences

169

Proof. For i < 2 let ϕi : Ci → B/E be such that for all s ∈ Ci , fi (t) = s for all t ∈ ϕi (s). Apply now Lemma II.4.10 and note that the alternative (2) is impossible since ϕ0 [C0 M ] = (B M )/E = ϕ1 [C1 M ]. The alternative II.4.10 (1) gives us the required conclusion.

Exercise II.4.13. Deduce Theorem II.4.2 from Theorem II.4.8.

II.5

Unconditional Subsequences of Weakly Null Sequences

Recall the notion of an unconditional basic sequence (xn ) in some Banach space X, a basic sequence for which we one can ﬁnd a constant C such that

k n=0

εn an xn ≤ C

k

a n xn

n=0

for every sequence (an )kn=0 of scalars and every sequence (en )kn=0 of signs. Note that every separable Banach space is spanned by an unconditional basic sequence. In fact there are separable Banach spaces that are not even embeddable into one with unconditional basic sequences. Thus as in the case of Schauder basic sequences one is left to analyzing the possibilities of having unconditional basic subsequences. It turns out that weakly null sequences (xn ) in Banach spaces do sometimes have basic unconditional subsequences (xnk ). First of all, note that they do always have Schauder basic subsequences: Lemma II.5.1. Every normalized weakly null sequence in some Banach space X has a subsequence (xnk ) that is a Schauder basis of its closed linear span. Proof. We may assume that X itself has a Schauder basis (ei ). Renorming X we may assume that the basis constant of (ei ) is 1. Choose an ε > 0. Let Pj be the sequence of projections on ﬁnite initial segments relative to the basis (ei ). Put n0 = 0 and choose j0 such that x0 − Pj0 (x0 ) < ε/2. Since limn→∞ e∗i (xn ) = 0 for all i, we can ﬁnd n1 and j1 > j0 such that ε Pj0 (xn1 ) + xn1 − Pj1 (xn1 ) < , 4 and so on, at stage k we can ﬁnd nk > nk−1 and jk > jk−1 such that ε Pjk−1 (xnk ) + xnk − Pjk (xnk ) < k+1 . 2 Proceeding in this way one obtains a subsequence (xnk ) that is a Schauder basis with constant at most 1 + 2ε.

170

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

We shall now see that the problem whether a given weakly null sequence (xn ) has an unconditional basic sequence may depend on the nature of the vectors xn as well as the Banach space X in which we are working. Lemma II.5.2 (Rosenthal). Suppose (xn ) is a sequence of characteristic functions of subsets of some set Γ and that (xn ) is weakly null inside the Banach space ∞ (Γ). Then (xn ) has an unconditional basic subsequence. Proof. For γ ∈ Γ, let Fγ = {n ∈ N : xn (γ) = 1}. Let F = {Fγ : γ ∈ Γ}. Then F is a pre-compact family of ﬁnite subsets on N, i.e., the pointwise closure F of F in 2N consists only of ﬁnite sets. By Theorem II.3.22 there is an inﬁnite set M ⊆ N such that F [M ] = B for some uniform barrier B on M . We claim that the subsequence (xm )n∈M is 2-unconditional. To see this let F be a ﬁnite initial segment of M and let an (n ∈ F ) be a given sequence of scalars from [−1, 1] and let εn (n ∈ F ) be a given sequence of signs. Fix γ ∈ Γ. Then εn an xn (γ)| = | εn an |. | n∈F ∩Fγ

n∈F

Let G = {n ∈ F ∩ Fγ : εn = 1} and H = {n ∈ F ∩ Fγ : εn = −1}. Then we can ﬁnd α and β ∈ Γ such that G = F ∩ Fα and H = F ∩ Fβ . Then |

εn an | ≤ |

n∈F ∩Fγ

n∈F ∩Fα

=|

an |

n∈F ∩Fβ

an xn (α)| + |

n∈F

≤ 2

an | + |

an xn (β)|

n∈F

an xn ∞

n∈F

It follows that n∈F εn an xn ∞ ≤ 2 n∈F an xn ∞ , as required. Note that the proof establishes the following slightly more general fact.

Lemma II.5.3. Suppose that (xn ) is a semi-normalized weakly null sequence in some Banach space of the form ∞ (Γ) such that there is some ﬁnite set V of positive real numbers such that the range of xn is contained in V , for every n. Then (xn ) has an unconditional basic sequence. Proof. For every γ ∈ Γ, let Fγ = {n ∈ N : xn (γ) = 0}. Let F = {Fγ : γ ∈ Γ}, which is a pre-compact set. Then there is some inﬁnite set M ⊆ N such that F[M ] = B for some uniform barrier B on M . We claim that the subsequence

II.5. Unconditional Subsequences of Weakly Null Sequences

171

(xm )n∈M is 4(max V / min V )-unconditional. To see this we are going to show that for every subset N ⊆ M and a sequence of scalars (am )m∈M

an xn ∞ ≤ 2

n∈N

max V am xm ∞ . min V

(II.19)

m∈M

Fix s ⊆ I M , and γ ∈ Γ. Set s+ = {n ∈ s ∩ Fγ : an > 0}, and s− = {n ∈ s ∩ Fγ : an < 0}. Then an xn (γ)| = an xn (γ) − |an |xn (γ). (II.20) | n∈s

n∈s−

n∈s+

+ − We s = Fβ ∩ I. Then can ﬁnd nowα, β ∈ Γ such thats = Fα ∩ I and n∈I an xn (α) = n∈s+ an xn (α) and n∈I an xn (β) = n∈s+ an xn (β). So, | an xn (γ)| = an xn (γ) − |an |xn (γ) ≤ n∈s

n∈s−

n∈s+

min V max V an xn (α) − |an |xn (β) = ≤ min V max V n∈I n∈I min V max V an xn (α) + an xn (β) ≤ = min V max V n∈I n∈I

min V

max V

≤ an xn (α) + an xn (β) ≤

max V

min V n∈I

≤2

max V an xn ∞ min V

(II.21)

n∈I

n∈I

(II.22)

Let us now extend Lemma II.5.3 to the case when the ﬁnite set V has both positive and negative values. To state the corresponding combinatorial result we need a piece of notation. For a given set V , let Fn(N, V ) be the set of functions f with domain a ﬁnite subset of N and range included in V , i.e. f : domf → V , where domf is a ﬁnite subset of N. From now on we ﬁx a ﬁnite set V ⊆ R. Every map Φ : B → Fn(N, V ) considered below will satisfy that domΦ(s) = s for every s ∈ B. Lemma II.5.4. Suppose that B is a uniform barrier on M and that Φ : B → Fn(N, V ). Then there is some inﬁnite set N ⊆ M such that Φ (B N ) is 1Lipschitz, i.e., for every s, t ∈ B N if s∩t s, t then Φ(s) (s∩t) = Φ(t) (s∩t).

172

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Proof. We ﬁnd a decreasing chain of inﬁnite sets (Mi ) such that, if we set mi = min Mi , then mi < Mi+1 and for every i and every u ⊆ {n0 , ..., ni }, if s, t ∈ B{u} then Φ(u ∪ s) u = Φ(u ∪ t) u. Then it is clear that the desired result will hold for N = {mi }. So, suppose we have deﬁned (Mj )j≤i . For a ﬁxed u ⊆ {n0 , ..., ni } consider the ﬁnite coloring cu : Bu → u V deﬁned for s ∈ Bu by cu (s) = Φ(u∪s) u. Then there will always be some P ⊆ Mi /mi such that cu is monochromatic on Bu P . Since there are only ﬁnitely many u ⊆ {n0 , ..., ni } we can ﬁnd a single inﬁnite set Mi+1 ⊆ Mi /mi such that for every u ⊆ {n0 , ..., ni }, the coloring cu is constant on Bu Mi+1 . Remark II.5.5. (a) Whenever Φ : B → Fn(N, V ) is 1-Lipschitz then Φ has a natural extension Φ : B → Fn(N, V ) deﬁned for t ∈ B we set Φ(t) = Φ(s) t, where s ∈ B is arbitrary member of B such that t s. (b) In this context the following notation is going to be useful. Fix Φ : B → Fn(N, V ), and m ∈ M . The corresponding section-mapping Φm : B{m} → Fn(N, V ) is deﬁned by Φm (s) = Φ({m} ∪ s) s, where s ∈ B{m} . Now for v ∈ V , we deﬁne ϕv : B → N[<∞] by ϕv (s) = Φ(s)−1 {v} = {n ∈ s : Φ(s)(n) = v}.

(II.23)

The corresponding section-mappings, denoted by ϕvm , are deﬁned by ϕvm (s) = {n ∈ s : Φ({m} ∪ s)(n) = v}, where s ∈ B{m} . (c) Suppose that B is a barrier on M , and that ϕ : B → Fn(N, V ) is 1-Lipschitz. Suppose that N is an inﬁnite subset of M such that M \ N is also inﬁnite. Then for every s ∈ B N there is some t ∈ B N such that Φ(t) N = Φ(s). Lemma II.5.6. Suppose that B is a barrier on M , and suppose that Φ : B → Fn(N, V ). There is an inﬁnite subset N ⊆ M such that for every s ∈ B N and every v ∈ V there is some t = t(s, v) ∈ B such that (a) Φ(t)”(t ∩ N ) ⊆ {v}12 , and (b) ϕv (s) = ϕv (t) ∩ N . Proof. Fix v ∈ V . We may assume that B is an α-uniform barrier on M , and that Φ is 1-Lipschitz. We may also assume that the mapping s → Φ(s)(min s) ∈ V is constant, with value v0 . The proof is by induction on the ordinal α. We ﬁrst note the following Claim. There is an inﬁnite subset N of M such that for every n ∈ N and every s ∈ B{n} N there is some t ∈ B{n} such that Φ{n} (t)”(t ∩ N ) ⊆ {v}, ϕvn (s) = ϕvn (t) ∩ N and min ϕvn (t) ≥ min ϕvn (s). Moreover every inﬁnite subset P of N has the same properties. 12 Notation:

For a mapping φ and a set X, we let φ”X = {φ(x) : x ∈ X}.

II.5. Unconditional Subsequences of Weakly Null Sequences

173

From now on, we ﬁx N ⊆ M with the properties stated in the claim. There are two cases to consider: Case 1: v0 = v. Then N satisﬁes the conclusion of the Lemma. Case 2: v0 = v. Then using Corollary II.3.23 we can ﬁnd a decreasing sequence (Ni ) of inﬁnite subsets of N such that if we set n0i = min Ni and n1i = min ∗ Ni then Ni+1 ⊆ Ni /n1i and there are j(i) = k(i) ∈ {0, 1} such that (ϕvnj(i) ”(B{nj(i) } (Ni /n1i )))[P ] ⊆ (ϕvnk(i) ”(B{nk(i) } (Ni /n1i )))[P ] i

i

(II.24)

i

i

j(i)

with both traces hereditary and with P = {ni of the Lemma.

}. Then P satisﬁes the conclusion

Theorem II.5.7. Suppose that (xn ) is a normalized weakly-null sequence of ∞ (Γ) with the property that inf{|xn (γ)| : n ∈ N, γ ∈ Γ} = δ > 0.

(II.25)

Then there is an inﬁnite set N and the closure C of a α-uniform barrier on N for some ordinal α > 0 such that (xn )n∈N is δ/4-equivalent to the natural basis (ti ) of the corresponding Schreier space SC and so is, in particular, unconditional. Proof. Consider the corresponding family F = {Fγ : γ ∈ Γ}, where Fγ = {n ∈ N : xn (γ) = 0}. Notice that condition (II.25) implies that F is a pre-compact family. Use Theorem II.3.22 to ﬁnd an inﬁnite M and a uniform barrier on M such that Note that since each xn is normalized, for every M F[M ]max = B and F[M ] = B. and every m ∈ M there is some γ ∈ Γ such that m ∈ F (γ), so F[M ] contains the singletons {m} (m ∈ M ). Hence B is α-uniform with α > 0. For every s ∈ B we choose γ(s) ∈ Γ such that s = F (γ(s)) ∩ M . Deﬁne Φ : B → Fn(N, {−1, 1}) for s ∈ B by

xn (γ(s))/|xn (γ(s))| if n ∈ s Φ(s)(n) = (II.26) 0 otherwise. Let N ⊆ M be the result given by the use of Theorem II.5.6 to Φ and B. Let Θ : N → N be the order-preserving bijection between N and N, and let C = Θ”B N = {Θ”(s) : s ⊆ t ∈ B N }. It is clear that C is the closure of a uniform barrier on N. We claim that the subsequence (xn )n∈N is δ/4-equivalent to the natural basis of the correspondingSchreier space SC . Fix a sequence of scalars (an )n∈N . Let γ ∈ Γ be such that | n∈N an xn (γ)| ≈ n∈N an xn . Note that an xn (γ)| = | an xn (γ)| ≤ |an | ≤ an tΘ(n) SC , (II.27) | n∈N

n∈Fγ ∩N

n∈Fγ ∩N

n∈N

]=B N . Now let s ∈ B N be such since Fγ ∩ N ∈ F[N ] = F[M ][N ] = B[N that an tΘ(n) SC ≈ |an |. (II.28) n∈N

n∈s

174

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

For i = 0, 1, let si = {n ∈ s : (−1)i an ≥ 0},

(II.29)

Now we can ﬁnd γ(i, j) ∈ Γ ((i, j) ∈ {0, 1}2 ) such that F (γ(i, j)) ∩ N ={n ∈ F (γ(i, j)) ∩ N : |xn (γ(i, j))| = (−1)j xn (γ(i, j))} j

=ϕ(−1) (si ) = s(i, j). Note that

(II.30)

s=

s(i, j).

(II.31)

(i,j)∈{0,1}2

It can be shown that 4 a n xn ≥

(i,j)∈{0,1}2

n∈N

≥δ

|

an xn (γ(i, j))|

n∈N

|an | = δ

(i,j)∈{0,1}2 n∈s(i,j)

|an |,

(II.32)

n∈s

which gives the desired result.

More details about the above proof can be found in [52]. Similar results are proved in [8] and [33] and the reader is also referred to these sources for a more complete information about the problem of ﬁnding unconditional subsequences of weakly null sequences. Remark II.5.8. Besides the separation from 0 of the set of values of a given weakly null sequence (xn ), one can have various other simplifying conditions such as for example conditions on the corresponding pre-compact family of ﬁnite sets F = {{n : xn (γ) = 0} : γ ∈ Γ}. If we assume for example that there is an inﬁnite set M such that F[M ] = B for some uniform barrier B on M of some ﬁnite rank, then the sequence (xn ) has an unconditional basic subsequence. This is essentially the content of parts (a) and (b) of the next theorem; the part (c) says that some restriction on the uniform rank of B is necessary (and that some restrictions altogether are necessary!). Theorem II.5.9 (Maurey–Rosenthal). (a) For every ε > 0, every weakly null normalized sequence (xn ) ⊆ C(α + 1) for some α < ω ω has a (2 + ε)unconditional subsequence. (b) For every ε > 0, every weakly null normalized sequence (xn ) ⊆ C(ω ω + 1) has a (4 + ε)-unconditional subsequence. 2

(c) There is a weakly null normalized sequence (xn ) ⊆ C(ω ω + 1) with no unconditional subsequence.

II.5. Unconditional Subsequences of Weakly Null Sequences

175

Proof. One can view (a) and (b) as natural application of the theory of fronts and barriers developed above. Given a weakly null sequence (xn ) ⊆ C(α + 1) for some α < ω ω , since α + 1 is a countable index-set we can ﬁnd a sequence (Ck ) of clopen subsets of α + 1 and a subsequence (xnk ) of (xn ) such that (1) χ Ck → 0 pointwise on α + 1 (2) ∞ k=0 xnk − xnk χCk < ∞. Thus, we may replace our sequence (xn ) with the sequence (yk ) where yk = xnk χCk which now has the property that Dk = {γ : yk (γ) = 0} is a sequence of clopen subsets of α + 1 with no γ ∈ α + 1 belonging to inﬁnitely many of the Dk ’s. It follows in particular that (3) γ → Fγ = {k : yk (γ) = 0} is a continuous map from α into N[<∞] . Hence F = {Fγ : γ ∈ α + 1} is a compact family of ﬁnite subsets of N, so applying Theorem II.3.22 we can ﬁnd a barrier B on M such that F[M ] = B. Since F[M ] is a continuous image of α + 1, its Cantor–Bendixon rank must be ﬁnite. It follows that B = M [l] for some positive integer l. Pick an ε > 0 and choose a ﬁnite set V of real numbers and for each k continuous zk : α + 1 → V such that yk − zk ∞ < ε/l. It follows that for every positive integer m, every sequence (ak )m k=0 ⊆ [−1, 1], and every γ < α + 1, |

m

ak (yk − zk )(γ)| = |

k=0

k∈Fγ

ε ak (yk − zk )(γ)| ≤ sup |ak ||Fγ | ≤ ε. l k

So if (zki ) is a (2 + ε)-unconditional subsequence of (zk ) then the corresponding (yki ) will be (2 + δ)-unconditional subsequence of (yk ) with the possibility that δ, depending on ε, be arbitrarily small. Hence, we have reduced the part (a) of Theorem II.5.9 to Theorem II.5.7. To prove the part (b) i.e., treat the case α = ω ω one proceeds as above to the subsequence (xnk ) for which one has a perturbation (yk ) supported by a sequence (Ck ) of clopen subsets of ω ω + 1 converging pointwise to 0 and then diagonalizes applying the part (a) to the restriction of the tail sequence to various compact sets Cki that have ﬁnite Cantor–Bendixon rank. The idea behind the example witnessing Theorem II.5.9 (c) is also quite interesting so let us examine it. Pick an 0 < ε < 1 and choose a strictly increasing sequence (mi ) such that ∞ i=0 j=i

min((

mi 1/2 mj 1/2 ) ,( ) ) ≤ ε/2. mj mi

176

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Choose a 1-1 function

13

σ : FIN[<∞] → M = {mi } such that ϕ(E0 , . . . , Ek ) > Ek for all (E0 , . . . , Ek ) ∈ FIN[<∞] . A ﬁnite or inﬁnite sequence (Ei ) of elements of FIN is special if (1) |E0 | = 1, (2) E0 < E1 < · · · < Ei < · · · , (3) |Ei+1 | = σ(E0 , . . . , Ei ) whenever i + 1 is in the domain of the sequence. Let Sσ be the collection of all special sequences of ﬁnite nonempty subsets of N. For each s = (Ei ) ∈ Sσ we associate a special functional

fs =

i∈dom(s)

χEi |Ei |1/2

and a special sequence (xsi ) of vectors from c00 deﬁned by xsi =

χEi . |Ei |1/2

Let Fσ be the collection of all special functions associated to ﬁnite or inﬁnite special block sequences of ﬁnite nonempty subsets of N. For s = (Ei ) ∈ Sσ and I ⊆ N let χE ∩I i f sI = , |Ei |1/2 i∈dom(s)

the restriction of f s to I. Let F σ be the collection of functionals of the form f sI

(s ∈ Sσ , I ⊆ N interval ).

Using F σ one deﬁnes the Maurey–Rosenthal space XM R as the completion of c00 under the norm x = sup{f, x : f ∈ F σ }. Take an arbitrary inﬁnite s = (Ei )∞ i=0 in Sσ . The corresponding sequence of vectors (xsi ) is semi-normalized and weakly null, and max | k≤n

k i=0

ai | ≤

n i=0

ai xsi

≤ (1 + ε) max | k≤n

k

ai |

(II.33)

i=0

13 FIN[<∞] is the collection of all ﬁnite block sequences E < E < · · · < E of nonempty ﬁnite 0 1 k subsets of N; see Section III.3.

II.6. Topological Ramsey Theory

177

for every n and every choice (ai )ni=0 of scalars from [−1, 1]. These two inequalities result from the observation that up to a small error the norm of the vector of the term k y= ai xsi i=0

is essentially equal to the absolute value of the evaluation of this vector at the corresponding special functional f

sk+1

k χEi = 1/2 |E i| i=0

(which happens to be equal to | ki=0 ai |). To see this note that for any other (inﬁnite) special sequence t = (Fi ) ∈ Sσ , there is j = j(s, t) such that: (4) Ei = Fi for i < j, (5) |Ei | = |Fi | for i > j. So our assumption about the growth of the sequence (mi ) will give us that, up to a small error, the evaluation of f t on a vector of the form y = ki=0 ai xsi will be equal to the evaluation of f sj on y. Note that the inequalities (II.33) show that the summing basis of c0 , s0 = (1, 0, 0, . . . ), s1 = (1, 1, 0, 0, . . . ), s2 = (1, 1, 1, 0, 0, . . . ), . . . is (1 + ε)-equivalent to any special sequence (xsi )∞ i=0 of vectors of XM R . Note that every inﬁnite subset N of N contains an inﬁnite special block sequence s = (Ei )∞ i=0 . This shows that every subsequence (ei )i∈N of the standard basis (ei )∞ i=0 of XM R contains a block subsequence (xsi )∞ i=0 which is (1 + ε)-equivalent to the summing basis of c0 . So in particular, no subsequence of (ei )∞ i=0 is unconditional. Let XM R be the bounded version of the Maurey–Rosenthal space, where one takes only the special functionals over special sequences s = (Ei ) ∈ Sσ of ﬁnite lengths j determined by their starting sets E0 in the sense that E0 = {j}. It is not diﬃcult to show that the bounded version XbM R of the Maurey–Rosenthal space 2 is isomorphic to C(ω ω + 1) and that every subsequence of its unit vector basis ﬁnitely block represents the summing basis of c0 .

II.6

Topological Ramsey Theory

Deﬁnition II.6.1. A set X ⊆ N[∞] is Ramsey if for every s and M there is N ∈ [s, M ] such that [s, N ] ⊆ X or [s, N ] ∩ X = ∅.

178

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Example II.6.2 (Baumgartner, Mathias). Choose a nonprincipal ultraﬁlter U on N, and let X = {M = (mi )i∈N ∈ N[∞] : [m2i , m2i+1 ) ∈ U}. i∈N

Then X is not Ramsey, since M ∈ X iﬀ M \ {min(M )} ∈ / X. Exercise II.6.3. Show that there is a non-Ramsey X ⊆ N[∞] which has the property of Baire relative to the product topology of N[∞] . (Hint: Intersect X of Example II.6.2 with {M ∈ N[∞] : ∀m, n ∈ M m ≡ n (mod 2)}.) Lemma II.6.4 (Galvin). Metrically open subsets of N[∞] are Ramsey. Proof. Fix an open set O ⊆ N[∞] , and consider F = {s ∈ N[<∞] : [s, N] ⊆ O}. Applying Lemma II.3.8 we obtain N ⊆ M such that F N is either empty or it contains a barrier. In the ﬁrst case [∅, N ] ∩ O = ∅, and in the second [∅, N ] ⊆ O. Relativizing to an arbitrary [s, M ] in place of [∅, N ] we get the conclusion of the lemma. Corollary II.6.5 (Nash–Williams). For every pair O0 and O1 of disjoint metrically open subsets of N[∞] there exists M ⊆ N such that M [∞] ∩Oi = ∅ for some i = 0, 1. Theorem II.6.6 (Galvin–Prikry). Metrically Borel subsets of N[∞] are Ramsey. Proof. Use induction on Borel rank to show that for every [s, M ] and every Borel X ⊆ N[∞] there is N ∈ [s, M ] such that X ∩ [s, N ] is relatively clopen in [s, N ] . Suppose X = k∈N Xk is a Borel set written as union of a sequence of Borel sets of smaller rank for which the conclusion holds. Build a decreasing sequence M0 ⊇ M1 ⊇ · · · ⊇ Mk ⊇ · · · of inﬁnite sets such that mk = min(Mk ) < mk+1 = min(Mk+1 ) and such for all k and s ⊆ {m0 , . . . , mk−1 }, [s, Mk ] ⊆ Xk or [s, Mk ] ∩ Xk = ∅. Let M = {mk : k ∈ N}. Then M ∩ [∅, M ] is a relatively open set, so we ﬁnish by Lemma II.6.4. We ﬁnish this section with Farahat’s proof of Rosenthal’s 1 -theorem that uses Galvin–Prikry theorem. Theorem II.6.7 (Rosenthal). Suppose that (xn ) is an inﬁnite normalized sequence of elements of some Banach space X. Then (xn ) has an inﬁnite subsequence (xnk ) which is either weakly Cauchy or is equivalent to the natural basis of 1 . It suﬃces to show that for every set Γ and an uniformly bounded sequence (fn ) of real-valued functions on Γ there is a subsequence (fnk ) which is either pointwise convergent on Γ or is equivalent (relative to the supremum norm on Γ) to the unit vector basis of 1 .

II.6. Topological Ramsey Theory

179

Recall the notion of independence for a sequence ((An , Bn ))∞ n=0 of pairs of disjoint subsets of Γ: ⎞ ⎛ Ai ∩ ⎝ Bj ⎠ = ∅ i∈I

j∈J

for all ﬁnite nonempty I, J ⊆ N with I ∩ J = ∅. The relevance of this notion to Rosenthal’s theorem is explained by the following lemma. Lemma II.6.8. Suppose that for some uniformly bounded (fn )∞ n=0 ⊆ ∞ (Γ) and some ε < δ, the sequence ({ξ ∈ Γ : fn (ξ) < ε}, {ξ ∈ Γ : fn (ξ) > δ})∞ n=0 is independent. The (fn ) is equivalent to the natural basis of 1 . Proof. Set An = {ξ ∈ Γ : fn (ξ) < ε}, Bn = {ξ ∈ Γ : fn (ξ) > δ} (n = 0, 1, . . . ). Choose a ﬁnite sequence (ai )ki=0 ⊆ [−1, 1]. It suﬃces to show that

k

ai fi ∞ ≥ (

i=0

k δ−ε |ai |. ) 2 i=0

(II.34)

Let I + = {i ≤ k : ai ≥ 0}, I − = {i ≤ k : ai < 0}. By independence we can choose ξ ∈ ( Ai ) ∩ ( Bj ) (II.35) i∈I

and η∈(

j∈J

Bi ) ∩ (

i∈I

Then

k i=0

and

k

ai fi (ξ) ≥

Aj ).

(II.36)

j∈J

|aj |δ −

j∈J

ai fi (η) ≤ −

i=0

|ai |ε

(II.37)

i∈I

|aj |δ +

i∈I

|aj |ε.

(II.38)

j∈J

From (II.38)–(II.37) we get k

ai (fi (ξ) − fi (η)) ≥ (δ − ε)

i=0

since the left-hand side is bounded by 2

k

|ai |

(II.39)

i=0

k i=0

ai fi ∞ , we are done.

180

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences Call a sequence ((An , Bn ))∞ n=0 of pairs of disjoint subsets of Γ convergent if ⎛ ⎞ ⎛ ⎞ ⎝ An ⎠ ∩ ⎝ Bn ⎠ = ∅. m n≥m

m n≥m

Lemma II.6.9. Any sequence ((An , Bn ))∞ n=0 of pairs of disjoint subsets of Γ contains an inﬁnite independent subsequence or an inﬁnite convergent sequence. Proof. Let X be the collection of all inﬁnite M = (mi )∞ i=0 ⊆ N such that for all k, (

k i=0

Am2i ) ∩ (

k

Bm2i+1 ) = ∅.

(II.40)

i=0

Clearly, X is a closed subset of N[∞] . Applying the Galvin–Prikry theorem we get an M ∈ N[∞] such that either M [∞] ⊆ X or else M [∞] ∩ X = ∅. Note that if the second alternative holds, the sequence ((Am , Bm ))m∈M is convergent. So assume that M [∞] ⊆ X. Let M = (mi )∞ i=0 be the increasing enumeration of M . Then it is easily checked that the sequence ((Am2i+1 , Bm2i+1 ))∞ i=0 is independent. We are now ready to ﬁnish the proof of Rosenthal’s 1 -theorem: Let (εk , δk ) (k ∈ N) be an enumeration of all pairs (ε, δ) of rationals such that ε < δ. By a successive application of Lemma II.6.9 we build a decreasing sequence (Mk )∞ k=0 of inﬁnite subsets of N such that (mk = min(Mk ))∞ is strictly increasing and such k=0 that for each k the sequence ({ξ ∈ Γ : fn (x) < εk }, {ξ ∈ Γ : fn (x) ≥ δk })n∈Mk

(II.41)

is either independent or convergent. If for some k the corresponding sequence is independent, Lemma II.6.8 gives us that (fn )n∈Mk is equivalent to the natural basis of 1 . On the other hand if this never happens the diagonal sequence (fmk )∞ k=0 is pointwise convergent on Γ.

II.7

The Theory of Better-Quasi-Orderings

Recall the notion of better-quasi-ordering introduced above in Section II.1. We shall now see how topological Ramsey theory can be eﬀectively used in proofs of some of the deepest results of b.q.o. theory. Lemma II.7.1. A quasi-ordering Q is b.q.o. iﬀ for every Borel function f : N[∞] −→ Q there is an inﬁnite X ⊆ N such that f(X)≤ f (X \ {min(X)}). Proof. Find ﬁrst M ∈ N[∞] such that the image f M [∞] is countable (see Theorem II.8.7 below) and then apply the Galvin–Prikry theorem to shrink even further to make the restriction f M [∞] continuous.

II.7. The Theory of Better-Quasi-Orderings

181

Exercise II.7.2. Prove that if Q is b.q.o., then so is Qω quasi-ordered by : (xi ) ≤ (yk ) iﬀ there is a strictly increasing sequence k0 < k1 < · · · < ki < · · · of integers such that xi ≤ yki for all i. (Hint: Assume there is a bad function f : N[∞] −→ Qω , i.e., f (M ) f (M \ {min(M )}) for all M ∈ N[∞] and ﬁnd M ∈ N[∞] and g : M [∞] −→ Q such that for every N ∈ M [∞] , g(N ) is a term of f (N ) and g(N ) g(N \ {min(N )}). Exercise II.7.3. Let Xi (i ∈ I) be a given uncountable family of Borel subsets of N[∞] . Show that there is inﬁnite M ⊆ N such that, either (a) M [∞] ⊆ Xi for inﬁnitely many i ∈ I, or (b) M [∞] ∩ Xi = ∅ for inﬁnitely many i ∈ I. A partial ranking of a quasi-ordered set Q is a well-founded partial ordering ≤ on Q such that x ≤ y implies x ≤ y. Given a set q.o. Q, a bad Q-array is a Borel map of the form f : M [∞] −→ Q where M is an inﬁnite subset of N and f (X) f (X \ {min(X)}) for all X in M [∞] . Given a pair f : M [∞] −→ Q and g : N [∞] −→ Q of bad Q-arrays, let f ≤∗ g mean that M ⊆ N and that f (X) ≤ g(X) for all X in M [∞] , and let f <∗ g mean that M ⊆ N and f (X) < g(X) for all X in M [∞] . A Q-array g is minimal bad (relative to the partial ranking ≤ ) if there is no bad Q-array f <∗ g. The following is one of the key results of b.q.o. theory. Theorem II.7.4 (Nash–Williams). For every bad Q-array g : M [∞] −→ Q (relative to the ﬁxed partial ranking of Q) there is a minimal Q-array f ≤∗ g. Proof. Starting with g0 = g we build for as long as it is possible a ≤∗ -decreasing transﬁnite sequence gα : Mα[∞] −→ Q(α ≤ γ) of bad Q-array as follows. [∞] Suppose we have constructed gα : Mα −→ Q. If gα is a minimal bad Q-array [∞] we stop the construction. Otherwise, we choose a bad Q-array fα : Nα −→ Q ∗ such that fα < gα . By the Galvin–Prikry theorem we may assume that fα is [∞] actually continuous on its domain Nα . We may assume that Mα \ Nα is inﬁnite and we can pick an integer nα ≥ 0 such that fα is constant on the basic open neighborhood [nα , Nα ]14 . Let Mα+1 = Nα ∪ ({0, 1, . . . , nα } ∩ Mα ) 14 [k, P ]

= {X ∈ P [∞] : X ∩ {0, 1, . . . , k − 1} = P ∩ {0, 1, . . . , k − 1}}.

182

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences [∞]

and deﬁne gα+1 : Mα+1 −→ Q by letting gα+1 Nα[∞] =fα [∞]

[∞]

gα+1 Mα+1 \ Nα[∞] =gα Mα+1 \ Nα[∞] . Then gα+1 is a bad Q-array and gα+1 ≤∗ gα . If β is a countable limit ordinal such that gα is deﬁned for all α < β, let Mβ = Mα α<β

Claim. Mβ is inﬁnite. Proof. Suppose not and pick m such that Mβ ⊆ {0, 1, . . . , m − 1}. For α < β, let mα = min(Mα \ m). Clearly mα < mα whenever α ≤ α < β. By the choice of m there must be inﬁnitely many α < β such that mα < mα+1 , or in other words, mα ∈ / Mα+1 . Going back to the deﬁnition of fα , Nα , Mα+1 and gα+1 [∞] from gα : Mα −→ Q we see that mα < mα+1 implies nα ≤ m So we can ﬁnd an inﬁnite set A ⊆ β, an integer 0 ≤ n ≤ m and s ⊆ {0, 1, . . . , n−1} such that for all α ∈ A, nα = n and Nα ∩ {0, 1, . . . , n} = s Consider a pair α < α of elements of A. Note that Nα ∈ [nα , Nα ], and therefore gα (Nα ) ≤ gα+1 (Nα ) = fα (Nα ) = fα (Nα ) < gα (Nα ). This gives us an inﬁnite < -decreasing sequence contradicting the well-foundedness of ≤ . [∞]

Given that Mβ is inﬁnite deﬁne gβ : Mβ

−→ Q by

gβ (X) = lim gα (X) α→β

Since gα (X) is ≤ -decreasing in α, the limit exists. Note also that gβ is a Borel function. Since Mα ’s are strictly decreasing, the process must stop at some countable ordinal γ in which case we have reached a minimal bad Q-array.

II.7. The Theory of Better-Quasi-Orderings

183

For a given quasi-ordered set Q let QOrd =

Qα

α∈Ord

quasi-ordered by: (xξ )ξ<α ≤1 (yη )η<β iﬀ there is a strictly increasing sequence ηξ (ξ < α) of ordinals < β such that xξ ≤ yηξ for all ξ < α. Theorem II.7.5 (Nash–Williams). If Q is b.q.o., then so is QOrd . Proof. For (xξ )ξ<α , (yη )η<β ∈ QOrd , set (xξ )ξ<α ≤ (yη )η<β iﬀ α ≤ β and xξ = yξ for all ξ < α. Suppose QOrd is not b.q.o. By Theorem II.7.4 we can consider a minimal bad QOrd -array g : M [∞] −→ QOrd relative to this particular partial ranking. For a given X ∈ M [∞] we know that g(X) 1 g(X \ {min(X)}). So there must be γX such that g(X) γX ≤1 g(X \ {min(X)}) but g(X) (γX + 1) 1 g(X \ {min(X)}). Let f : M [∞] −→ QOrd deﬁned by f (X) = g(X) γX Note that f is a Borel function and that f (X) < g(X) for all X ∈ M [∞] . Applying the Galvin–Prikry theorem to the Borel set X = {X ∈ M [∞] : f (X) 1 f (X \ {min(X)})}, we get N ∈ M [∞] such that N [∞] ⊆ X or N [∞] ∩ X = ∅. Note that the second alternative is not possible since this would make f N [∞] a bad QOrd -array contradicting the minimality of g. To simplify the notation, for X ⊆ N, let ∗ X denote X \ {min(X)}. Then the inclusion N [∞] ⊆ X means g(X) γX ≤1 g(∗ X) γ∗ X

(X ∈ N[∞] ).

Combining this with the choice of X −→ γX , we conclude that for all X ∈ N[∞] , g(X) γX ≤1 g(∗ X) γ∗ X but g(X) (γX + 1) 1 g(∗ X) (γ∗ X + 1).

184

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

It follows that g(X)γX g(∗ X)γ∗ X for all X ∈ N[∞] . So starting from a minimal bad QOrd -array g : M [∞] −→ QOrd we have found a bad Q-array X −→ g(X)γX (X ∈ N[∞] ) which for every X picks a term out of the transﬁnite sequence g(X). This ﬁnishes the proof. The following result provides a converse for Theorem II.7.5. Theorem II.7.6 (Pouzet). The following are equivalent for a quasi-ordered set Q: 1. Q is b.q.o.

2. Q<ω1 is w.q.o.

Recall that a linearly ordered set L is scattered if it has no subset isomorphic to the rationals. The class S of scattered orders can be generated by the following recursive procedure: Let S0 be the class of one-point linear orders. Given an ordinal ξ > 0 let Sξ be the class of linear orderings isomorphic to either well-ordered or conversely wellordered sums of linear orders from η<ξ Sη . The well-known result of Hausdorﬀ shows that S = ξ∈Ord Sξ . Exercise II.7.7. Prove Hausdorﬀ’s Theorem, that S = ξ∈Ord Sξ . Theorem II.7.8 (Laver). The class of scattered linear orderings is b.q.o. under embeddability. Proof. For L ∈ S = ξ∈Ord Sξ let ξ(L) be the minimal ξ such that L ∈ Sξ . Deﬁne L < M iﬀ L is embeddable into M and ξ(L) < ξ(M ). Let L ≤ M iﬀ L < M or L = M . Note that ≤ is a partial ranking on S. If S is not b.q.o., by Theorem II.7.4 there is a minimal bad array g : M [∞] → S, relative to this partial ranking. For each X ∈ M [∞] we write X X g(X) = LX 0 + L1 + · · · + Lα + . . .

(α < βX )

or X X g(X) = · · · + LX α + · · · + L1 + L0 ,

such that ξ(LX α ) < ξ(g(X)) for all α < βX . Shrinking M we may assume that g(X) has, say, the ﬁrst form for all X ∈ M [∞] . Then as in the proof of Theorem II.7.5 we ﬁnd N ∈ M [∞] and for each X ∈ N [∞] and ordinal γX < βX such that X\{min(X)} LX Lα α ≤ α<γX X\{min(X)}

but LX γX ≤ LγX

α<γX

. It follows that f (X) = LX γX

(X ∈ N[∞] )

deﬁnes a bad array such that f <∗ g, a contradiction.

II.8. Ellentuck’s Theorem

185

The following corollary was conjectured by R. Fra¨ıss´e in the 1950s and it represents one of the ﬁnest applications of Nash–Williams b.q.o. theory. Corollary II.7.9 (Laver). The class of countable linearly ordered sets is w.q.o. Remark II.7.10. The reader interested in a more complete picture of this interesting subject is referred to the excellent introductions to this area given by Laver [47], Milner [62] and Simpson [85].

II.8

Ellentuck’s Theorem

Consider the reﬁnement of the usual metrizable topology on N[∞] by letting the sets of the form [s, M ] (s ∈ N[<∞] , M ∈ N[∞] ) be open. This is the Ellentuck topology on N[∞] . Lemma II.8.1. Every Ellentuck-open subset of N[∞] is Ramsey. Proof. Let X be a given Ellentuck-open set, and let [s, M ] be a given basic-open set. We simplify notation by letting s = ∅. Shrinking M , we assume that M has the following property. (1) If for some ﬁnite s ⊆ M one can ﬁnd an inﬁnite N ⊆ M such that [s, N ] ⊆ X then already [s, M ] ⊆ X. Let F = {s ∈ M [<∞] : [s, M ] ⊆ X}. Applying Galvin’s Lemma to F we ﬁnd N ⊆ M such that either (a) F ∩ P(N ) = ∅, or (b) Every inﬁnite X ⊆ N contains an initial segment in F. If (b) holds, then clearly [∅, N ] ⊆ X. If on the other hand (a) holds then by (1) and the fact that X is Ellentuck-open one gets that [∅, N ] ∩ X = ∅ Deﬁnition II.8.2. A subset X of N[∞] is Ramsey-null if for every basic-open set [s, M ] there is N ∈ [s, M ] such that [s, N ] ∩ X = ∅ Lemma II.8.3. Every Ellentuck-nowhere-dense subset of N[∞] is Ramsey-null. Proof. Let X be a given subset of N[∞] which is nowhere dense relative to the Ellentuck topology. Clearly we may assume X is Ellentuck-closed. Let [s, M ] be a given basic-open set. By Lemma II.8.1, there is N ∈ [s, M ] such that [s, N ] ⊆ X or [s, N ] ∩ X = ∅

Note that the ﬁrst alternative is impossible Lemma II.8.4. Every Ellentuck-meager subset of N

[∞]

is Ramsey-null.

186

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

Proof. Let X = ∪i∈N Xi be a given Ellentuck-meager set written as increasing union of Ellentuck-nowhere-dense sets. Let [s, M ] be a given basic set. We simplify the notation by assuming s = ∅. Starting with M0 = M we build an inﬁnite decreasing sequence M0 ⊇ M1 ⊇ · · · ⊇ Mk ⊇ · · · of inﬁnite subsets of N such that the sequence {mi = min(Mi )}i∈N is strictly increasing and such that (applying Lemma II.8.3) [s, Mk+1 ] ∩ Xk = ∅ for all k and s ⊆ {m0 , . . . , mk }. Let N = {mi : i ∈ N}. Then N ⊆ M and [∅, N ] ∩ X = ∅ . Theorem II.8.5 (Ellentuck). The following conditions are equivalent for a subset X of N[∞] : (a) X has the Ramsey property (b) X has the Baire property relative to the Ellentuck topology. Moreover, X is Ramsey-null if and only if X is Ellentuck-meager. Proof. Suppose an X = O M is a given subset of N[∞] written as the symmetric diﬀerence of an Ellentuck-open set O and an Ellentuck-meager set M. Let [s, M ] be a given basic set. By Lemma II.8.3. there is an N ∈ [s, M ] such that [s, N ] ∩ M = ∅ It follows that X ∩ [s, N ] = O ∩ [s, N ] is an Ellentuck-open subset of N[∞] so by Lemma II.8.1 there is P ∈ [s, N ] such that [s, P ] ⊆ X or [s, P ] ∩ X = ∅. Corollary II.8.6 (Silver). The ﬁeld of Ramsey subsets of N[∞] is closed under Souslin operation, and so in particular every metrically analytic subset of N[∞] is Ramsey. Theorem II.8.7 (Louveau–Simpson). Suppose f is a Baire-measurable map from the Ellentuck space N[∞] into some metric space Z. Then there is M ∈ N[∞] such that f M [∞] is continuous and has separable range in Z. Proof. Otherwise we can assume to have a discrete family U of size continuum of open subsets Z such that for every inﬁnite set M ⊆ N, UM = {U ∈ U : U ∩ f [M [∞] ] = ∅} is uncountable. Choose a Borel map g : Z −→ R such that (a) g (Z \ U) ≡ 0 (b) for every U ∈ U, g U takes only one value xU ∈ R such that xU = xV whenever U = V . (c) {xU : u ∈ U } is a set of reals which contains no perfect subset.

II.8. Ellentuck’s Theorem

187

Consider the composition h = f ◦ g : N[∞] −→ R. Then h is Baire-measurable relative to the Ellentuck topology on N[∞] . Using Ellentuck’s theorem, a simple diagonalization argument over a countable basis of R will give us an inﬁnite M ⊆ N such that h M [∞] is continuous even relative to the metrizable topology of M [∞] . It follows that the range of h M [∞] is an uncountable analytic set of reals without a perfect subset, a contradiction. Lemma II.8.8. Every analytic equivalence relation E on N[∞] either has a perfect set of pairwise inequivalent elements, or there is an inﬁnite set M ⊆ N such that E on M [∞] has at most countably many classes. Proof. We shall supply the proof working under the assumption that the Baire property of the Ellentuck topology on N[∞] is ℵ1 -additive. A simple absoluteness argument (which we leave to the reader) shows that the conclusion of II.8.8 cannot depend on this sort of assumptions. So, suppose E is an analytic equivalence relation on N[∞] that has no perfect set of pairwise non-equivalent elements of N[∞] . By Burges’ theorem (see Kechris [42]) we conclude that the quotient N[∞] /E has size at most ℵ1 . Taking N[∞] /E with the discrete metric our assumption yields that the quotient map π : N[∞] → N[∞] /E is Baire measurable relative to the Ellentuck topology on N[∞] . Then Theorem II.8.7 gives us M ∈ N[∞] such that π[M [∞] ] is countable, as required. Exercise II.8.9. Show how the assumption in the proof of Lemma II.8.8 can be avoided. Lemma II.8.10. Let (en ) be a normalized basic sequence in some Banach space X. Then either there exist continuum many pairwise non isomorphic closed subspaces of X of the form (enk ), or else there is a subsequence (enk ) of (en ) such that there exists only countable many isomorphism types among spaces of the form (enkl ) where (enkl ) is a subsequence of (enk ). Proof. For M, N ∈ N[∞] put M E N iﬀ the closed subspaces (em )m∈M and (en )n∈N are isomorphic. Note that E is an analytic relation on N[∞] . Now, apply Lemma II.8.8. Going in this direction one has the following more reﬁned result. Theorem II.8.11 (Rosendal). Let (en ) be a basic sequence in some Banach space X. Then either there exist continuum many pairwise non isomorphic subspaces among spaces spanned by subsequences of (en ) or there is a subsequence (enk ) of (en ) such that all further subsequences of (enk ) span isomorphic spaces. We ﬁnish this section by mentioning yet another result along these lines. Theorem II.8.12 (Ferenczi-Rosendal). Let X be a Banach space with an unconditional basis (en ). Then either:

188

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

(I) There is continuum many isomorphism types among subspaces of X of the form (enk ) where (enk ) is an inﬁnite subsequence of (en ), or (II) There is a subspace Y spanned by a subsequence of (en ) such that Y is isomorphic to its hyperplanes, to its square, and it admits a Schauder decomposition into countably many uniformly isomorphic copies of itself.

II.9

Summability in Banach Spaces

Recall that a sequence (xn ) of elements of some Banach space X is Ces` arosummable if the corresponding sequence x1 + · · · + xn n of arithmetic means converges in norm. A classical result reads that every convergent sequence (xn ) ⊆ R is Ces`aro-summable in R to the same limit and in fact the same is true for any other Banach space. A Banach space X is said to have the Banach–Saks property if every bounded sequence (xn ) in X has a Ces`aro-summable subsequence. Example of such spaces are the spaces Lp [0, 1], (1 < p < ∞) and this is what Banach and Saks have originally established. Example II.9.1 (Schreier). The basis (en ) of the Schreier space S (see Example I.2.13 above) is weakly null but has no Ces` aro-summable subsequence since for every ﬁnite set s ∈ FIN, we have that 1 1 ei S ≥ . |s| i∈s 4 The Ces`aro-summability is just one of the methods of summability in Banach spaces. Thus, an ω × ω-matrix (aij ) is called a regular method of summability if given a sequence (xn ) of elements of some Banach space X converging in norm to some x ∈ X, then the sequence yi =

∞

aij xj

j=0

also converges in norm to x. It is well known that (aij ) is a regular method of summability if and only if: ∞ (a) The sequence ( j=0 aij )i is uniformly bounded in R. ∞ (b) limi→∞ j=0 aij = 1. (c) limi→∞ aij = 0 for every j. Theorem II.9.2 (Erd¨ os–Magidor). Let (xn ) be a bounded sequence in some Banach space X and let (aij ) be a given method of summability. Then there is a subsequence (xnk ) of (xn ) such that either:

II.9. Summability in Banach Spaces

189

(1) Every subsequence of (xnk ) is (aij )-summable and all of them have the same limit. (2) No subsequence of (xnk ) is (aij )-summable. Proof. Let X = {M ∈ N[∞] : (xn )n∈M is (aij )-summable}. Note that X is a Borel subset of N[∞] , so by the Galvin–Prikry theorem there is M ∈ N[∞] such that M [∞] ⊆ X or M [∞] ∩ X = ∅. The second alternative gives us the alternative (2) of the theorem. So let us assume M [∞] ⊆ X and reﬁne M further and obtain a subsequence satisfying the alternative (a). Let Y be the closed subspace of X spanned by the subsequence (xn )n∈M . Note that Y is separable, so we can ﬁx a sequence (Bk ) of open balls of Y forming a basis of its norm topology. For each k let Xk = {P ∈ M [∞] : (xn )n∈P is (aij )-summable to a point of Bk }. Then each Xk is a Borel subset of M [∞] so by a sequence of successive applications of the Galvin–Prikry Theorem we can ﬁnd a decreasing sequence M ⊇ M0 ⊇ M1 ⊇ · · · ⊇ Mk ⊇ . . . of inﬁnite subsets of M such that [∞]

Mk

[∞]

⊆ Xk or Mk

∩ Xk = ∅

for all k. We may assume that the sequence (mk = min(Mk )) is strictly increasing, and this will give us an inﬁnite set N = {mk : k ∈ N} which diagonalizes the sequence (Mk ). Note that for an inﬁnite set P = (pi ) ⊆ N the limit of the sequence ∞ yiP = aij xpj j=0

does not change if we make a ﬁnite change to the set P (see conditions (a), (b), (c) on (aij )). It follows that for every k and every inﬁnite P, Q ⊆ N the sequence (yiP ) and (yiQ ) either both converge to a point in Bk or both converge to a point outside Bk . Hence we must have lim

i→∞

∞ j=0

aij xpj = lim

i→∞

∞

aij xqj

j=0

for every pair P = (pi ) and Q = (qi ) of inﬁnite subsets of N . It follows that the subsequence (xn )n∈N satisﬁes the alternative (2). Corollary II.9.3. If a Banach space X has the Banach–Saks property, then every bounded sequence (xn ) in X has a subsequence (xnk ) such that every subsequence of (xnk ) is Ces` aro-summable.

190

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

os–Magidor Exercise II.9.4. Prove that the set X ⊆ N[∞] from the proof of Erd¨ Theorem is Gδσδ . What is the complexity of sets Xk from the same proof? Exercise II.9.5. Prove that in a ﬁnite-dimensional Banach space X, a bounded sequence each subsequence of which is Ces`aro-summable must itself be convergent in X. Exercise II.9.6 (Nishiura–Waterman). Show that every Banach space with the Banach–Saks property is reﬂexive. Lemma II.9.7. Suppose (xn ) is a weakly null basic subsequence in some Banach space X such that no subsequence of (xn ) is Ces` aro summable. Then (xn ) has a subsequence (xnk ) with spreading model isomorphic to 1 . Proof. We think of (xn ) as a sequence of continuous functions on the dual ball BX ∗ with the weak∗ topology. Our assumption about (xn ) in this setting states that (xn ) converges pointwise to the constant 0 function and that for no subsequence (xnk ) the sequence xn1 + · · · + xnk k converges uniformly on BX ∗ . For M ∈ N[∞] , q ∈ R and x∗ ∈ BX ∗ , let M (x∗ ≥ ε) = {m ∈ M : x∗ (xm ) ≥ ε}. Similarly one deﬁnes M (x∗ ≤ ε). Let F≥ε (M ) = {M (x∗ ≥ ε) : x∗ ∈ BX ∗ }. Similarly one deﬁnes F≤ε (M ). Let X≥ε = {M ∈ N[∞] : F≥ε (M ) contains sets of arbitrary large size}. Similarly one deﬁnes X≤ε . Note that assuming as we may that the Banach space X inside which we are working is separable, all these sets X≥ε and X≤ε are analytic. We claim that there must be ε > 0 such that either X≥ε or X≤−ε contains a set of the form M [∞] for some M ⊆ N. Otherwise, by Silver’s theorem we can choose a decreasing sequence M1 ⊇ · · · ⊇ Mk ⊇ . . . of inﬁnite subsets of N such that [∞]

Mk

∩ (X≥1/k ∪ X≤−1/k ) = ∅

for all k. Pick inﬁnite M such that k ∈ M for all , k ∈ M with < k. Then for all ε > 0 there is a positive integer n(ε) such that the set M (x∗ , ε) = {m ∈ M : |x∗ (xm )| ≥ ε} has size at most n(ε). From this one easily concludes that if (mk ) is the increasing enumeration of M , then the sequence xm1 + · · · + xmk k

II.9. Summability in Banach Spaces

191

converges uniformly on BX ∗ contradicting our assumption. So, ﬁx an ε > 0 such that X≥ε or X≤−ε contains a set of the form M [∞] for some inﬁnite M ⊆ N. By symmetry we may consider only the ﬁrst case and to save on notation, we assume that for every inﬁnite M ⊆ N the family F≥ε (M ) contains sets of arbitrarily large cardinality. Note that this means that for every k > 0 and M ∈ N[∞] there is N ∈ M [∞] such that N [k] ⊆ F≥ε (N ). So diagonalizing we obtain an inﬁnite set M ⊆ N such that F≥ε (M ) = S(M ) = {E ⊆ M : |E| ≤ |{m ∈ M : m < min(E)}|}. Pick an E ∈ S(M ) and a sequence an (n ∈ E) of non negative scalars. Choose an x∗ ∈ BX ∗ such that E ⊆ M (x∗ ≥ ε). Then an xn ≥ |x∗ ( an xn )| n∈E

n∈E

≥ ε|

an | = ε

n∈E

an .

n∈E

Let (mk ) be the sequence which increasingly enumerates M . Applying the Brunel– Sucheston Theorem I.2.12 we conclude that some subsequence (xmki ) of (xmk ) has a spreading model (ei ) that is unconditional. It follows that

k−1

ai ei ≥ (ε/2)

i=0

k−1

|ai |,

i=0

for every k and every choice ai (i < k) of positive scalars. Since (ei ) is unconditional this inequality (with ε/2 replaced by some other constant C > 0) remains true for every choice of scalars ai (i < k). Hence (ei ) equivalent to the unit vector basis of 1 . This ﬁnishes the proof. Combining Corollary II.9.3 and Lemma II.9.7 we obtain the following. Theorem II.9.8 (Rosenthal). Let (xn ) be a weakly null sequence in some Banach space X. Then either: (1) (xn ) has a subsequence all of whose subsequences are Ces` aro-summable, or (2) (xn ) has a subsequence with spreading model isomorphic to 1 .

Exercise II.9.9. Show that for every uniformly bounded sequence in some space of the form ∞ (Γ) the following are equivalent: (1) There is f ∈ ∞ (Γ) such that lim [

n→∞

k1 <···
1 fk )∞ ] = 0. n i=1 i n

(

192

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

(2) Every subsequence of (fn ) is Ces`aro-summable in ∞ (Γ) and each to the same limit. Remark II.9.10. For further results in this area we refer the reader to the paper of Argyros, Mercourakis and Tsarpalias [7] which generalizes Ces`aro-summability into a transﬁnite sequence of summability methods and prove the corresponding analogue of Rosenthal’s theorem.

II.10 Summability in Topological Abelian Groups In this section we examine the summability problem for series not only in Banach spaces but more generally in (Hausdorﬀ) ∞ topological abelian groups. Given such a group G, we say that a series n=0 xn of elements of G is unconditionally convergent if ∞ x is convergent in G for every subsequence (xnk ) of (xn ). n k k=0 ∞ Lemma II.10.1. Let n=0 xn be an unconditionally convergent series in a topo∞ logical abelian group G. Suppose that (An )∞ n=0 and (Bn )n=0 are sequences of Borel subsets of G such that An ∪ (B n + xn ) = G for all n. Then there is a subsequence (xnk ) of (xn ) such that y = ∞ k=0 xnk belongs to inﬁnitely many An or inﬁnitely many Bn . Proof. For M ∈ N[∞] , let xM =

xm .

m∈M

For k ∈ N let / An ∪ Bn for all n ≥ k}. Nk = {M ∈ N[∞] : xm ∈ Note that all Nk ’s are Borel subsets of N[∞] , so by the Galvin–Prikry or Ellentuck theorem it suﬃces to show that all of them are Ramsey null. For suppose that [l, M ] ⊆ Nk for some l and k. Pick n > max{k, l} in M . Let N = M \ {n}. Then N ∈ [l, M ] ⊆ Nk so in particular neither xM nor xN belongs to An ∪ Bn . However, note that by the assumption of the lemma, xM ∈ / An implies that xM ∈ Bn + xn , or equivalently xM − xn = xN ∈ Bn , a contradiction. Exercise II.10.2. (a) Show that Lemma II.10.1 is true if the sets An and Bn are assumed to be sequentially open in G. (b) Show that if the sets An and Bn in Lemma II.10.1 are assumed only to have the property of Baire in G, then m n≥m (An ∪ Bn ) is comeager in G. (c) Show that if G is locally compact andif the sets An and Bn of Lemma II.10.1 are assumed to be Haar measurable, then m n≥m (An ∪Bn ) has the full measure. Corollary II.10.3 (Uniform boundedness principle). Suppose that F is a pointwise bounded family of continuous linear operators from a Banach space X into a normed space Y . Then F is uniformly bounded.

II.10. Summability in Topological Abelian Groups

193

Proof. If supT ∈F T = ∞, then we can ﬁnd two inﬁnite sequences (xn ) ⊆ X and (Tn ) ⊆ F such that xn ≤ 2−n and Tn (xn ) ≥ n for all n. For n ∈ N, let An = Bn = {x ∈ X : Tn (x) ≥

n }. 2

Then A n∞(and Bn ) is a closed subset of X and An ∪ (Bn + xn ) = X for all n. The series n=0 xn is clearly unconditionally convergent, so by Lemma II.10.1 there is x ∈ X such that Tn (x) ≥ n/2 for inﬁnitely many n, so the family F is not pointwise bounded. Corollary II.10.4 (Automatic continuity principle). Suppose that T : X → Y is a Borel linear map from a Banach space X into a normed space Y . Then T is continuous. Proof. Otherwise we can ﬁnd an inﬁnite sequence (xn ) ⊆ X such that xn ≤ 2−n and T (xn ) ≥ n for all n. For n ∈ N, let An = Bn = {x ∈ X : T (x) ≥ n/2}.

(II.42)

Then An and Bn are Borel sets such that An ∪ (Bn + xn ) = X for all n. By Lemma II.10.1 there is x ∈ X such that T (x) ≥ n/2 for inﬁnitely many n, a contradiction. Corollary II.10.5 (Schur’s 1 -theorem). In the space 1 every weakly convergent sequence is norm convergent. Proof. Suppose (xn ) ⊆ 1 is weakly null but for some ε > 0, xn ≥ ε for all n. Since (xn ) is weakly null, there is an inﬁnite subsequence (xnk ) of (xn ) and a corresponding inﬁnite sequence of disjointly and ﬁnitely supported functionals (fk ) ⊆ S∞ such that |fk (xnk )| > ε/2 for all k. For k ∈ N, set Ak = Bk = {f ∈ ∞ : |f (xnk )| > ε/4}. The sets Ak and Bk are w∗ -open and ∞ = Ak ∪ (Bk + fk ) for all k. Since (fk ) is normalized and disjointly supported the series ∞ k=0 fk is unconditionally convergent in (∞ , w∗ ). By Lemma II.10.1 there is f ∈ ∞ such that |f (xnk )| > ε/4 for inﬁnitely many k. This contradicts the fact that (xn ) is weakly null. Exercise II.10.6 (Von Neumann). Find a weakly null sequence in 2 which is not norm null. ∞ Corollary II.10.7 (Orlicz–Pettis Theorem). A series n=0 xn of some normed space X is unconditionally convergent relative to the norm topology iﬀ it is unconditionally convergent relative to the weak topology. Proof. First of all note that we may assume that X is a separable Banach ∞space and that it suﬃces to show that xn →n 0 under the assumption that n=0 xn is unconditionally convergent relative to the weak topology of X. Otherwise, there

194

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

is an ε > 0 and an inﬁnite subsequence (xnk ) of (xn ) such that xnk ≥ ε for all k. By Lemma II.5.1, going to a subsequence we may assume that (xnk ) is a seminormalized basis of its closed linear span Y . The corresponding sequence (fk ) ⊆ Y ∗ of the evaluation functionals relative to the basis (xnk ) is also semi-normalized, is w∗ -null and has the property that |fk (xnk )| > ε for all k. For k ∈ N, let

Ak = Bk = {x ∈ Y : |fk (x)| > δ/2}.

Then Ak and Bk are weakly open in Y and Y = Ak ∪ (Bk + xnk ) for all k. By Lemma II.10.1 there is x ∈ Y such that |fk (x)| > ε/2 for inﬁnitely many k, contradicting the fact that (fk ) is w∗ -null. Corollary II.10.8 (Nikodym boundedness principle). Let F be a family of countable additive set functionals deﬁned on a measurable space (X, B) such that bE = sup |µ(E)| < ∞ µ∈F

for every E ∈ B. Then

sup bE < ∞.

E∈B

Proof. Suppose that supE∈B bE = ∞. Pick E0 ∈ B and µ0 ∈ F such that |µ0 (E0 )| > 1 + bX . Then |µ0 (E0 )| > 1 and |µ0 (X \ E0 )| ≥ |µ0 (E0 )| − |µ0 (X)| > 1. At least one of supE⊆E0 bE or supE⊆X\E0 bE must be inﬁnite. Renaming X \ E0 as E0 if needed, we may assume that supE⊆X\E0 bE = ∞. Then we can repeat the same argument inside X \ E0 getting E1 ⊆ X \ E0 such that µ(E1 ) > 2 and supE⊆X\(E0 ∪E1 ) bE = ∞, and so on. This gives us an inﬁnite sequence (En )∞ n=0 of disjoint elements of B and a sequence (µn )∞ such that |µ (E )| > 1 + n for all n n n=0 n. Let G be the group of all bounded scalar-valued B-measurable functions on X with the topology of pointwise convergence on the family of all countably additive set-functionals on B. Then the series ∞ n=0 χEn is unconditionally convergent in G and the limits are characteristic functions of members of B. Let An = Bn = {f ∈ G : | f dµn | > (n + 1)/2}. Then An and Bn are open in G, and An ∪ (Bn + χn ) = G for all n. By Lemma II.10.1 there exists an E ∈ B such that |µn (E)| > n + 1/2 for inﬁnitely many n, contradicting the assumption that bE < ∞.

II.10. Summability in Topological Abelian Groups

195

Corollary II.10.9 (Vitali–Hahn–Saks Theorem). Suppose that (X, B, µ) is a measure space (i.e., µ is non-negative on B) and let (νn )∞ n=0 be a sequence of µcontinuous additive measures deﬁned on B. Suppose that limn→∞ νn (E) exists for all E ∈ B. Then the sequence (νn )∞ n=0 is uniformly µ-continuous. Proof. Otherwise, one can ﬁnd an inﬁnite subsequence (νnk )∞ k=0 of (νn ) and a sequence (Ek )∞ k=0 ⊆ B such that µ(Ek ) → 0 but for some ε > 0, |νnk (Ek )| > ε for all k. Using the µ-continuity of the νn ’s, and going to a subsequence of (νnk ) if needed, we may assume that for every k |νnk (E)| < ε/3 for all E ⊆ Ek in B. l>k

Let Fk = Ek \ l>k Ek for k ∈ N. Then (Fk ) is an inﬁnite sequence of disjoint sets from B such that |νnk (Fk )| > (2/3)ε and |νnk (Fl )| < ε/3 for all k < l. In particular, |νnk+1 (Fk+1 ) − νnk (Fk )| > ε/3 for all k. As in the previous proof applying Lemma II.10.1 we get a set E ∈ B such that |νnk+1 (E) − νnk (E)| > ε/3 for inﬁnitely many k’s. This means that (νn (E))∞ n=0 is not a convergent sequence, a contradiction. Remark II.10.10. We have taken Lemma II.10.1 (but not its proof) from a paper of Matheron [53]. We refer the reader to papers [15] and [19] for some further uses of these ideas. Deﬁnition II.10.11. A topological group G is Mazur–Orlicz complete if every null ∞ sequence (xn ) in G contains a subsequence (x ) such that the series n k k=0 xnk ∞ is convergent in G. If in addition k=0 xnk is unconditionally convergent then we say that G is unconditionally Mazur–Orlicz complete . Clearly every completely metrizable abelian group is unconditionally Mazur– Orlicz complete. The following result is a partial converse to this implication. Theorem II.10.12 (Burzyk–Klis–Lipecki). Every metrizable Mazur–Orlicz complete abelian group G is a Baire space. Proof. Consider a decreasing sequence (Un )∞ n=0 of dense open subsets of G. We shall show that 0 is in the closure of the intersection from which it follows easily that ∞ n=0 Un is dense in G. Let d be a ﬁxed translation invariant metric on G giving us its topology and let ε > 0. A simple recursive construction will give us ∞ a sequence (xn )∞ n=0 ⊆ G and a sequence (Vn )n=0 of open subsets of G such that for all n: (1) d(0, xn ) ≤ ε · 2−n−1 , (2) Vn ⊆ Un , and n n (3) k=0 σk xk ∈ k=0 Vkσk for all (σk )nk=0 ⊆ {0, 1}, where for a subset S of G we let S 0 = X and S 1 = S. Applying the Mazur–Orlicz completeness of G to (xn ) we get a subsequence (xni ) such that the series ∞ i=0 xni

196

Chapter II. Ramsey Theory of Finite and Inﬁnite Sequences

converges in G. It follows that required.

∞ i=0

xni ∈

∞ n=0

Un and d(0,

∞ i=0

xni ) ≤ δ, as

Corollary II.10.13. Suppose that G is a metrizable abelian group which has the property of Baire in its completion. Then G is complete if and only if it is Mazur– Orlicz complete. Corollary II.10.14. Let Y be a subspace of a Banach space X. If Y is an analytic subset of X and if it is Mazur–Orlicz complete then it must in fact be closed. Remark II.10.15. This corollary has been ﬁrst observed by A.R.D. Mathias who was using Silver’s theorem to prove it (see [39] and [88]). Note that there are interesting groups that are unconditionally Mazur–Orlicz complete but are not Baire spaces. One such example is the group G = (1 , w). They of course cannot be metrizable. It turns out however that non-metrizable group do share some of the properties of the metrizable ones as the following result shows. Theorem II.10.16. Let G be an unconditionally Mazur–Orlicz complete abelian group. Then every Borel subgroup H of G which is Mazur–Orlicz complete must in fact be closed in G. Proof. Suppose H is not closed in G and pick a sequence (xn ) ⊆ H converging to a point of G \ H. Applying the unconditional Mazur–Orlicz completeness of G and going to a subsequence of (xn ) we may assume that the series ∞ x n=0 n is unconditionally convergent in G. For n ∈ N let y = x − x . Note that n n+1 n ∞ [∞] y ∈ / H. For M ∈ N , let n n=0 yM =

∞ M (2k+1)−1

yn ,

k=0 n=M (2k)

where (M (k))∞ k=0 is the increasing enumeration of M . Let XM be the collection of all M ∈ N[∞] such that yM ∈ H. Then XH is a Borel subset of N[∞] , so Galvin–Prikry theorem applies to XH . Pick an M ∈ N[∞] such that M [∞] ⊆ XH or M [∞] ∩ XH = ∅. Note that for N ∈ N[∞] and n ∈ N at most one of the elements yN and yN \{n} can belong to XH since ∞ / H. It follows that n=0 yn ∈ M [∞] ∩ XH = ∅. For k ∈ N, set

M (2k+1)−1

zk =

n=M (2k)

yk .

The fact that the series yM = ∞ k=0 zk is convergent in G means in particular that the sequence (zk ) of elements of H converges to 0. Applying the Mazur–Orlicz ∞ completeness of H to (zk ) we get that a subsequence (zki ) such that i=0 zki is ∞ convergent in H. Note however that i=0 zki is equal to yN for some N ∈ M [∞] , so M [∞] ∩ XH = ∅, a contradiction.

Chapter III

Ramsey Theory of Finite and Inﬁnite Block Sequences III.1 Hindman’s Theorem Hindman’s theorem in its ﬁnite unions form (rather than non-repeating sums form) can be considered as the ﬁrst non-trivial result of so-called block Ramsey theory. To state this theorem, let FIN denote the collection of all ﬁnite nonempty subsets of N. A ﬁnite or inﬁnite sequence X = (xi ) of elements of FIN is a block sequence if xi < xj 1 whenever i < j. For a block sequence X = (xi ) set [X] = {xi0 ∪ · · · ∪ xik : k ∈ N, i0 < · · · < ik < |X|}. We call [X] a partial subsemigroup of (FIN, ∪) generated by X = (xi ). Note that when X = (xi )∞ i=0 is inﬁnite then ([X] , ∪) is isomorphic to (FIN, ∪) via isomorphism which sends {i} to xi . So the following well-known result gives us a basic pigeon-hole principle for the semigroup FIN. Theorem III.1.1 (Hindman). For every ﬁnite coloring of FIN there is an inﬁnite block sequence X in FIN such that [X] is monochromatic. Proof. Let γFIN be the collection of all ultraﬁlters U on FIN such that for all n ∈ N, {x ∈ FIN : n < x} ∈ U. For U, V ∈ γFIN let U ∪ V = {A ⊆ FIN : {x : {y : x ∪ y ∈ A} ∈ V} ∈ U}. 1 For

x, y ∈ FIN by x < y we denote the fact that max(x) < min(y).

198

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Note that U ∪ V is an ultraﬁlter and that it belongs to γFIN. So we have deﬁned an operation ∪ on γFIN which is easily seen to be associative, i.e., (U ∪ V) ∪ W = U ∪ (V ∪ W). ˇ One can view γFIN as a closed subset of the Stone–Cech compactiﬁcation βFIN of FIN taken with its discrete topology. Thus a basic open set of γFIN is determined by a subset A of FIN as below: A∗ = {U ∈ γFIN : A ∈ U}.

Lemma III.1.2. For each V ∈ γFIN the function U → U ∪ V is a continuous function from γFIN into γFIN. Proof. It suﬃces to show that for every A ⊆ FIN, the set T = {U ∈ γFIN : A ∈ U ∪ V} is open in γFIN. Let B = {x : {y : x ∪ y ∈ A} ∈ V}. Then T = B ∗ , and so T is open in γFIN. Lemma III.1.3 (Ellis). Suppose (S, ∗) is a compact semigroup such that x → x ∗ y is a continuous map for each y ∈ S. Then (S, ∗) has an idempotent. Proof. By compactness, S contains a minimal closed nonempty subsemigroup T . Pick t ∈ T . By continuity of x → x ∗ t the subsemigroup T ∗ t is compact and therefore closed subsemigroup of T. It follows that T ∗ t = T so V = {x ∈ T : x ∗ t = t}. is nonempty. It is also closed being a preimage of the point t under x → x ∗ t. It follows that V = T and therefore t ∗ t = t, as required. The following lemma is the last step in the proof of Theorem III.1.1. Lemma III.1.4. Suppose U is an idempotent of (γFIN, ∪). Then every A ∈ U contains a subsemigroup of (FIN, ∪) generated by an inﬁnite block sequence. Proof. Since A ∈ U = U ∪ U we can ﬁnd x0 ∈ A such that A1 = {y ∈ A : x0 < y and x0 ∪ y ∈ A} ∈ U. Then again since A1 ∈ U ∪ U we can ﬁnd x1 ∈ A1 such that A2 = {y ∈ A1 : x1 < y and x1 ∪ y ∈ A1 } ∈ U, and so on. This procedure gives us a decreasing sequence A = A0 ⊇ A1 ⊇ · · · ⊇ An ⊇ · · · of elements of U and an inﬁnite block sequence X = (xn ) such that xn ∈ An and An+1 = {y ∈ An : xn < y and xn ∪ y ∈ An }.

III.1. Hindman’s Theorem

199

Inductively on k ∈ N one shows that for every sequence n0 < n1 < · · · < nk of elements of N, xn0 ∪ xn1 ∪ · · · ∪ xnk ∈ An0 . This is clearly so for k = 0. To see the recursive step, let y = xn1 ∪ · · · ∪ xnk . Then by the inductive hypothesis, y ∈ An1 . Since An1 ⊆ An0 +1 we get that y ∈ An0 +1 and therefore xn0 ∪ y ∈ An0 as required. Exercise III.1.5. Show that V → U ∪ V is not necessarily a continuous function on γFIN, when V is taken to be an arbitrary member of γFIN. For an integer k ≥ 1, let FIN[k] denote the collection of all block sequences of elements of FIN of length k. Similarly, let FIN[∞] denote the set of all inﬁnite block sequences of elements of FIN. The following is an analogue of the Ramsey theorem in the context of FIN. Theorem III.1.6 (Taylor). For every integer k ≥ 1 and every ﬁnite coloring of [k] FIN[k] there exists an inﬁnite block sequence X = (xi ) such that [X] is monochromatic. Before proving the theorem, we give some deﬁnitions: For two block sequences X = (xi ) and Y = (yj ) of ﬁnite subsets of N we put X ≤ Y iﬀ xi ∈ [Y ] for all i. For a block sequence X = (xi ) ⊆ FIN and s ∈ FIN, let X/s = (xm+j )j≥1 where m = min{i : xi > s}. Thus X/s is the block sequence that enumerates the tail of X determined by s. It is clear that Theorem III.1.6 follows from the next lemma. Lemma III.1.7. For every P ⊆ FIN[k] and every inﬁnite block sequence Y of elements of FIN there is an inﬁnite block sequence X ≤ Y such that [X][k] ⊆ P or [k] [X] ∩ P = ∅. Proof. The proof is by induction on the integer k. The case k = 1 is given by the proof of Hindman’s theorem restricted to the semigroup [Y ] instead of FIN. So we assume k > 1 and that the Lemma is true for k − 1. Recursively on n, we build a block sequence Z = (zn ) in [Y ] and a decreasing sequence (Yn ) of members of FIN[∞] such that Y0 = Y and such that: (1) zn is the ﬁrst term of Yn . k−1 (2) For every n and every (x0 , . . . , xk−2 ) ∈ [(zi )ni=0 ] either (x0 , . . . , xk−2 , x) ∈ P for all x ∈ [Yn+1 ], or (x0 , . . . , xk−2 , x) ∈ / P for all x ∈ [Yn+1 ]. There is no problem in constructing (zn ) and (Yn ) since (2) is provided by the case k = 1 of the lemma.

200

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Having constructed Z = (zn ) we note (see (2)) that for (x0 , . . . , xk−2 , xk−1 ) ∈ [Z] , the sentence (x0 , . . . , xk−2 , xk−1 ) ∈ P [k]

does not depend on xk−1 , or in other words, P ∩ [Z][k] is a cylinder over its [k−1] projection on [Z] . So we are done using the inductive hypothesis.

III.2 Canonical Equivalence Relations on FIN Consider the following ﬁve equivalence relations on FIN: xE0 y xE1 y xE2 y xE3 y

iﬀ iﬀ iﬀ iﬀ

x=y x=x min(x) = min(y) max(x) = max(y)

xE4 y iﬀ min(x) = min(y) and max(x) = max(y). The purpose of this section is to show that these are all essential equivalence relations on FIN provided one is willing to shrink to a subsemigroup of FIN generated by an inﬁnite block sequence. Theorem III.2.1 (Taylor). For every equivalence relation E on FIN there is i < 5 such that E [X] = Ei [X] for some inﬁnite block sequence X = (xi ). Proof. An equation (in 4 variables v0 , v1 , v2 , v3 ) is a formula ϕ(a, b, v0 , v1 , v2 , v3 ) of the form ϕ(a, b, v0 , v1 , v2 , v3 ) ≡ a ∪ vi0 ∪ · · · ∪ vi3 E b ∪ vj0 ∪ · · · ∪ vj3 where i0 , j0 , ..., i3 , j3 ∈ {0, 1, 2, 3} and a, b ∈ FIN ∪ {∅}. Given a block sequence Z and an equation ϕ ≡ ϕ(a, b, v0 , ..., v3 ) we say that ϕ is true in Z iﬀ ϕ(a0 , . . . , a3 ) holds for every (a0 , . . . , a3 ) ∈ [Z/(a ∪ b)]4 . We say that ϕ is false in Z iﬀ ¬ϕ(a0 , . . . , a3 ) holds for every (a0 , . . . , a3 ) ∈ [Z/(a ∪ b)]4 . We say that ϕ is decided in Z if either ϕ is true or ϕ is false in Z. By Theorem III.1.6, and using a non diﬃcult diagonal procedure, we can ﬁnd a block sequence Z = (zn )n such that all equations ϕ(a, b, v0 , ..., v3 ) with a, b ∈ [Z] ∪ {∅} are decided in Z. Our aim is to show that there is some i < 5 such that E [Z] = Ei [Z]. Case 1. v0 E v1 is true in Z. Then E [Z] = E1 [Z]: Let s, t ∈ [Z] and pick u ∈ [Z] such that u > s, t. Then s E u and t E u, and hence s E t. Case 2. v0 E v1 is false in Z, v0 ∪ v1 E v0 is true in Z and v0 ∪ v1 E v1 is false in Z. Let us check that E is equal to E2 on [Z]. Fix s, t ∈ [Z]. Suppose ﬁrst that s E2 t, and write s = zn ∪ s and t = zn ∪ t , with s , t ∈ [Z] and zn < s , t . Using that v0 ∪ v1 E v0 is true in Z, we obtain that s, t E zn , and hence s E t.

III.3. Fronts and Barriers on FIN[<∞]

201

Assume that s E t and suppose now that s E2 t. Without loss of generality, we may assume that min(s) < min(t). Let n be such that s = zn ∪ s , s ∈ [Z] and zn < s . Then s E zn , zn < t, and hence zn E t. Since zn < t, we obtain that v0 E v1 is true in Z, a contradiction. Case 3. v0 E v1 and v0 ∪ v1 E v0 are false in Z, and v0 ∪ v1 E v1 is true in Z. Similar proof than in Case 2 shows that E is equal to E3 on [Z]. For the last two cases we use the following fact Claim. If v0 ∪ v1 E v0 is false in Z, then E ⊆ E4 when restricted to [Z]: Suppose that max s = max t and s E t. We may assume that max s < max t. Let n be such that t = t ∪ zn with t < zn and t ∈ [Z]. Since s E t, the equation s E t ∪ v0 is true in Z and hence the equation t ∪ v0 ∪ v1 E t ∪ v0 also is true in Z. This implies that v0 ∪ v1 E v0 is true in Z. Case 4. v0 E v1 , v0 ∪v1 E v0 and v0 ∪v1 E v1 are false in Z, and v0 ∪v1 ∪v2 E v0 ∪v2 is true in Z. Then E is equal to E4 on [Z]: It is not diﬃcult to prove that E4 ⊆ E on [Z]. For the converse use the previous claim. Suppose now that max s = max t but min s = min t, say min s < min t. Suppose that s E t and we work for a contradiction. Write s = zn0 ∪ s ∪ zn , t = zm0 ∪ t ∪ an with zn0 < s < zn , zm0 < t < an , n0 < m0 , and all in [Z]. Using that the equation v0 ∪v1 ∪v2 E v0 ∪v2 is true in Z, we may assume that s = t = ∅. Since n0 < m0 ≤ n, one of the equations v0 ∪ v2 E v1 ∪ v2 or v0 ∪ v1 E v1 must be true in Z. The second case is impossible by hypothesis. In the ﬁrst case we obtain that v0 ∪ v3 E v1 ∪ v2 ∪ v3 and v0 ∪ v3 E v2 ∪ v3 are true in Z and hence so is v0 E v0 ∪ v1 , a contradiction. Case 5. v0 E v1 , v0 ∪ v1 E v0 , v0 ∪ v1 E v1 , and v0 ∪ v1 ∪ v2 E v0 ∪ v2 are all false in Z. Then E [Z] = E0 [Z]. For suppose that s E t and that s = t. Since v0 ∪ v1 E v0 is false in Z, we obtain that max s = max t (previous claim). Write s = i∈F zi , t = i∈G zi with max F = max G. Let k = max(F G). Without loss of generality we assume that k ∈ F \ G. This implies that s = s ∪ zk ∪ s and t = t ∪ s with s , t < zk < s , s = ∅ and all in [Z]. Therefore the equation s ∪v0 ∪v1 E t ∪v1 is true in Z, which implies that s ∪v0 ∪v1 ∪v2 E t ∪v2 and s ∪ v0 ∪ v2 E t ∪ v2 are both true in Z. So, the equation v0 ∪ v1 ∪ v2 E v0 ∪ v2 is true in Z, a contradiction.

III.3 Fronts and Barriers on FIN[<∞] Let FIN[<∞] denote the collection of all ﬁnite block sequences of elements of FIN and let FIN[∞] denote the collection of all inﬁnite block sequences of elements of FIN. Recall the ordering ≤ on FIN[<∞] ∪ FIN[∞] : X = (xi ) ≤ Y = (yi ) iﬀ xi ∈ [Y ] for all i.

202

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

For a family F of ﬁnite block sequences of FIN and an inﬁnite block sequence Y of members of FIN in analogy with the classical case of N[<∞] and N[∞] it is natural to consider the following form of restriction F [Y ] = {s ∈ F : s ≤ Y } based on the ordering ≤ of being a block subsequence rather than a subset. As in the classical case, there is another natural ordering on FIN[<∞] ∪ FIN[∞] : X Y iﬀ X = Y n for some n, or in other words X Y iﬀ the block sequence X is an initial segment of the block sequence Y . By we denote the strict version of this ordering. Using the orderings ≤ and one can go on and deﬁne analogues of Nash–Williams’ notions of fronts and barriers in the present context. In what follows, we shall reserve the variable s, t, u, v, . . . for members of FIN[<∞] , i.e., for ﬁnite block sequences and variables X, Y, Z, . . . for members of FIN[∞] , i.e., for inﬁnite block sequences of elements of FIN. Deﬁnition III.3.1. (i) A family F ⊆ FIN[<∞] is thin if s t for distinct s, t ∈ F. (ii) A family F ⊆ FIN[<∞] is Sperner if s t for all s = t in F. (iii) A family F ⊆ FIN[<∞] is Ramsey if for every ﬁnite partition F = F0 ∪ · · · ∪ Fn there is X ∈ FIN[∞] such that at most one of the restrictions F0 [X] , . . . , Fn [X] is nonempty. Example III.3.2. By Theorem III.1.6, for each positive integer k, the family FIN[k] of all block sequences of length k is thin, Sperner, and Ramsey. Deﬁnition III.3.3. Given Y ∈ FIN[∞] a family F of ﬁnite block subsequences of Y is a front on Y if F is thin and has the property that every inﬁnite block subsequence of Y has an initial segment in F. Deﬁnition III.3.4. For a given Y ∈ FIN[∞] , a family F of ﬁnite block sequences of Y is a barrier on Y if F is Sperner and has the property that every inﬁnite block subsequence of Y has an initial segment in F. In order to deﬁne something analogous to the lexicographical ordering of N[<∞] that makes fronts and barriers well-ordered we use the following well ordering on FIN: x
III.3. Fronts and Barriers on FIN[<∞]

203

Using

We are now ready to state and prove the analogue of Galvin’s lemma in this context that will play a crucial role in the further development of the theory of fronts and barriers of FIN[<∞] . Lemma III.3.7 (Milliken). For every B ⊆ FIN[<∞] and every Y ∈ FIN[∞] there exists X ≤ Y such that either B ∩ [X] = ∅ or B ∩ [X] contains a barrier on X. Proof. The proof of this Lemma is quite analogous to the proof of Lemma II.3.8 above. The crucial to the proof is the following notion of combinatorial forcing. First of all we make the analogous deﬁnition of a basic set: [s, Y ] = {X ∈ FIN[∞] : s X ≤ s (Y /s)}.2 Deﬁnition III.3.8. Fix a family B ⊆ FIN[<∞] . We say that Y accepts s if every X ∈ [s, Y ] has an initial segment in B. We say that Y rejects s if there is no X ≤ Y accepting s. We say that Y decides s if Y either accepts or rejects s. Deﬁnition III.3.9. A subset S of FIN is small if it contains no set of the form [X] for some inﬁnite block sequence X of members of FIN. Note the following immediate properties of the forcing relation: (i) If Y accepts (rejects) s then every X ≤ Y accepts (rejects) s. (ii) For every Y and s there is X ≤ Y which decides s. (iii) If Y accepts s then Y accepts s x for every x ∈ [Y /s]. (iv) If Y decides of all its ﬁnite block subsequences but it rejects s then the set {x ∈ [Y ] : Y does not reject s x} is small. Starting with Y ∈ FIN[∞] and using (i) and (ii) we build a decreasing sequence (Yn ) of inﬁnite block subsequences of Y and an inﬁnite block sequence Z = (zn ) such that for all n, zn is the ﬁrst term of Yn and Yn decides every s ≤ (zi )i
204

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

on. At stage n, assuming that Z rejects all s ≤ x0 , . . . , xn we ﬁnd Zn+1 ≤ Zn /xn such that Z rejects s x for all s ≤ x0 , . . . , xn and x ∈ [Zn+1 ]. This procedure gives us an inﬁnite block sequence X = (xn ) which rejects all of its ﬁnite block subsequences. Clearly, this X satisﬁes the other alternative of Lemma III.3.7, i.e., B ∩ [X] = ∅.

This ﬁnishes the proof of Lemma III.3.7. Corollary III.3.10. Suppose F is a front on some Y ∈ FIN Z ≤ Y in FIN[∞] such that F ∩ [Z] is a barrier on Z.

[∞]

. Then there exists

We are now ready to describe the notion of uniformity for fronts and barriers on FIN[<∞] that is likely to play a role in applications analogous to that of the classical case. Deﬁnition III.3.11. Let α be a countable ordinal and let Y ∈ FIN[∞] . We say that a family B ⊆ FIN[<∞] is α-uniform on Y provided that: (a) α = 0 implies B = {∅}. (b) α = β + 1 implies ∅ ∈ / B and B(x) 3 is β-uniform on Y /x for all x ∈ [Y ]. (c) α > 0 limit implies that for all x ∈ [Y ] there is (unique) ordinal αx < α such that B(x) is αx -uniform on Y /x and such that the set {x ∈ [Y ] : αx ≤ β} is small for all β < α. Note the following consequences of this deﬁnition proved by a simple induction on α. Lemma III.3.12. If B is an uniform family on Y , then B is a front on Y .

Lemma III.3.13. If B is α-uniform on Y , then B is α-uniform on any X ≤ Y .

Lemma III.3.14. If B is an uniform family on Y (i.e., α-uniform on Y for some α), then B is a maximal thin family of ﬁnite block subsequences of Y . Example III.3.15. For every positive integer k the family FIN[k] of all block sequences of length k is k-uniform on the maximal inﬁnite block sequence Y = ({n})∞ n=0 of members of FIN. Exercise III.3.16. Construct an ω-uniform and an (ω + 1)-uniform family on FIN. One of the main advantages of uniform families over arbitrary fronts is the possibility of proving results about them by typically easy induction arguments on their uniformity indexes α. The following lemma shows that our notion of uniformity is abundant among fronts of FIN[<∞] so the induction arguments applicable in the uniform case will typically cover most general cases. Lemma III.3.17. For every family B ⊆ FIN[<∞] and for every Y ∈ FIN[∞] there is X ≤ Y such that either B ∩ [X] = ∅, or else B ∩ [X] contains an uniform family on X. 3B (x)

= {s ∈ FIN[<∞] : x s ∈ B}.

III.4. Milliken’s Theorem

205

Proof. By Lemma III.3.7 we may assume to have a Z ≤ Y such that B ∩ [Z] contains a barrier on Z. Shrinking B we assume that B is actually a barrier on Z. Now the second alternative of the lemma is achieved by induction on the lexicographical rank of B since clearly, rank(B(x) ) < rank(B) for all x ∈ [Z].

Theorem III.3.18. The following are equivalent for every family F ⊆ FIN[<∞] : (a) F is Ramsey. (b) There is Y ∈ FIN[∞] such that F ∩ [Y ] is Sperner. (c) There is Y ∈ FIN[∞] such that F ∩ [Y ] is uniform on Y . (d) There is Y ∈ FIN[∞] such that F ∩ [Y ] is thin. (e) There is Y ∈ FIN[∞] such that for every X ≤ Y the restriction F ∩ [X] does not contain two disjoint uniform families on X. Proof. (a) → (b) Apply the Ramsey property of F to the partition F = F0 ∪ F1 where F0 is the set of all ≤-minimal members of F. (b) → (c) This follows from Lemma III.3.17. (c) → (d) This follows from Lemma III.3.12. (d) → (e) The union of two disjoint fronts on the same X ∈ FIN[∞] cannot be thin. (e) → (a) Consider a ﬁnite partition F = F0 ∪ · · · ∪ Fk . By Lemma III.3.17 there is X ≤ Y in FIN[∞] such that for all i ≤ k, the restriction Fi ∩ [X] is either empty or it contains a uniform family. Since by our assumption F ∩ [X] contains no two disjoint uniform families at most one of the restrictions Fi ∩ [X] (i ≤ k) is non-empty. Remark III.3.19. Note that once we have the analogue of Galvin’s Lemma for the space FIN[∞] (i.e. Lemma III.3.7) the theories of fronts and barriers on N[<∞] and FIN[<∞] become strikingly similar. The reader is urged to complete the theory of fronts and barriers on FIN[<∞] and examine the further analogues from the theory of N[<∞] . For example, it is quite natural to try to ﬁnd the descriptions of all canonical equivalence relations on uniform fronts FIN[k] of ﬁnite rank, and more generally to describe all equivalence relations on all uniform fronts of FIN[<∞] .

III.4 Milliken’s Theorem In this section we prove an inﬁnite-dimensional version of Hindman’s theorem, a theorem about coloring of the set FIN[∞] of all inﬁnite block sequences of ﬁnite nonempty subsets of N.

206

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Deﬁnition III.4.1. A set X ⊆ FIN[∞] is Ramsey if for every [s, Y ] there is X ∈ [s, Y ] such that [s, X] ⊆ X or [s, X] ∩ X = ∅. We say that X is Ramsey-null if the second alternative always holds. The sets of the form [s, Y ] = {X ∈ FIN[∞] : s X ≤ s (Y /s)} where s ∈ FIN[<∞] and Y ∈ FIN[∞] form a base for a topology on FIN[∞] which we call Milliken’s topology on FIN[∞] which is obviously richer than the metrizable topology induced from the Tychonoﬀ power FINN with FIN taken with its discrete topology. It is this topology that we usually refer to when we think of FIN[∞] as a topological space. The weaker topology of FIN[∞] will be referred to as the metric topology of FIN[∞] . Lemma III.4.2. Every open subset of FIN[∞] is Ramsey. Proof. Let X be a given open subset of FIN[∞] and let [s, Y ] be a given basic open set. We give argument only in case s = ∅ since the general case is a simple relativization of this one. We ﬁrst perform an already standard recursive diagonalization procedure to obtain Z ≤ Y such that if for some ﬁnite block sequence t ≤ Z there is X ∈ [t, Z] such that [t, X] ⊆ X, then already [t, Z] ⊆ X. Let F = {t ∈ FIN[<∞] : t ≤ Z and [t, Z] ⊆ X}. Applying Milliken’s lemma (Lemma III.3.7) we get an inﬁnite block subsequence X of Z such that either: (a) F ∩ [X] = ∅, or (b) Every inﬁnite block subsequence of X contains an initial segment in F. Note that if (b) holds, then [∅, X] ⊆ X. If on the other hand (a) holds, since X is open, we must have that [∅, X] ∩ X = ∅. A simple diagonalization argument gives us the following fact which shows that the ideal of Ramsey-null subsets of FIN[∞] is a σ-ideal. Lemma III.4.3. The union of countably many Ramsey-null subsets of FIN[∞] is Ramsey-null. Lemma III.4.4. Every meager subset of FIN[∞] is Ramsey-null. Proof. By Lemma III.4.3 it suﬃces to show that every nowhere-dense subset of FIN[∞] is Ramsey-null. Let X be a given nowhere-dense subset of FIN[∞] and let [s, Y ] be a given basic-open set of FIN[∞] . Since the closure of a nowhere-dense

III.4. Milliken’s Theorem

207

set is nowhere-dense we may assume X is closed. By Lemma III.4.2, X is Ramsey, so there is X ∈ [s, Y ] such that [s, X] ⊆ X or [s, X] ∩ X = ∅. We ﬁnish by noting that since X is nowhere-dense the ﬁrst alternative is impossible. Corollary III.4.5. A subset of FIN[∞] is meager iﬀ it is Ramsey-null.

Theorem III.4.6 (Milliken). The following are equivalent for an arbitrary subset X of FIN[∞] : (a) X has the Ramsey property. (b) X has the Baire property. Moreover, the ideals of meager and Ramsey-null subsets of FIN[∞] coincide. Proof. Let X = O M be a given property of Baire subset of FIN[∞] written as symmetric diﬀerence of an open set O and a meager set M. Let [s, Y ] be a given basic-open set. By Corollary III.4.5 there is Z ∈ [s, Y ] such that [s, Z] ∩ M = ∅. It follows that X ∩ [s, Z] = O ∩ [s, Z] . By Lemma III.4.2 there is X ∈ [s, Z] such that [s, X] ⊆ O or [s, X] ∩ O = ∅. It follows that [s, X] ⊆ X or [s, X] ∩ X = ∅ as required. Corollary III.4.7. Every metrically analytic subset of FIN[∞] is Ramsey.

Corollary III.4.8 (Silver). Every metrically analytic subset of N[∞] is Ramsey.

Proof. Consider the projection π : FIN

[∞]

→ N[∞] deﬁned by

∞ π((xi )∞ i=0 ) = (min(xi ))i=0 .

Corollary III.4.9 (Parametrized Perfect-Set Theorem). For every ﬁnite Borel coloring of N[∞] × R there is M ∈ N[∞] and a perfect P ⊆ R such that M [∞] × P is monochromatic. Proof. Let X be a given Borel subset of N[∞] × P(N). Let [∞] X∗ = {X = (xi )∞ : (min(x2i+1 ))∞ i=0 ∈ FIN i=0 ,

∞

x2i ) ∈ X}.

i=0

Then X∗ is a (metrically) Borel subset of FIN[∞] , so by Corollary III.4.7 there is Y = (yi ) ∈ FIN[∞] such that [Y ][∞] ⊆ X∗ or [Y ][∞] ∩ X∗ = ∅. Choose {xσ : σ ∈ 2<∞ } ⊆ {y2i : i ∈ N} such that: (1) x0 = y0 and xσ = xτ whenever σ = τ , (2) xσ < yτ whenever |σ| < |τ |.

208

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Let M = (min(y2ϕ(i)+1 ))∞ i=0 where ϕ : N → N is a strictly increasing map such that xσ < yϕ(|σ|) for all σ ∈ 2<∞ , and let P ={

∞

xai : a ∈ 2∞ }

i=0

Then P is a perfect subset of P(N) with the topology obtained by identifying P(N) with 2N . Note that for every N ∈ M [∞] and q ∈ P there exists X = (xi ) ≤ Y such that ∞ N = (min(x2i+1 ))∞ and q = x2i . i=0 i=0

It follows that M × P ⊆ X or M [Y ][∞] ⊆ X∗ or [Y ][∞] ∩ X∗ = ∅. [∞]

[∞]

× P ∩ X = ∅ depending on whether

Corollary III.4.10 (Stern). For every analytic set X of inﬁnite chains of the complete binary tree 2<∞ there is a perfect subtree T ⊆ 2<∞ (not necessarily downwards closed) such that either every inﬁnite chain of T is in X or every inﬁnite chain of T is not in X. Proof. Let C∞ be the collection of all inﬁnite chains of 2<∞ with the natural topology of pointwise convergence. Consider the map π : N[∞] × 2∞ → C∞ deﬁned by π(M, a) = {a m : m ∈ M }. Clearly, π is a continuous onto map, so for a given analytic set X ⊆ C∞ the set X∗ = {(M, a) ∈ N[∞] × 2∞ : π(M, a) ∈ X} is an analytic subset of N[∞] × 2∞ . By Corollary III.4.9 there is M ∈ N[∞] and a perfect set P ⊆ 2∞ such that M [∞] × P ⊆ X∗ or (M [∞] × P ) ∩ X∗ = ∅. Let T = {a m : m ∈ M, a ∈ P }. Then T is a perfect subtree of 2<∞ such that C∞ (T ) ⊆ π M [∞] × P . It follows that C∞ (T ) ⊆ X or C∞ (T ) ∩ X = ∅.

Corollary III.4.11 (Parametrized 1 -theorem). Suppose xσ (σ ∈ 2 ) is a normalized sequence of elements of some Banach space X indexed by the complete binary tree 2<∞ . Then there exist M ∈ N[∞] and perfect P ⊆ 2∞ such that either (a) For all a ∈ P the sequence (xan )n∈M is equivalent to the unit vector basis of 1 , or (b) For all a ∈ P the sequence (xan )n∈M is weakly Cauchy in X. <∞

III.5. An Approximate Ramsey Theorem

209

Proof. Color N[∞] ×P according whether for a given pair (M, a) the corresponding sequence (xan )n∈M is weakly Cauchy, equivalent to the unit vector basis of 1 , or neither of the two. Apply Corollary III.4.9 and Rosenthal’s 1 -theorem.

III.5 An Approximate Ramsey Theorem An inﬁnite dimensional Banach space (X, · ) is distortable if there is λ > 1 and an equivalent norm | · | on X such that for every inﬁnite-dimensional subspace Y of X, there exists x, y ∈ Y such that x = y = 1 and |x| > λ · |y|. When this happens we say that (X, · ) is λ-distortable. We say that (X, · ) is arbitrarily distortable if it is λ-distortable for every λ > 1 . The ﬁrst known example of a distortable Banach space is the Tsirelson space T discussed above in Section 2. However it is still unknown if T is arbitrarily distortable. The following is the ﬁrst known example of an arbitrarily distortable Banach space. Example III.5.1 (Schlumprecht). Let XS be the completion of c00 under the norm · deﬁned by the following implicit formula 1 Ei x} log2 (n + 1) i=1 n

x = max{x∞ , sup sup n

where E1 < E2 < · · · < En is a sequence of intervals of non-negative integers and where for x ∈ c00 and an interval E by Ex we denote the restriction of x to E, i.e., Ex(k) = 0 if k ∈ / E and Ex(k) = x(k) if k ∈ E. Then XS is arbitrarily distortable. The distortion problem for classical spaces c0 , p (1 ≤ p < ∞) is solved by the following two results. Theorem III.5.2 (James). The spaces c0 and 1 are not distortable.

The fact that Schlumprecht space XS is arbitrarily distortable was one of the key ingredients in the proof of the following result which solved a long-standing distortion problem. Theorem III.5.3 (Odell–Schlumprecht). The spaces p (1 < p < ∞) are all arbitrarily distortable. Hence, in particular, the Hilbert space 2 is arbitrarily distortable. The Odell– Schlumprecht method of proof is to transfer distortion from one space to another via uniform homeomorphism between spheres of the spaces. For example the Mazur map Mp : S(1 ) → S(p ) deﬁned by 1/p ∞ )i=0 Mp ((xi )∞ i=0 ) = (sgn(xi ) · |xi |

210

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

is an uniform homeomorphism which preserves block basic sequences as well as asymptotic sets. Recall that a subset A of some Banach space X is asymptotic if it intersects an arbitrary closed inﬁnite dimensional subspace of X. For 1 0. The Mazur map Mp preserves asymptotic sets so p is distortable iﬀ S(1 ) contains a pair of asymptotic sets on positive distance from each other. Thus, the distortion problem for the Hilbert space was solved by transferring separated asymptotic sets from Schlumprecht space into S(1 ) and then to S(2 ) . The distortion problem is related to the problem of oscillation stability of functions f : S(X) → R. Deﬁnition III.5.4. Given a Banach space X, a function f : S(X) → R is oscillation stable on X if for all inﬁnite dimensional closed subspaces Y of X and ε > 0 there exists a closed inﬁnite dimensional subspace Z of Y such that sup{|f (x) − f (y)| : x, y ∈ S(Z)} < ε. Lemma III.5.5. The following are equivalent for every Banach space X: (1) Every Lipschitz function f : S(X) → R is oscillation stable. (2) Every uniformly continuous function f : S(X) → R is oscillation stable. Proof. Suppose f : S(X) → R is an uniformly continuous function which is not oscillation stable. Then there exists ε < δ and an inﬁnite dimensional closed subspace Y of X such that A = {x ∈ S(Y ) : f (x) < ε} and B = {x ∈ S(Y ) : f (x) > δ} are two asymptotic sets in Y such that dist(A, B) > 0. Consider g : S(X) → R deﬁned by g(x) = dist(x, A). Then g is a Lipschitz function which is not oscillation stable on X. A combination of results of Gowers, Milman and Odell and Schlumprecht gives us the following remarkable synthesis. Theorem III.5.6. The following are equivalent for every inﬁnite dimensional Banach space X: (1) Every Lipschitz function f : S(X) → R is oscillation stable. (2) Every closed inﬁnite dimensional subspace Y contains an isomorph of c0 . We shall prove here the implication (2) → (1) of this theorem, or more precisely the following result that is inherently Ramsey theoretic in nature. Theorem III.5.7 (Gowers). Every Lipschitz function f : S(c0 ) → R is oscillation stable. Let K be the Lipschitz constant of the given Lipschitz map f : S(c0 ) → R and let ε > 0 be a given number. Choose a suﬃciently large positive integer k such that if (1 + δk )1−k = δk then δk · K ≤ ε/3. Let ∆±k be the collection of all ﬁnitely supported maps p : N → {0, ±(1 + δk )1−k , ±(1 + δk )2−k , . . . , ±(1 + δk )−1 , ±1}

III.5. An Approximate Ramsey Theorem

211

which achieves at least once one of the values ±1. Then ∆±k forms an (ε/3)-net on the unit sphere S(c0 ) of c0 , and Theorem III.5.7 gets transferred to the following approximate Ramsey theoretic result. Theorem III.5.8. For every ﬁnite coloring of ∆±k there exists an inﬁnite block sequence X = (xi ) ⊆ ∆±k and a color P such that [X] ⊆ (P )δ . Here [X] denotes the intersection of the subspace of c0 generated by (xi ) with the set ∆±k and (P )δ = {x ∈ ∆±k : ∃p ∈ P p − x∞ ≤ δ}. One can make Theorem III.5.8 even more close to Hindman’s Theorem by identifying (1 + δk )−k with for 0 < ≤ k which amounts to considering the set FIN±k of all ﬁnitely supported maps p : N → {0, ±1, ±2, . . . , ±k − 1, ±k} that obtain one of the values ±k at least once. To get the analogous of the subspace [X] generated by a block sequence besides the partial semigroup operation + we need to consider the operation of multiplying by −1 as well as multiplying by the scalar (1 + δk )−1 . The operation on FIN±k corresponding to the scalar multiplication by (1 + δk )−1 is the operation T : FIN±k → FIN±(k−1) deﬁned by

⎧ ⎨ p(n) − 1 if p(n) + 1 if T (p)(n) = ⎩ 0 if

p(n) > 0 p(n) < 0 p(n) = 0.

Then for an (inﬁnite) block sequence X = (xi ) of elements of FIN±k we let4 [X] = { ±T (i0 ) (xn0 ) ± · · · ± T (i ) (xn ) : n0 < · · · < n , i0 , . . . , i ∈ {0, . . . , k − 1}, and ij = 0 for some j ≤ }. Then Theorem III.5.7 has yet another Ramsey-theoretic reformulations as follows. Theorem III.5.9. For every positive integer k and every ﬁnite coloring of FIN±k there is an inﬁnite block sequence X = (xi ) ⊆ FIN±k and a color P such that [X] ⊆ (P )1 .5 Remark III.5.10. Note that the approximate constant 1 in Theorem III.5.9 can be replaced with any other ﬁxed positive integer m (i.e, in the conclusion [X] ⊆ (P )1 can be replaced by [X] ⊆ (P )2 or [X] ⊆ (P )3 , etc.) without changing the strength of the result. 4 T (i) 5 (P )

1

denotes the ith iterate of T with T (0) = identity, T (1) = T , T (i+1) = T (T (i) ), etc. = {x ∈ FIN±k : ∃p ∈ P x − p∞ ≤ 1}.

212

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Proof of Theorem III.5.9. The proof follows closely the proof of Hindman’s theorem applied to the (partial) semigroup FIN[0,±k] =

k

FIN± .

=0

ˇ comAs before, we start with the closed subspace γFIN[0,±k] of the Stone–Cech pactiﬁcation βFIN[±k] consisting of all ultraﬁlters U on FIN[0,±k] such that {p ∈ FIN[0,±k] : p(i) = 0 for all i ≤ n} ∈ U for all n. As before we consider γFIN[0,±k] as a semigroup with the operation + deﬁned by A ∈ U + V iﬀ {p : {q : p < q and p + q ∈ A} ∈ V} ∈ U.6 Then U → U + V is a continuous map from γFIN[0,±k] into γFIN[0,±k] for all V ∈ γFIN[0,±k] . So, as before, every nonempty closed subsemigroup of γFIN[0,±k] contains an idempotent. We shall need the following general fact about idempotents and homomorphisms in compact semigroups. Lemma III.5.11. Let (S, +) be a compact semigroup, let I be a closed subsemigroup of S, and let k be a non negative integer. Suppose H : S → S is a continuous homomorphism that stabilizes after k steps 7 . Then there is an e ∈ I such that e + H () (e) = H () (e) + e = e, for all = 0, 1, . . . , k. Proof. For 0 ≤ ≤ k let

I = H (k−) [I] .

Thus Ik = I, Ik−1 = H [I], Ik−2 = H (2) [I], etc. Then I0 , I1 , . . . , Ik are all compact subsemigroups of S0 . By Ellis’ lemma we can pick an idempotent e0 in I0 . Suppose that for some 0 < < k, ei ∈ Ii (i < ) have been selected such that: (1) H (j−i) (ej ) = ei whenever i ≤ j < , (2) ei + ej = ej + ei = ej whenever i ≤ j < . Let R = {x ∈ I : H(x) = e−1 }. Then R is a nonempty closed subsemigroup of S. Note also that R + e−1 is also a nonempty closed subsemigroup of S so by Ellis’ lemma we can ﬁx an idempotent b = a + e−1 ∈ R + e−1 . Let e = e−1 + b + e−1 . 6p < q

means that supp(p) < supp(q) and p + q is deﬁned by taking pointwise addition. H (k+) = H (k) for all , where for a given integer , H () is the th iterate of H with taken as the identity map and H (1) = H.

7 i.e.,

H (0)

III.5. An Approximate Ramsey Theorem

213

Then (1) and (2) continue to be valid for i ≤ j ≤ . Proceeding this way we get ei ∈ Ii (0 ≤ i ≤ k) satisfying the equations of (1) and (2) for every choice of 0 ≤ i ≤ j ≤ k. So ek satisﬁes the conclusion of the Lemma. We apply the lemma to S = γFIN[0,±k] , I = FIN±k , and H : γFIN[0,±k] → γFIN[0,±k] deﬁned by H(U) = −T (U). Note that H is indeed a continuous homomorphism being continuous extension of the partial homomorphism8 H : FIN[0,±k] → FIN[0,±k] deﬁned by H(p) = −T (p) = T (−p). This gives us an ultraﬁlter U ∈ γFIN±k such that (3) U + (−T )() (U) = (−T )() (U) + U = U for all = 0, 1, . . . , k. From now on we ﬁx U ∈ γFIN±k satisfying (3). For an (inﬁnite) block sequence X = (xn ) of members of FIN±k , and a homomorphism H : FIN[0,±k] → FIN[0,±k] , [X]H = {H (0 ) (x0 ) + · · · + H (t ) (xt ) : n0 < · · · < nt , 0 , . . . , t ≤ k, i = 0 for some i ≤ t}. Note that the proof of Lemma III.1.4 gives us the following. Lemma III.5.12. For every P ∈ U there is an inﬁnite block sequence X = (xn ) ⊆ FIN±k such that [X](−T ) ⊆ P . We shall also need the following property of the ultraﬁlter U satisfying (3). Lemma III.5.13. −(P )1 ∈ U for all P ∈ U. Proof. This follows from the following sequence of equivalent statements where we use the fact that U = (−T )(U) + U = U + (−T (U)): −(P )1 ∈ U + (−T )(U) = U (Up )(Uq ) p − T (q) ∈ −(P )1 (Up )(Uq ) − p + T (q) ∈ (P )1 (Up )(Uq ) − T (p) + q ∈ P P ∈ −T (U) + U = U.

8 Partial homomorphism means that the equation H(p + q) = H(p) + H(q) holds for disjointly supported elements p and q of γFIN[0,±k] .

214

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences

Back to the proof of Theorem III.5.9. We shall actually show that there is an inﬁnite block sequence X = (xn ) ⊆ FIN±k and a color P such that [X] ⊆ (P )2 . As noted earlier (see Remark III.5.10) this will complete the proof. The paper of Kanellopoulos [41] from where we have taken some of the above arguments contains a direct combinatorial reductions of the conclusion [X] ⊆ (P )1 from the conclusion [X] ⊆ (P )2 . Pick a color P such that P ∈ U. By Lemma III.5.13, −(P )1 ∈ U. Let Q = −(P )1 ∩ (P )1 . Then Q ∈ U and Q = −Q. By Lemma III.5.12 there is an inﬁnite block sequence X = (xn ) such that [X](−T ) ⊆ Q. So the desired conclusion [X] ⊆ (P )2 follows from the following general fact. Lemma III.5.14. Suppose X = (xn ) is an inﬁnite block sequence of members of FIN±k such that [X](−T ) ⊆ Q for some Q ⊆ FIN±k such that Q = −Q. Then [X] ⊆ (Q)1 . Proof. Consider a p ∈ [X](−T ) . Then there exist n0 < · · · < nt , 0 , . . . , t ≤ k with i = 0 for at least one i ≤ t and signs ε0 , . . . , εt ∈ {±} such that p = ε0 T (0 ) (xn0 ) + · · · + εt T (t ) (xnt ). We need to ﬁnd a q ∈ Q such that p − q∞ ≤ 1. Case 1: There is i ≤ t such that εi = + and i = 0. For i ≤ t, let yi = εi T (i ) (xni ). We shall ﬁnd yi ∈ [X](−T ) (i ≤ t) such that yi = (−T )(i ) (xni ) for some i ∈ {i + 1, i }. Then yi − yi ∞ ≤ 1 for all i ≤ t and therefore we have that

q = (−T )(0 ) (xn0 ) + · · · + (−T )(t ) (xnt ) is a member of [X](−T ) such that p − q∞ ≤ 1, as required. (i) If εi = − and i is odd, or if εi = + and i is even, we put yi = yi . Note that in both cases: yi = yi = (−T )(i ) (xni ). (ii) If εi = − and i is even, or if εi = + and i is odd, we put yi = T (yi ). Note that in both cases yi = (−T )(i +1) (xni ). Note that these choices and the assumption of Case 1 ensures that the sum q = y0 + · · · + yt belongs to [X](−T ) . Case 2. For every i ≤ t, if i = 0 then εi = −. Let p¯ = −p. Then the representation p¯ = (−ε0 )T (0 ) (xn0 ) + · · · + (−εt )T (t ) (xnt ) p − q¯∞ ≤ falls into Case 1, so working as in Case 1 we ﬁnd q¯ ∈ [X](−T ) such that ¯ q belongs to Q. Finally note that 1. Since [X](−T ) ⊆ Q = −Q we have that q = −¯ p − q¯∞ ≤ 1. This completes the proof of Lemma III.5.14. p − q∞ = ¯

III.5. An Approximate Ramsey Theorem

215

For a positive integer k, let FINk = {p : N → {0, 1, . . . , k} : supp(p) is ﬁnite and k ∈ rang(p)}. Thus (FINk , +) is a partial subsemigroup of FIN±k so one can consider the partial homomorphism T : FINk → FINk−1 deﬁned as before: T (p)(n) = p(n)−1 if p(n) > 0; T (p)(n) = 0 if p(n) = 0. Given an (inﬁnite) block sequence X = (xn ) of members of FINk the partial subsemigroup of FINk generated by X is deﬁned as before: [X] = {T (0 ) (xn0 ) + · · · + T (t ) (xnt ) : n0 < · · · < nt , 0 , . . . , t ≤ k and i = 0 for some i ≤ t}. Then we have the following exact Ramsey-theoretic result. Theorem III.5.15 (Gowers). For every ﬁnite coloring of FINk there is an inﬁnite block sequence X = (xn ) ⊆ FINk such that [X] is monochromatic. k Proof. Let FIN[0,k] = =0 FIN and the corresponding compact semigroup γFIN[0,k] and its continuous homomorphism T : γFIN[0,k] → γFIN[0,k] extending T . Applying Lemma III.5.11 to γFIN[0,k] , γFINk and T we get an ultraﬁlter U ∈ γFINk such that U + T () (U) = T () (U) + U = U for all = 0, 1, . . . , k. Choose P ∈ U that is monochromatic relative to the given coloring of FINk . The proof of Lemma III.1.4 (see also Lemma III.5.12) gives us an inﬁnite block sequence X = (xn ) of members of FINk such that [X] ⊆ P , as required. Note that FIN1 can be identiﬁed with the space FIN of all ﬁnite nonempty subsets of N, so Hindman’s theorem (Theorem III.1.1) is an immediate consequence of Theorem III.5.15. [] For = 1, 2, 3, . . . , ∞ let FINk be the collection of all block sequences of [1] elements of FINk of length . Thus FINk is naturally identiﬁed with FINk . The proof of Theorem III.1.6, replacing the use there of Hindman’s Theorem by the use of Gowers’ Theorem, lead us to the following more general result. Lemma III.5.16. For every pair of positive integers k and and for every ﬁnite [] coloring of FINk there is an inﬁnite block sequence X = (xn ) such that the set [X][] of all block subsequences of X of length is monochromatic.

216

Chapter III. Ramsey Theory of Finite and Inﬁnite Block Sequences [<∞]

One can go on and deﬁne a notion of a (uniform) barrier on the set FINk of all ﬁnite block sequences in FINk and prove the following generalization of Lemma III.3.7. Lemma III.5.17. For every family F of ﬁnite block sequences of elements of FINk there is an inﬁnite block sequence X = (xn ) such that either F ∩ [X] = ∅, or every inﬁnite block subsequence of X has an initial segment in F. Starting from this lemma one goes on and proves the analogue of the Ellentuck theorem and therefore obtains the following generalization of Milliken’s. Theorem. [∞]

Theorem III.5.18. For every analytic set X ⊆ FINk there is an inﬁnite block sequence X = (xn ) such that either all inﬁnite block-subsequences of X belongs to X or all inﬁnite block-subsequences of X fall outside X. Taylor’s theorem (Theorem III.2.1) also allows to be generalized. Theorem III.5.19 (Lopez–Abad). For each positive integer k there is a ﬁnite list Ei (k) (i < ϕ(k)) of equivalence relations on FINk with the property that for every equivalence relation E on FINk there is i < ϕ(k) and an inﬁnite block sequence X = (xn ) such that E [X] = Ei (k) [X]. The number ϕ(k) of the smallest list of equivalence relations on FINk satisfying this conclusion allows to be expressed using standard enumerating functions so the ﬁrst few values of ϕ(k) can be computed. For example ϕ(1) = 5, ϕ(2) = 43, ϕ(3) = 619, etc.

Chapter IV

Approximate and Strategic Ramsey Theory of Banach Spaces IV.1

Gowers’ Dichotomy

Fix a Banach space X with a Schauder basis (ei ). Recall the notion of a block sequence of (ei ) is a sequence (xn ) again typically formed of norm 1 vectors, such that for each n there is a ﬁnite set Dn ⊆ N such that (a) xn = i∈Dn ai ei for some sequence ai (i ∈ Dn ) of scalars. (b) Dm < Dn 1 whenever m < n. Note that if (xn ) is a (normalized) block sequence of (ei ) then (xn ) is a Schauder basis of its closed linear span Y = (xn ). Theorem IV.1.1 (Bessaga–Pelczynski). Let X be a given inﬁnite-dimensional Banach space with a Schauder basis (ei ). Then for every ε > 0 and every inﬁnitedimensional closed subspace Y of X there is a (normalized) block sequence (xn ) and a sequence (yn ) ⊆ Y such that ∞

xn − yn < ε.

n=0

Proof. Going to an equivalent norm we may assume that all projections PI (x) = i∈I ai xi on intervals I ⊆ N have norm 1. Pick a normalized y0 ∈ Y . Then there is k0 such that P[k0 ,∞) (y0 ) < ε0 , so if we put x0 = Pk0 (y0 ), then x0 − y0 < ε0 . Since Y is inﬁnite-dimensional there is a normalized y1 ∈ Y such that Pk0 (y1 ) = 0. 1 For subsets D and E of some ordered set, the inequality D < E means that every element of D is smaller than every element of E.

218 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Choose k1 > k0 such that P[k1 ,∞) (y1 ) < ε1 and let x1 = P[k0 ,k1 ) (y1 ). Then x1 − y1 < ε1 , and so on. Continuing this way we get the desired block-sequence (xn ). In what follows a typical use of Theorem IV.1.1 is in reducing problems about arbitrary inﬁnite-dimensional subspaces of X to subspaces spanned by inﬁnite block-subsequences of (ei ). For example, suppose we are interested whether X contains an inﬁnite-dimensional hereditarily indecomposable subspace, i.e., a subspace Y with the property that no inﬁnite-dimensional closed subspace Z of Y can be decomposed as a sum Z0 ⊕ Z1 of two closed-inﬁnite-dimensional subspaces. Then this is same as asking for the existence of a hereditarily indecomposable block-subspace of X, i.e., a closed subspace Y of X spanned by a blocksubsequence of (ei ). Namely, if Y ⊆ X is hereditarily indecomposable and if (xn ) and (yn ) satisfy the conclusion of Theorem IV.1.1 then the closed linear span of the block-subsequence (xn ) is also hereditarily indecomposable. There are Banach spaces with no unconditional bases. For example C[0, 1] is one such space. However, the problem whether every inﬁnite-dimensional Banach has an unconditional basic sequence remained open for quite some time until was answered negatively by Gowers and Maurey in 1992. The Gowers–Maurey example has turned out to be hereditarily indecomposable and the following subsequent result of Gowers explains why this always must be so. Theorem IV.1.2 (Gowers). Let X be a Banach space with a Schauder basis (ei ). Then every block subsequence (xn ) of (ei ) has a block subsequence (xnk ) which is either unconditional or its closed linear span is hereditarily indecomposable. The proof of this result uses a combinatorial forcing argument quite analogous to those presented in Sections 4 and 5. Note that the dichotomy is really about normed spaces since completeness plays no role either in the deﬁnition of an unconditional basic sequence or a hereditarily indecomposable space. Note that for example an inﬁnite-dimensional space X is hereditarily indecomposable (H.I.) if for any pair Y and Z of inﬁnite-dimensional subspaces of X, dist(SY , SZ ) = inf{y − z : y ∈ SY , z ∈ SZ } = 0.2 Note also that we may restrict ourselves to normed spaces over the rationals rather than reals. This will make the forcing argument a bit more natural. So from now on a “space” or a “subspace”, unless otherwise speciﬁed, refers to inﬁnite dimensional normed spaces over the rationals. For ε > 0 we say that a space X is ε-H.I. if for all subspaces Y, Z ⊆ X there exists y ∈ Y, z ∈ Z such that y − z < εy + z. Clearly, X is H.I. iﬀ X is ε-H.I. for all ε > 0. So the Gowers dichotomy will follow once we prove the following. Theorem IV.1.3. For a given ε > 0 an inﬁnite-dimensional normed space X either contains a (2/ε)-unconditional basic sequence or an inﬁnite-dimensional subspace Y which is (2ε)-H.I. 2 For

a normed space X, we denote by SX its unit sphere.

IV.1. Gowers’ Dichotomy

219

Proof. We assume that 0 < ε < 1 and ﬁx an inﬁnite-dimensional normed space X over the rationals, its basis (ei ) and assume that the projections on intervals of N are uniformly bounded by 1. The variables U, V, Y, Z will run over inﬁnitedimensional block-spaces of X. For ε > 0 we set P (ε) = {(x, y) ∈ X : x − y < εx + y}. Deﬁnition IV.1.4 (Combinatorial forcing). We say that Z ⊆ X accepts a pair (x, y) ∈ X 2 if for all U, V ⊆ Z there exists (u, v) ∈ U ×V such that (x + u, x + v) ∈ P (ε). We say that Z rejects a pair (x, y) if no subspace Y of Z accepts (x, y). We say that Z decides (x, y) if Z either accepts or rejects (x, y). Note the following immediate properties of these notions: (1) If Z accepts (rejects) (x, y), so does any Y ⊆ Z. (2) For every Z and (x, y) there is Y ⊆ Z deciding (x, y). (3) If Z accepts (0, 0), then Z is (2ε)-H.I. (4) If Z decides all (x, y) ∈ X 2 but rejects (x0 , y0 ), then for all Y ⊆ Z there is V ⊆ Y such that Z rejects all (x0 + v, y) for v ∈ V . Only (4) needs some argument. Assume (4) fails and ﬁx a subspace Y ⊆ Z such that for all V ⊆ Y there is v ∈ V such that Z accepts (x0 + v, y0 ). Choose arbitrary U, V ⊆ Y . Pick v0 ∈ V such that Z accepts (x0 + v0 , y0 ). Then we can ﬁnd (u, v) ∈ U × V such that (x0 + v0 + v, y0 + u) ∈ P (ε) This checks that Y accepts (x0 , y0 ) contradicting (1). Using (1) and (2) we build an inﬁnite decreasing sequence X ⊇ X1 ⊇ · · · ⊇ Xn ⊇ . . . of block-subspaces of X such that for all (x, y) ∈ X 2 there is some n such that Xn decides (x, y). Let X∞ be the block subspace which diagonalizes (Xn ). Then X∞ decides all pairs (x, y). By (3) we may assume X∞ rejects (0, 0). n For every n, let Nn be a ﬁxed (1/4)-net of the unit ball of 1 . For a ﬁnite block sequence (zi )i
ai zi ,

i=0

where (ai )n−1 i=0 ∈ Nn . A pair x, y of vectors of (zi )i
i∈I

ai zi ,

y=

aj zj

j∈J

and I ∩ J = ∅. Recursively on i we build an inﬁnite block sequence(zi ) such that for all n, X∞ rejects all pairs of disjointly supported vectors from (zi )i
220 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces So, suppose we have constructed a block sequence (zi )i
IV.2

1 3 2 + = . 2ε 2ε ε

Approximate and Strategic Ramsey Sets

Gowers’ dichotomy presented in the previous section suggests the corresponding Ramsey theory of ﬁnite and inﬁnite block sequences in Banach spaces with Schauder bases. The purpose of this and the following sections is to give an overview of such a theory. In what follows E is a ﬁxed Banach space with a [k] Schauder basis (ei ). For k = 0, 1, 2, . . . , ∞ by B1 (E) we denote the collection of all normalized block subsequences of (ei ) of length k. Let [<∞]

B1

(E) =

∞ k=0

[k]

B1 (E)

IV.2. Approximate and Strategic Ramsey Sets

221

and [≤∞]

B1

[<∞]

(E) = B1

[∞]

(E) ∪ B1 (E). [≤∞]

We shall identify a given block sequence (zn ) ∈ B1 (E) with the corresponding closed linear span Z = zn n and use the capital letter Z to denote also the block sequence itself. When talking about topological properties of a subset X of some [k] B1 (E) we refer to the topology induced from E k with the product topology. [k] [l] For s ∈ B1 (E), X ∈ B1 (E) and k ≤ m ≤ l, let [m]

[s, X][m] = {Y ∈ B1 (E) : s = Y k and Y ≤ s (X/s)}.3 We shall frequently suppress the exponent [m] from [s, X][m] when it is clear in which dimension we are working at a particular moment. Let [X][m] = [∅, X][m] . We shall frequently suppress even this exponent [m] especially when it is equal to the length of the block sequence X. The sets [s, X] of block sequences will be frequently called basic sets for reason that will be clear later in this chapter. The lack of a true pigeon-hole principle in this context requires that we consider cubes of the form [X][m] which are ‘approximately monochromatic’ rather than ‘monochromatic’. It is for this reason that we need to consider the ‘errors’ of monochromaticity. It turns out that these ‘errors’ are naturally described by sequences ∆ = (δn )∞ n=0 ⊆ R+ of nonnegative real numbers. For this it is useful to [≤∞] have the following piece of notation, where X ⊆ B1 (E) and ∆ = (δn )∞ n=0 ⊆ R+ , [≤∞]

(X)∆ = {(zn )k0 ∈ B1

(E) : ∃(yn )k0 ∈ X ∀n zn − yn ≤ δn }.

For ∆ = (δn ), Γ = (γn ) ∈ R∞ + , we write Γ < ∆ whenever δn < γn for all n. Unless explicitly speciﬁed, we always work with Γ ∈ R∞ + which are decreasing and strictly positive. In this context, we shall frequently perform coordinatewise addition and scalar multiplication among such sequences. We are now ready to deﬁne two analogues of the classical notion of a Ramsey set in the context of sets of inﬁnite normalized block sequences of a given Banach space E with a Schauder basis (ei ). [∞]

Deﬁnition IV.2.1. A subset X of B1 (E) is approximately Ramsey if for every [∞] [<∞] ∆ = (δn )∞ (E) there is Y ∈ [s, X] 0 ⊆ R+ , every X ∈ B1 (E) and every s ∈ B1 such that either [s, Y ] ∩ X = ∅ or [s, Y ] ⊆ (X)∆ . Note that Theorem III.5.6 says that it makes sense to study this notion only when E is a c0 -saturated space, i.e., when every closed inﬁnite-dimensional subspace of E contains an isomorphic copy of c0 . In other words, the notion of approximately Ramsey sets of inﬁnite block sequences make sense studying only 3 Recall that Y ≤ X denotes the fact that Y is a block subsequence of X and that X/s denotes the maximal tail of X that lies entirely above s.

222 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces in the case E = c0 . When the given Banach space E is not c0 -saturated one needs to weaken the second alternative [Y ] ⊆ (X)∆ [∞]

in the deﬁnition of Ramsey set X ⊆ B1 (E) using the terminology of inﬁnite games. As customary in this context in order to avoid repeating the same words over and over again, we reserve the letters X,Y ,Z, etc for denoting inﬁnite normalized block sequence and the letters s,t,u, etc for denoting ﬁnite normalized block sequences. [∞]

[∞]

Deﬁnition IV.2.2. Given Y ∈ B1 (E) and X ⊆ B1 (E), we deﬁne an inﬁnite perfect-information game GX (Y ) between two players I and II as follows: • I plays an inﬁnite block subsequence Y0 of Y • II plays z0 ∈ S[Y0 ] • I plays another inﬁnite block subsequence Y1 of Y • II plays z1 ∈ S[Y1 ] such that z1 > z0 • I plays an inﬁnite block subsequence Y2 of Y , and so on. We say that II wins the resulting inﬁnite play Y0 , z0 , Y1 , z1 , . . . , Yn , zn , . . . if the sequence (zn ) belongs to X. Otherwise I wins the play. [∞]

Deﬁnition IV.2.3. (1) We say that X ⊆ B1 (E) is large for [s, X] if for every inﬁnite Y ∈ [s, X] there is inﬁnite Z ∈ [s, Y ] such that Z ∈ X. We say that X is very large for [s, X] if [s, X] ⊆ X. [∞] (2) We say that X ⊆ B1 (E) is strategically large for [s, X] if player II has a [∞] winning strategy for the game G(Xs ) (X), where Xs = {x ∈ B1 (E) : s X ∈ X}. (3) We say that X ⊆ B1 (E) is strategically Ramsey if for every ∆ = (δn )n ∈ R∞ + [<∞] [∞] and every (s, Y ) ∈ B1 (E) × B1 (E) there is Z ∈ [s, Y ] such that either (a) [s, Z] ∩ X = ∅, or (b) X∆ is strategically large for [s, Z]. [∞]

Remark IV.2.4. (1) Note that what we call here strategically Ramsey should more properly be called strategically approximately Ramsey. The notion that deserves to be called strategically Ramsey would have its second alternative strengthened to the conclusion that player II has a winning strategy in the game GXs (Z) rather than in the game G(Xs )∆ (Z). While the notion that deserves the name strategically Ramsey is both well behaved and well studied in Ramsey theory, it would still be too strong in the context of inﬁnite block basic sequences in Banach spaces. Since we will never deal with the properly named notion of strategically Ramsey sets we have decided, in the context of these lecture notes only, to adopt this simpliﬁcation of terminology.

IV.2. Approximate and Strategic Ramsey Sets

223

(2) Note that in the terminology of various notions of largeness introduced above, a set X is approximately Ramsey iﬀ for every basic set [s, X] there is Y ∈ [s, X] such that either [s, Y ] ∩ X = ∅ or (X)∆ is very large for [s, Y ], i.e., [s, Y ] ⊆ (X)∆ . So the formal diﬀerence between these two notions is that very large is weakened to strategically large. From Theorem III.5.6 we learn however that the actual diﬀerence between these two notions is immense. The terminology ‘large’ and ‘strategically large’ is chosen to suggest the corresponding ideals of ‘non large’ and ‘non strategically large’ sets. It is therefore quite natural to investigate how much these ideals do resemble σ-ideals in their behavior. In the classical context, that smallness is a σ-complete notion corresponds to the possibility of being able to perform various diagonal arguments with basic sets. In the approximate Ramsey theory the diagonal arguments are typically facilitated by net-approximations. [<∞] Deﬁnition IV.2.5. Let X ∈ B1 (E), ∆ = (δi ) ∈ R∞ is a + . A ∆-net of [X] [<∞] such that countable subset N ⊆ [X] (a) for every (y0 , . . . , yn ) ∈ [X][<∞] there is some (z0 , . . . , zn ) ∈ N such that yi − zi ≤ δi and supp yi = supp zi for every i ≤ n . (b) N ∩ [(ej )i0 ][<∞] is ﬁnite, for every i ∈ N. For a single real number ε > 0, an ε-net is by deﬁnition an (ε, ε2 , . . . )-net. [<∞] Finally, for a sequence (∆k )k ∈ R∞ is any countable set N + , a (∆k )k -net of [X] of ﬁnite block subsequences of X which can be written as the union of a sequence (Nk )k such that Nk is a ∆k -net of [X][<∞] for all k ∈ N. [≤∞]

Clearly, countable ∆-nets in [X][<∞] do always exist. Proposition IV.2.6. Let (Xn )n be a given sequence of families of block subsequences of E. Let ∆ = (δn )n ∈ R∞ + , and let φ : N → N. Suppose that the union X = ∞ [<∞] X is large for some basic set [s, X]. Then there exist t ∈ B1 (X) with n=0 n t > s, an inﬁnite block sequence Y ∈ [s, X], and an n ∈ N such that |t| ≥ φ(n) and such that (Xn )∆ is large for [s t, Y ]. Proof. Otherwise, for every Y ≤ X, every n ∈ N and every t > s with |t| ≥ φ(n) there is some Z ∈ [s t, Y ] such that [s t, Z] ∩ (Xn )∆ = ∅. Using this, ﬁxing a ∆-net N of [X][<∞] , and then performing a simple diaginalization argument, we ﬁnd block subsequence Z = (zn ) ≤ Y and a decreasing sequence Zn (n ∈ N ) of block subsequences of X such that for every n, [1] (1) zn+1 ∈ Zn , and supp zn [φ(m)] (2) for every m ≤ n with φ(m) ≤ n, and every t ∈ N ∩ [(ei )max ] 0 [s t, Zn ] ∩ (Xm )∆ = ∅. Having obtained such a block subsequence Z = (zn ), we claim that X ∩ [s, Z] = ∅. Otherwise, there is W ∈ [s, Z] ∩ Xm for some m ∈ N. Let (wn ) be such that φ(m)−1 W = s (wn ). Choose minimal n such that u = (wi )0 ∈ [(z0 , . . . , zn )] and

224 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces supp zn [φ(m)] ] such that d(u, t) ≤ ∆. By the choice of Zn we pick t ∈ N ∩ [(ei )max 0 have that [s t, Zn ] ∩ (Xm )∆ = ∅.

Since s t (wi )i≥φ(m) is a member of this basic set on distance ≤ ∆ from W , we obtain that W ∈ / Xm , a contradiction. [∞]

Proposition IV.2.7. Let (Xn )n be a given sequence of subsets of B1 (E) and let X [∞] be a given sequence of B1 (E). Then there is inﬁnite Z ≤ X such that for every n, if there is Y ≤ Z such that Xn is strategically large (large) for [Y ], then Xn is also strategically large (large) for [Z]. Proof. We construct recursively on n a decreasing sequence (Zn ) (n ∈ N) of block subsequences of X such that for every n ∈ N if there is some Y ≤ Zn such that Xn is strategically large (large) for [Y ], then Xn is also strategically large (large) for [Zn+1 ]. Let Z = (zn ) be an inﬁnite normalized block sequence such that [∞] (zi≥n ) ≤ Zn for all n. Now observe a general fact that if some set Y ⊆ B1 (E) is strategically large (large) for some inﬁnite block sequence Y , then Y is also strategically large (large) for any inﬁnite normalized block sequence Y which has a tail that is a block subsequence of Y .

IV.3

Combinatorial Forcing on Block Sequences in Banach Spaces

The asymmetric nature of approximate and strategic Ramsey sets makes these notions behave diﬀerently from the classical notion of Ramsey sets considered above in Sections II.5 and II.4, though one can still prove some analogous results. For example, as in the classical case one has results that corresponds to Galvin’s lemma which deals with sets X of ﬁnite block sequences. In the strategical Ramsey context, the game GX (Y ) is then of course slightly modiﬁed to the eﬀect that the player II wins a play Y0 , z0 , Y1 , z1 , . . . , Yn , zn , . . . just in case there is a k such that (zn )k0 ∈ X. Lemma IV.3.1 (Gowers). Let X be an arbitrary set of ﬁnite normalized block subsequences of a ﬁxed Schauder basis (ei ) of some Banach space E, let X be an arbitrary inﬁnite normalized block subsequence of (ei ), and let ∆ = (δi )i ∈ R∞ +. Then there is an inﬁnite normalized block subsequence Y of X such that either [Y ][<∞] ∩ X = ∅ or else II has a winning strategy in the game G(X)∆ (Y ). Proof. We will in fact establish a stronger conclusion. For two (ﬁnite or inﬁnite) block sequences s and t of the same length k and ∆ = (δi ) ∈ R∞ + , we write d(s, t) ≤ ∆ whenever for all i < k, s(i) − t(i) ≤ δi and supp s(i) = supp t(i).

IV.3. Combinatorial Forcing on Block Sequences in Banach Spaces

225

Fix X and ∆ = (δi )i as in the hypothesis of the Lemma. Given two (ﬁnite or inﬁnite) block subsequences s = (yi ), t = (zi ), of the same length let Θs,t be the linear isomorphism between the span of (yi ) and the span of (zi ) deﬁned by extending the assignment Θ(yi ) = zi . Note that Θs,t maps in a natural way a given block subsequence u of s to a corresponding block subsequence Θs,t (u) of t. The reader can easily verify the following fact of future use. (1) There is Γ ∈ R∞ + such that Γ < ∆ and such that for every two block sequences s and t of the same length and with the property that d(s, t) ≤ Γ we have that d(u, Θs,t (u)) ≤ ∆ for every (ﬁnite) block subsequence u of s. (Hint: Let γn = min{δ0 , . . . , δn }/(2n+2 C), where C is the basis constant of the Schauder basis (ei ) of E.) Given an inﬁnite normalized block sequence X, a ﬁnite normalized block [<∞] subsequence s and a family R ⊆ B1 (E), we say that X R-rejects s iﬀ ∀Y ≤ X ∀n ∈ N Y n ∈ (R)c . We say that X R-accepts s iﬀ no Y ≤ X R-rejects s. Given ∆ as in the hypothesis of the Lemma, we let Γ = (γi ) be chosen to satisfy the conclusion of (1) for ∆/2. For n ∈ N, set Γn = (nγi /(n + 1))∞ i=0 . We are now ready to deﬁne the combinatorial forcing that corresponds to that appearing in standard proofs of Galvin’s lemma. For this we shall need one more piece of notation, X0 = {s ∈ (X)∆/2 : [s][<|s|] ∩ (X)∆/2 = ∅}. We say that X rejects s iﬀ X (X0 )Γ|s| -rejects s. We say that X accepts s iﬀ X (X0 )Γ|s|+1 -accepts s. We say that X decides s if it either accepts or rejects s. We [<∞]

(E) is small if it contains no subset of the form [Y ][<∞] for say that R ⊆ B1 any inﬁnite normalized block subsequence Y . The reader can easily verify the following basic facts about these notions of combinatorial forcing that will be freely used in the arguments that follow. (a) If X R-accepts (R-rejects) s, then every Y ≤ X R-accepts (R-rejects) s. (b) If X accepts (rejects) s, then every Y ≤ X accepts (rejects) s. (c) If R ⊆ R , then if X R -rejects s, then X R-rejects s. [<∞] (d) For every X, s and R ⊆ B1 (E) there is some Y ≤ X which R-decides s. (e) For every X and s there is some Y ≤ X which decides s. (f) If X rejects s, then X rejects s (x) for every x ∈ X. (g) If X accepts s, then {x ∈ X : X rejects s (x)} is small. As in the classical case we are now ready to state and prove the following claim. (2) There is Y ≤ X which decides every s ∈ [Y ][<∞] .

226 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces The inﬁnite normalized block subsequence Y = (yi ) is deﬁned recursively in i along with a decreasing sequence and Yi (i ∈ N) of inﬁnite normalized block subsequences of X with the requirement that for every i ∈ N and for every s ∈ [(y0 , . . . , yi )], Yi decides s. Suppose we have already deﬁned (y0 , . . . , yi ) and Yi . Choose yi+1 ∈ Yi such that yi+1 > yi . For each 1 ≤ k ≤ i + 1, let Fk be a ﬁnite Λk = (Γk+1 − Γk )/3-net of [(y0 , . . . , yi+1 )][k] , i.e. (Fk )(Γk+1 −Γk ) = [(y0 , . . . , yi+1 )][k] . i+1 Let Yi+1 ∈ [Yi ] be such that for every t ∈ k=1 Fk either Yi+1 (X0 )Γk +Λk -rejects t or Yi+1 (X0 )Γk+1 −Λk -accepts t, and Yi+1 decides ∅. We claim that Yi+1 decides every s ∈ [(y0 , . . . , yi+1 )]: Suppose that s ∈ [(y0 , . . . , yi+1 )][k] , and choose t ∈ Fk be such that d(s, t) ≤ Λk and supp s = supp t. If Yi+1 (X0 )Γk +Λk -rejects t, then Yi+1 rejects s. If not, there is some Z ∈ [s, Yi+1 ] and some n ∈ N such that Z n ∈ (X0 )Γk . Let u ∈ X0 be such that d(u, Z n) ≤ Γk . Then t u [|t|, |u| − 1] ∈ (X0 )Γk +Λk , in contradiction with the fact that Yi+1 (X0 )Γk +Λk -rejects t. If Yi+1 (X0 )Γk+1 −Λk accepts t then one can easily prove that Yi+1 accepts s. This describes our recursive construction. It should be clear that the resulting block sequence Y satisﬁes the conclusion of Claim (2). From now on we ﬁx Y satisfying the conclusion of Claim (2). Case 1: There is some Y ≤ Y such that Y X-rejects ∅. Then [Y ][<∞] ∩ X = ∅ and we are done. Case 2: There is no Y ≤ Y X-rejecting ∅. Then we have that ∀Y ≤ Y ∃Y ≤ Y ∃n ∈ N Y n ∈ X.

(IV.1)

Note that this also gives that Y accepts ∅. We now claim that player II has a winning strategy for the game G(X)∆ (Y ): Suppose that I plays Z0 ∈ [Y ]. By (f) and the fact that Y decides every s ∈ [Y ][<∞] , II can play z0 ∈ Z0 such that Y accepts (z0 ). In general, suppose that in the nth -run I plays Zn ∈ [Y ]. Then II can play zn > zn−1 such that Y accepts (z0 , . . . , zn ). We claim that the payoﬀ of this run (zi ) of the game is such that there must be some n ∈ N with the property that (z0 , . . . , zn ) ∈ (X)∆ : To see this note that by the above displayed property (IV.1) of Y from Case 2, we can ﬁnd an n ∈ N and some t ∈ [z0 , . . . , zn ] ∩ X . Since Y accepts (z0 , . . . , zn ) there is some W ∈ [(z0 , . . . , zn ), Y ] and some m such that W m ∈ (X0 )Γm+1 . If m ≤ n, then (y0 , . . . , ym ) ∈ (X0 )Γm+1 ⊆ (X0 )Γ ⊆ (X0 )∆/2 ⊆ (X)∆ , and we would be done. Otherwise, there is some non-empty u, and some ﬁnite block subsequence (y0 , . . . , yn ) such that d((y0 , . . . , yn ), (y0 , . . . , yn )) ≤ Γm+1 and such that (y0 , . . . , yn ) u ∈ X0 . Then d(t, Θ((y0 ,...,yn )u),(y0 ,...,yn )u (t)) ≤ ∆/2, which is in a contradiction with the fact that (y0 , . . . , yn ) u ∈ X0 , since t ∈ X.

IV.3. Combinatorial Forcing on Block Sequences in Banach Spaces

227

Exercise IV.3.2. Supply a proof of Gowers’ Banach space dichotomy that uses Lemma IV.3.1.(Hint: Let C be the basis constant of √ the Schauder basis (en ) of E and choose summable ∆ ∈ R∞ such that 1 + 2Cd < 2, where d is the sum of ∆. Let X be the set of all all ﬁnite block subsequences (xi )k0 of (en ) such that ∀K ∃n < m ∃λ0 , ..., λm−n

m

λi−n xi > K

n

m

(−1)i λi−n xi .

n

Show that if E contains no inﬁnite-dimensional subspace with an unconditional basis then X is large and therefore by Lemma IV.3.1 the corresponding set (X)∆ is strategically large below some inﬁnite block subsequence of (en ). Use the strategy to build an inﬁnite block sequence whose closed linear span is a hereditarily indecomposable space.) The corresponding lemma for the approximate Ramsey property for c0 saturated Banach spaces is also true. In fact, the corresponding notion of combinatorial forcing resembles considerably more the combinatorial forcing from the classical case than the one appearing in the proof of Lemma IV.3.1. It is for this reason that we choose to spell it out in a more explicit manner. So again we simplify the notation by letting the variables R, S, T , etc run over subsets of [<∞] [∞] B1 (c0 ), the variables X, Y , Z, etc run over elements of B1 (c0 ), and variables [<∞] s, t, u, etc run over elements of B1 (c0 ). Deﬁnition IV.3.3. For a given S, s and X as above, we say that X S-accepts s iﬀ for every Y ∈ [s, X] there is some n ∈ N such that Y n ∈ S. An inﬁnite block subsequence X S-rejects s iﬀ no Y ∈ [X] S-accepts s. An inﬁnite block subsequence X S-decides s iﬀ either X S-accepts s or X S-rejects s. More generally, [<∞] for a pair of sets R ⊆ S ⊆ B1 (c0 ), we say that X (S, R)-decides s iﬀ either X S-accepts s or X R-rejects s. The following three propositions summarize the facts about these notions that will be quite useful below. Proposition IV.3.4. (a) If R ⊆ S and X R-accepts s, then X S-accepts s. (b) If R ⊆ S and X S-rejects s, then X R-rejects s.

∞

Proposition IV.3.5. Let ε ≥ 0 and let Γ ∈ R be deﬁned by Γ(i) = ε if i = |s| and 0 otherwise. Suppose that for some X, R, and s we have that the block sequence X (R)Γ -rejects s. Then the set ({x ∈ X : X R-accepts s (x)})ε is small in the sense that it contains no set of the form [Y ][1] for Y an inﬁnite block subsequence of X. Proof. Otherwise, there is inﬁnite Y ≤ X such that for every y ∈ [Y ][1] there is some y such that y − y ≤ ε and X R-accepts s (y ). It follows that for every y ∈ [Y ][1] , the block sequence X (R)Γ -accepts s (y) contradicting the assumption that X (R)Γ -rejects s.

228 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Proposition IV.3.6. Suppose Γn < ∆n for n ∈ N is a given sequence of pairs of [∞] [∞] elements of R∞ + . Then for every X ∈ B1 (E) and R ⊆ B1 (E) there is inﬁnite Y ≤ X such that for every s ∈ [Y ][<∞] , the block sequence Y ((R)∆|s| , (R)Γ|s| )decides s. Proof. This is similar to the proof of Claim (2) from the proof of Lemma IV.3.1. Now we are ready to state and prove the analogue of Galvin’s Lemma for the approximate Ramsey property of the space c0 . Lemma IV.3.7 (Gowers). Let X be an arbitrary set of ﬁnite normalized block sequences of the Banach space c0 . Let X be an inﬁnite normalized block sequence of c0 and let ∆ = (δi )i ∈ R∞ + . Then there is inﬁnite Y ≤ X such that either [Y ][<∞] ∩ X = ∅, or else for every inﬁnite Z ≤ Y there is n ∈ N such that Z n ∈ (X)∆ . Proof. Fix X and ∆. For n ∈ N, we set ∆n = (1/(n + 1))∆. Using these objects, we now deﬁne the following notion of combinatorial forcing. Deﬁnition IV.3.8. We say that X accepts s iﬀ X (X)∆4|s| -accepts s. We say that X rejects s iﬀ X (X)∆4|s|+1 -rejects s. We say that X decides s iﬀ either X accepts s or X rejects s. Note the following properties of this notion of combinatorial forcing. Proposition IV.3.9. (a) If X accepts (rejects) s, then for every block subsequence Y of X accepts (rejects) s. (b) For every X and s there is some inﬁnite block subsequence Y of X which decides s. (c) If X accepts s, then for every x ∈ X, the sequence X accepts s (x). Proof. (a) This is immediate from the deﬁnitions. (b) If no Y ≤ X accepts s, then by the deﬁnition X must (X)∆4|s| -reject s. Since (X)∆4|s|+1 ⊆ (X)∆4|s| , we get that X rejects s. (c) This is immediate from the deﬁnition of acceptance. We are now ready to continue with the proof of Lemma IV.3.7. Using Proposition IV.3.6 and a simple diagonalization procedure we obtain X0 ∈ [X] which ((X)∆4|s|+i , (X)∆4|s|+j )-decides every s ∈ [X0 ][<∞] for 0 ≤ i < j ≤ 4. If X0 accepts ∅ then the ﬁrst alternative from the conclusion of the lemma holds. Otherwise, X0 rejects ∅. We are going to recursively ﬁnd Y = (yi ) ≤ X0 such that for every s ∈ [Y ][<∞] Y rejects s. Suppose we deﬁned (y0 , . . . , yn ) such that the minimal term of Yn starts above yn , Yn ≤ Yn−1 , and for every s ∈ [(y0 , . . . , yn )] Yn rejects s. Let ε = (∆4n+4 (n) − ∆4n+3 (n))/2.

IV.4. Coding into Approximate and Strategic Ramsey Sets

229

Fix s ∈ [(y0 , . . . , yn )]. Consider the coloring [Yn ][1] = Ca ∪ Cr where Ca ={y ∈ [Yn ][1] : Yn (X)∆4|s|+2 -accepts s (y)} Cr ={y ∈ [Yn ][1] : Yn (X)∆4|s|+3 -rejects s (y)}. By Gowers’ c0 -theorem III.5.7 we obtain that there is some Yn (s) ≤ Yn such that either [Yn (s)][1] ⊆ (Ca )ε or [Yn (s)][1] ⊆ (Cr )ε . Notice that ∆4|s|+2 + Γ ≤ ∆4|s|+1 where Γ ∈ R∞ + is given by Γ(i) = ε if i = |s| and 0 otherwise. So by our recursive assumption, the block sequence Yn (X)∆4|s|+2 +Γ rejects s. Then by Proposition IV.3.5 applied to R = (X)∆4|s|+2 we obtain that (Ca )ε is small. So, we have the second alternative. This gives us that for every y ∈ [Yn (s)][1] , the block sequence Yn (s) (X)∆4|s|+4 -rejects s (y). Now ﬁx a ﬁnite ε-net N of [(y0 , . . . , yn )]. Applying the above procedure successively for every s ∈ N we obtain a single Yn+1 ≤ Yn such that for every s ∈ N and every y ∈ [Yn+1 ][1] , Yn+1 (X)∆4|s|+4 -rejects s (y). Hence for every s ∈ [(y0 , . . . , yn )] and every y ∈ [Yn+1 ][1] , Yn+1 (X)∆4(|s|+1)+1 -rejects s (y), as desired. Let yn+1 be the minimal term of the sequence Yn+1 above yn . This ﬁnishes our recursive construction of the inﬁnite block sequence Y = (yi ). Now we claim that the existence of such sequence Y = (yi ) gives us the ﬁrst alternative of the Lemma. To see this note that since X ⊆ (X)∆n for every n, by the deﬁnition of rejection, we obtain that [Y ][<∞] ∩ X = ∅.

IV.4

Coding into Approximate and Strategic Ramsey Sets

In what follows, E is a ﬁxed Banach space with a Schauder basis (en ). It will be convenient to denote its unit sphere by S(E). More generally, given a ﬁnite or inﬁnite normalized block subsequence (xn )n , we use the notation S((xn )n ) for the unit sphere of the corresponding closed linear span of (xn )n . We shall use N ↑ to denote the set of strictly increasing sequences of positive integers as a topological subspace of the Baire space N = NN . Recall that all topological [k] properties of subsets of powers of the form B1 (E) refer to the topology induced k from the Tychonoﬀ cube S(E) where S(E) is taken with its separable metric norm-topology. Deﬁnition IV.4.1. A subset A of the sphere S(E) is called asymptotic iﬀ A ∩ S(xn ) = ∅ for every inﬁnite block sequence (xn )n of E. An asymptotic pair of E is a pair A = A0 , A1 where A0 and A1 are disjoint asymptotic sets of E. Note that asymptotic pairs always exist. For example, the sets Ai = {x ∈ S((en )n ) : (−1)i+1 e∗min supp x (x) > 0} for i = 0, 1 are clearly asymptotic and they form an asymptotic pair. Since our coding procedure will be based on asymptotic pairs and since we will be interested

230 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces in calculations of the descriptive complexity of various sets of block sequence it will be quite important to have asymptotic pairs of as low complexity as possible. This motivates our next deﬁnition. Deﬁnition IV.4.2. An asymptotic pair A = A0 , A1 is called discrete if A0 and A1 are Fσ subsets of S(E). An asymptotic pair A = A0 , A1 is called separated if A0 and A1 are closed subsets of the sphere S(E) and if δ(A) > 0, where δ(A) = d(A0 , A1 ) = inf{a0 − a1 : a0 ∈ A0 , a1 ∈ A1 }. Remark IV.4.3. (a) The asymptotic pair Ai = {x ∈ S((en )n ) : (−1)i+1 e∗min suppx (x) > 0} for i = 0, 1 is an example of a discrete asymptotic pair since 1 and (∀l < L) e∗l (x) = 0}, (IV.2) {x ∈ S(E) : (−1)i+1 e∗L (x) ≥ Ai = N 2 (L,N )∈N

gives us a representation of these sets as countable union of closed sets. (b) By Theorem III.5.6 separated asymptotic pairs exist if the Banach space E is not c0 -saturated. If the Banach space E does not contain an isomorphic copy of c0 then for every block normalized block sequence X there is a normalized block subsequence Y of X such that S([Y ]) contains a separated asymptotic pair. From now on we ﬁx an asymptotic pair A = A0 , A1 and study the following coding procedure based on it. Deﬁnition IV.4.4. A sequence (xk )k ∈ B1 (E)[∞] codes the pair ((yn )n , (kn )n ), provided that yn = x2n for all n and provided that {kn }n is the increasing enumeration of the set {k : x2k+1 ∈ A1 }. Let BA (E) ={(xn )n ∈ B1 (E) : (∀n) x2n+1 ∈ A0 ∪ A1 and (∃∞ n) x2n+1 ∈ A1 }. [∞]

[∞]

[∞]

Then every member(xk )k of BA (E) codes the pair ΛA ((xk )k ) = ((x2n )n , (kn )n ), where (kn )n is the increasing enumeration of {k : x2k+1 ∈ A1 }. This gives us a mapping [∞] [∞] ΛA : BA (E) → B1 (E) × N ↑ which we will study in some detail below. But let us ﬁrst show that our coding procedure is in fact possible. Proposition IV.4.5. For every block sequence X there is some block subsequence Y of X such that every pair from [Y ][∞] × N ↑ is coded by some block subsequence of X.

IV.4. Coding into Approximate and Strategic Ramsey Sets

231

Proof. Fix a block sequence X = (xn )n . Let n0 be the ﬁrst integer n such that n−1 S(xi n−1 i=0 ) ∩ A0 and S(xi i=0 ) ∩ A1 are non empty, which is well deﬁned since A0 and A1 are asymptotic sets and, by deﬁnition, every element of them has ﬁnite support. With nk deﬁned, let nk+1 be the ﬁrst integer n > nk such that n−1 S(xi n−1 i=nk +1 ) ∩ A0 and S(xi i=nk +1 ) ∩ A1 are non empty. Set Y = (xnk )k . We

are going to show that [Y ] × N ↑ ⊆ (ΛA ”[X][∞] ) ∩ BA (E). Fix a block sequence Z = (zn )n ∈ [Y ] and ε = (εn )n ∈ N ↑ . Let kr be the minimal integer k such that zr ∈ S(xn0 , ..., xnk ) for every r. Choose nkr +1 −1 ) ∩ A1 if r ∈ {εn }n S(xj j=n k +1 (IV.3) wr ∈ nkr−1r+1 S(xj j=nkr +1 ) ∩ A0 if r ∈ / {εn }n [∞]

for every r. Then the block sequence W = (z0 , w0 , z1 , w1 , ...) is in [X] ∩ BA (E), and clearly ΛA (W ) = (Z, ε). [∞]

[∞](E)

Recall the notions of large and strategically large for subsets of B1 above in Deﬁnition IV.2.3. Let

given

π0 : B1 (E) × N ↑ → B1 (E) [∞]

[∞]

be the projection on the ﬁrst coordinate. Corollary IV.4.6. Let C be a subset of B1 (E) × N ↑ and X be an inﬁnite block sequence. If π0 ”C is large for [X] then Λ−1 A (C) is also large for [X]. [∞]

Proof. Suppose that π0 ”C is large for [X] and ﬁx some inﬁnite block sequence X of X. From Proposition IV.4.5 there is inﬁnite block subsequence Y of X [∞] such that [Y ] × N ↑ ⊆ ΛA ”(B1 (E)) ∩ [X ][∞] . Since π0 ”C is large for [X] there [∞] is some Z ∈ [Y ] ∩ π0 ”C. Fix ε ∈ N ↑ such that (Z, ε) ∈ C and choose W ∈ [∞] [∞] B1 (E) ∩ [s, X ] such that ΛA (W ) = (Z, ε). Clearly W ∈ Λ−1 . A (C) ∩ [X ] Proposition IV.4.7. Let C and X be as in the hypothesis of the previous propo−1 sition, and let ∆ ∈ R∞ + . If (ΛA (C))∆ is strategically (very) large for [X], then (π0 ”C)∆ is also strategically (very) large for [X]. Proof. Suppose that there is some X such that Player II has a winning strategy Φ for the game G(Λ−1 (C))∆ [X]. Let us describe a winning strategy Φ for Player II A for the game G(π0 ”C)∆ [X]: Start the game with Player I choosing X0 ∈ [X]. Then Player II splits X0 into two subsequences Y0 and Z0 and he picks y0 = Φ (X0 ) = Φ(Y0 ). Suppose that the next choice of Player I is X1 ∈ [X]. Then Player II splits X1 into two subsequences Y1 and Z1 and he chooses y1 = Φ (X0 , X1 ) = Φ(Y0 , Z0 , Y1 ), and so on. At the end of the game the block sequence

since

(y0 , y1 , ...) = Φ ∗ (Yn )n ∈ (π0 ”C)∆

(IV.4)

(y0 , z0 , y1 , z1 , ...) = Φ ∗ (Y0 , Z0 , Y1 , Z1 , ...) ∈ (Λ−1 A (C))∆ .

(IV.5)

232 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces

The proof for the case of the very large property is similar.

Corollary IV.4.8. Let SRE denote the family of strategically Ramsey subsets of B1 (E)[∞] . Then for every family family C of subsets of B1 (E) × N ↑ , the inclusion {Λ−1 A (C) : C ∈ C} ⊆ SRE implies the inclusion {π0 ”C : C ∈ C} ⊆ SRE . A quite analogous argument gives a corresponding result about the approximate Ramsey property in the class of c0 -saturated spaces E. Proposition IV.4.9. Let E be a c0 -saturated space, and let ARE be the family of approximate Ramsey subsets of B1 (E)[∞] . Then for every family family C of subsets of B1 (E) × N ↑ , the inclusion {Λ−1 A (C) : C ∈ C} ⊆ ARE implies the inclusion {π0 ”C : C ∈ C} ⊆ ARE . [∞]

Next we discuss the Borel complexity of the set BA (E) and the mapping ΛA depending on the complexity of the asymptotic pair A. We have the following two simple facts that describe these complexities. [∞]

Lemma IV.4.10. Suppose that A is a discrete asymptotic pair. Then BA (E) is a Fσδ -set and the mapping ΛA is a Baire-class-1 function. Hence Λ−1 A (C) is an Fσδ [∞] ↑ set for every closed set C ⊆ B1 (E) × N . More precisely, for every closed set [∞] [∞] C ⊆ B1 (E) × N ↑ , the set Λ−1 A (C) is the intersection of BA (E) and a Gδ subset [∞] of B1 (E). Corollary IV.4.11. Let E be a given Banach space with a Schauder basis (en ) and [∞] let A be a discrete asymptotic pair of E. If all relatively Gδ -subsets of BA (E) are [∞] approximately Ramsey then so are all analytic subsets of B1 (E). It follows that the study of the approximate Ramsey property in a given c0 saturated Banach space E reduces to the study of this property on relatively Gδ [∞] subsets of BA (E) for conveniently chosen asymptotic pairs A of E. For example, in the case of the Banach space c0 itself, we may restrict ourselves to Gδ -subsets [∞] of BA (c0 ), where A is the asymptotic pair given in Remark IV.4.3(a) above. [∞]

Lemma IV.4.12. Suppose that A is a separated asymptotic pair. Then BA (E) is [∞] a Gδ subset of B1 (E) and ΛA is a continuous mapping on this domain. Hence [∞] ↑ Λ−1 A (C) is a Gδ set for every closed set C ⊆ B1 (E) × N . More precisely, for [∞] [∞] every closed set C ⊆ B1 (E) × N ↑ , the set Λ−1 A (C) is the intersection of BA (E) [∞] and a closed subset of B1 (E). Corollary IV.4.13. If a Banach space E does not contain an isomorphic copy of c0 [∞] and if all Gδ subsets of B1 (E) are strategically Ramsey, then so are all analytic [∞] subsets of B1 (E). It follows that the notions of approximate and strategic Ramsey sets behave strikingly diﬀerent from the corresponding classical notion of Ramsey sets where results about analytic sets are typically considerably deeper.

IV.5. Topological Ramsey Theory of Block Sequences in Banach Spaces

233

Recall that subsets of Polish spaces can be classiﬁed according to their topological complexity. This yields the so-called projective (or Lusin) hierarchy of pointclasses. We shall use the following standard notation(see [42]): Σ11 is the class of analytic sets, i.e., the continuous images of Borel sets; Π11 is the class of coanalytic sets, i.e., the complements of analytic sets; Σ1n+1 is the class of the continuous images of Π1n sets, and Π1n+1 is the class of complements of Σ1n+1 sets. Then we have proved also the following more general result. Theorem IV.4.14 (Lopez–Abad). Let E be a given Banach space with a Schauder [∞] basis (en ). If every coanalytic subset of B1 (E) is approximately (strategically) [∞] Ramsey, then every Σ12 subset of B1 (E) is also approximately (strategically) [∞] Ramsey. More generally, for every n ≥ 1, if every Π1n subset of B1 (E) is ap[∞] proximately (strategically) Ramsey, then every Σ1n+1 subset of B1 (E) is approximately (strategically) Ramsey .

IV.5

Topological Ramsey Theory of Block Sequences in Banach Spaces

We start again with a Banach space E with a Schauder basis (en ) and study the [∞] corresponding family B1 (E) of inﬁnite normalized block sequences. The separa[∞] ble metric topology of B1 (E) is reﬁned by letting the sets of the form [<∞]

[s, X] (s ∈ B1

[∞]

(E), X ∈ B1 (E)) [∞]

be open. This is what we are going to call the Gowers topology of B1 (E). The following result relates this topology to the notions of approximately and strategically Ramsey sets introduced above. [∞]

Lemma IV.5.1. Every Gowers-open subset O of B1 (E) is strategically Ramsey. Proof. Let [s, X] and ∆ ∈ R∞ be given. Clearly we may assume that s is empty. Claim. There is X0 ≤ X be such that ∀s ≤ X0 ∀Y ≤ X0 ([s, Y ] ⊆ (O)∆/2 → [s, X0 ] ⊆ (O)∆ ).

(IV.6)

Proof of the claim. Fix a ∆/4-net N of [X][<∞] . The required block sequence X0 = (xi ) ≤ X will be deﬁned recursively on i together with a descending sequence (Yi ) of block subsequences of X such that for every i, (a) xi < Yi and xi+1 ∈ [Yi ][1] , supp xi (c) ∀s ∈ N ∩ [(ei )max ] ∀Y ≤ Yi ([s, Y ] ⊆ (O)3∆/4 −→ [s, Yi ] ⊆ (O)3∆/4 ). 0 We claim that the resulting X0 = (xi )i has the desired property. For suppose that we have s ∈ [X0 ][<∞] and s < Y ≤ X0 with the property that [s, Y ] ⊆ (O)∆/2 .

234 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Let i be the minimal j such that s ∈ [(x0 , . . . , xj )]. Choose t ∈ N such that d(s, t) ≤ ∆/4 and supp s(j) = supp t(j) for every j < |s|. (Note that t can always be found by the deﬁnition of a ∆/4-net). Hence [t, Y ] ⊆ (O)3∆/4 , and Y ≤ Yi . So we obtain that [t, Yi ] ⊆ (O)3∆/4 , and hence [t, X0 ] ⊆ (O)3∆/4 , which implies that [s, X0 ] ⊆ (O)∆ , as promised. From now on we ﬁx X0 ≤ X as in the claim. Let X = {s ∈ [X0 ][<∞] : ∀t ∈ [X0 ][<∞] (d(s, t) ≤ ∆/2 → [t, X0 ] ⊆ (O)∆ )}. Applying Lemma IV.3.1 we obtain X1 ≤ X0 that falls into one of the following two cases: Case 1: [X1 ][<∞] ∩ X = ∅. We claim that in this case, [X1 ][∞] ∩ O = ∅. Otherwise, ﬁx Y ∈ [X1 ][∞] ∩ O. Since O is Gowers-open there is some n ∈ N such that [Y n, Y ] ⊆ O. Set s = Y n. Consider an arbitrary ﬁnite block sequence t such that d(s, t) ≤ ∆/2. Let m ∈ N be such that t < Y /m. Then [t, Y /m] ⊆ (O)∆/2 and by (IV.6) we obtain that [t, X0 ] ⊆ (O)∆ . This checks that s ∈ X ∩ [X1 ][<∞] and this is in direct contradiction with the alternative of Case 1. Case 2: The player II has a winning strategy for the game G(X)∆/2 (X1 ). We claim that in this case (O)∆ is strategically large for [X1 ]. The winning strategy for player II in the game G(O)∆ (X1 ) is described as follows. He ﬁrst chooses a winning strategy σ for the game G(X)∆/2 (X1 ). Then he plays according to σ until he reaches some t ∈ (X)∆/2 . After that he plays arbitrarily z0 < z1 < . . . above t. We claim that after any such inﬁnite run, t (zi ) ∈ (O)∆ . To see this note that since t ∈ (X)∆/2 there is some s ∈ X such that d(t, s) ≤ ∆/2. By the deﬁnition of X, this means that [t, X0 ] ∈ (O)∆ . Hence we obtain that t (zi ) ∈ (O)∆ as promised. An almost identical net-approximation argument based on Lemma IV.3.7 will give us the following. [∞]

Lemma IV.5.2. Every Gowers-open subset O of B1 (c0 ) is approximately Ramsey. Unfortunately, this is as much as we can do with the Gowers topology in this context. For example, we will not be able to mimic the classical setup and relate the notion of, say, Gowers-nowhere-dense sets with the notion of strategic or approximate Ramsey property. Recall, however, that one always implicitly assumes (unless [∞] otherwise stated) the topology on B1 (E) to be the separable metric topology inN duced from (SE ) where the sphere SE is taken with its norm topology. Since this topology is weaker than the Gowers topology, the two previous lemmas say also that open sets relative to the metric topology are strategically and approximately Ramsey, respectively. We can indeed proceed to generalize these two lemmas to [∞] more complex subsets of B1 (E) but the complexity is expressed relative to this metric topology. Recall that in the previous Section we have learned that crucial [∞] results will be about Gδ -sets relative to the metric topology of B1 (E).

IV.5. Topological Ramsey Theory of Block Sequences in Banach Spaces

235

We start with the following technical result that will be used in the proof that Gδ -sets are strategically Ramsey. Proposition IV.5.3. Let (Xn )n be a given sequence of families of block ∞ subsequences of E. Let ∆ = (δn )n ∈ R∞ + , and let φ : N → N. If the union X = n=0 Xn is large for some [s, X], then there exists Y ∈ [s, X] such that the set X(∆, s, Y, φ) = {(zm )m :∃n ∃k ≥ φ(n) (Xn )∆ is large already for [s (z0 , . . . , zk ), Y ]}, is strategically (very) large for [s, Y ].

Proof. Fix all data as in the statement, and suppose that the union X = ∞ n=0 Xn is large for some [s, X]. Notice that the sets X(∆, s, Y, φ) are always open. For a set of block subsequences Z, set (Z)s = {W : s W ∈ Z}. Let N be a countable [<∞] ∆/4-net of B1 (E). Apply Proposition IV.2.7 to the family ((Xn )∆/2 )t (t ∈ N, n ∈ N) and block sequence X and obtain X0 such that, if for some t ∈ N, some n ∈ N, and some Y ≤ X0 , the family (Xn )∆/2 is large for [s t, Y ], then (Xn )∆/2 is large already for [s t, X0 ]. Since N was a ∆/4-net, for every t and n, we obtain that if (Xn )∆/4 is large for [s t, Y ] for some Y ≤ X0 , then (Xn )3∆/4 is large already for [s t, X0 ].

(IV.7)

Applying Proposition IV.2.6, we obtain that for every Y ∈ [s, X0 ] there is some n ∈ N some t ∈ [Y ][<∞] such that |t| ≥ φ(n) and some Z ≤ X0 such that (Xn )∆/4 is large for [s t, Z]. Now by (IV.7) one concludes that the set X(3∆/4, s, X0 , φ) is large for [X0 ]. Since this is an open set, by Lemma IV.5.1, we obtain that there is some Y ≤ X0 such that (X(3∆/4, s, X0 , φ))∆/4 is strategically large for [Y ]. This easily implies the desired result. We are now ready for the key result about the strategic Ramsey property. [∞]

Lemma IV.5.4. All Gδ -subsets of B1 (E) are strategically Ramsey. [∞] Proof. Suppose that X = ∞ n=0 Xn is a given Gδ -subset of B1 (E) written as the intersection of a decreasing sequence of open sets. We deﬁne Φ : X → NN by letting Φ(X)(n) = min{k : B(X k, 1/k)4 ⊆ Xn }. The mapping Φ is well deﬁned by our assumption that every Xn is an open subset [∞] of B1 (E). For σ ∈ N<∞ , let [σ] = {a ∈ NN : a |σ| = σ}, and let Yσ = Φ−1 [σ]. It is clear that for every σ ∈ N<∞ , Yσ =

∞

Yσn .

n=0 4 B((x , ..., x ), ε) n 1

= {(yn )n : ∀i ≤ k xi − yi < ε}.

(IV.8)

236 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Fix also a bijection ψ : N → N<∞ such that n ≥ max ψ(n) for every n. For σ ∈ N<∞ , let φσ : N → N be deﬁned by φσ (n) = ψ −1 (σ (n)). Suppose that X is large for [s, X]. We may assume that s = ∅ and that ∆ = (δn )n is decreasing. We assume also that for every n, δn < 1/C, where C is the basic constant of the ﬁxed Schauder basis (ei ) of E. For m ∈ N, let ∆m = ∆(2m − 1)/2m and Γm = (∆m − ∆m−1 )/4. [<∞]

(X) closed under concatenation. Let Let {sn }n be a countable (Γm )-net of B1 X0 be the result of the application of Proposition IV.2.7 to X and the family {((Yσ )∆m +2Γm+1 )sn : σ ∈ N<∞ , n, m ∈ N}. Now let X1 be the result of the application of Proposition IV.2.7 to X0 and the family {(Yσ )∆k (3Γm+1 , sn , X0 , φτ ) : σ, τ ∈ N<∞ , k, n, m ∈ N}. Claim. For every σ, τ ∈ N<∞ , k, n, m ∈ N, s ∈ [X][<∞] and every Y ≤ X1 , if (Yσ )∆k (Γm+1 , s, Y, φτ ) is strategically large for [Y ], then (Yσ )∆k (4Γm+1 , s, X1 , φτ ) is strategically large for [X1 ]. Proof of the claim. Suppose that (Yσ )∆k (Γm+1 , s, Y, φσ ) is strategically large for [Y ]. This means that player II has a strategy to produce some t, such that for some r ∈ N with |t| ≥ φσ (r) we have that (Yσr )∆k +Γm+1 is large for [s t, Y ]. Let sn = sn1 sn2 ∈ N be such that d(sn , s t) ≤ Γm . Then (Yσr )∆k +2Γm+1 is large for [sn , Y ]. Hence by the property of X0 , we get (Yσr )∆k +2Γm+1 is large for [sn1 sn2 , X0 ], which implies that (Yσr )∆k +3Γm+1 is large for [sn1 t, X0 ].

(IV.9)

So we have just shown that indeed player II has a winning strategy to produce t such that (IV.9) holds, or in other words, (Yσr )∆k )(3Γm+1 , sn1 , X0 , φσ ) is strategically large for [Y]

(IV.10)

By the property of X1 , we obtain from (IV.10), and the relationship X1 ≤ X0 , that (Yσr )∆k (3Γm+1 , sn1 , X1 , φσ ) is strategically large for [X1 ].

(IV.11)

IV.5. Topological Ramsey Theory of Block Sequences in Banach Spaces

237

So, (Yσr )∆k )(4Γm+1 , s, X1 , φσ ) is strategically large for [X1 ],

(IV.12)

as desired, thus ﬁnishing the proof of the claim. We now describe a winning strategy for player II in the game GX∆ (X1 ). It will consist of concatenations of inﬁnitely many strategies ofdiﬀerent games, all of them played in [X1 ]. We begin by noting that since X = n Yn is large for [X1 ][∞] , by Proposition IV.5.3, and the claim, we get that (Y∅ )∆0 (4Γ1 , ∅, X1 , φ∅ ) is strategically large for [X1 ]. So the player II can play according to a winning strategy τ0 for the game G(Y∅ )∆0 (4Γ1 ,∅,X1 ,φ∅ ) (X1 ) until he produces a ﬁnite block sequence t0 such that for some integer n0 the set (Yn0 )∆1 is large for [t0 , X0 ] and |t0 | ≥ φ∅ (n0 ). Since (Yn0 )∆1 =

∞

(Yn0 ,n )∆1 ,

n=0

by Proposition IV.5.3 and the claim, we know that (Yn0 )∆1 (4Γ2 , t0 , X1 , φ(n0 ) ) is strategically large for [X1 ]. The player II now continues playing according to a winning strategy τ1 for the game G(Yn0 )∆1 (4Γ2 ,t0 ,X1 ,φ(n0 ) ) (X1 ) until he produces a ﬁnite block sequence t1 > t0 such that for some integer n1 , the set (Yn0 ,n1 )∆2 is large for [t0 t1 , X1 ] and |t1 | ≥ φn0 (n1 ), and so on. We have to show that any sequence (yn ) = t0 t1 · · · produced by player II by playing according to this strategy belongs to X∆ . Fix (nk ) ∈ N ↑ such that for every k, we have that |tk | ≥ ψ −1 (n0 , . . . , nk ) and Yn0 ,...,nk )∆k+1 is large for [t0 · · · tk , X1 ]. For every k ﬁx Zk ∈ [t0 · · · tk , X1 ] and Wk ∈ Yn0 ,...,nk

(IV.13)

such that d(Zk , Wk ) ≤ ∆k+1 . Note that (IV.13) is equivalent to saying that (∀i ≤ k) B(Zk ni , 1/ni ) ⊆ Xi .

(IV.14)

Set rk = |t0 . . . tk | (k ∈ N). Note that for every k ∈ N and every k ≥ k d(Wk rk , t0 . . . tk ) ≤ ∆k+1

(IV.15)

238 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Set Wk = (wk,i )i (k ∈ N). We claim that (IV.15) implies that max supp yk+1

for every l ≥ k + 1 wl,k ∈ [ei ]0

.

(IV.16)

Otherwise, ﬁx l ≥ k + 1 such that for some i ∈ supp wl,k , we have that i > max supp yk+1 . Hence wl,k+1 > yk+1 , which implies that wl,k+1 − yk+1 ≥ 1/C, where C is the basic constant of (ei ). But then wl,k+1 − yk+1 > δk+1 , contradicting (IV.15). Fix a nonprincipal ultraﬁlter U on N. Note that the condition (IV.16) implies that the U-limit of the sequence (Wk ) exists. Set W∞ = U − lim Wk . It is clear also that d((yn ), W∞ ) ≤ ∆. Let us check that W∞ = (wn ) ∈ X: Fix k, and let k ≥ k be such that max{wi − wk ,i : i < nk } < 1/nk . Since B(Wk nk , 1/nk ) ⊆ Xk , (IV.17) implies that W∞ ∈ Xk .

(IV.17)

Our next goal is to show the corresponding result for the approximate Ramsey property for the Banach space c0 . As pointed out before, in this case there are no closed asymptotic pairs A but only discrete ones. Hence the mapping ΛA is [∞] not continuous but only of Baire-class-1, for a closed set C ⊆ BA (c0 ) × N ↑ the −1 corresponding set ΛA (C) is not a Gδ -set but an Fσδ -set, or more precisely, the [∞] intersection of a Gδ -set X and BA (c0 ). Hence the key result in this context is the [∞] approximate Ramsey property for all relatively Gδ -subsets of BA (c0 ) for some conveniently chosen discrete asymptotic pair A. This explains our next lemma. Lemma IV.5.5. Let A = A0 , A1 be the discrete asymptotic pair: Ai = {x ∈ S(c0 ) : (−1)i+1 e∗min suppx (x) > 0} (i = 0, 1) of the Banach space c0 . Then for every Gδ set X of block subsequences [∞] of c0 , the intersection X ∩ BA (c0 ) is approximately Ramsey. In particular, for [∞] every closed set C ⊆ B1 (c0 ) × N ↑ the set Λ−1 A (C) is approximately Ramsey. Proof. The proof will follow the proof Lemma IV.5.4 except that at places where the Proposition IV.2.7 (which fails if ‘large’ is replaced by ‘very large’) is used, we use instead the Claim A below. Let X = n Xn be a Gδ set of block sequences of c0 written as a decreasing intersection of open sets Xn . Suppose that X is large for some basic set [s, X] and that ∆ = (δn )n ∈ R∞ + is decreasing. We assume that s = ∅. For m ∈ N, let ∆m = ∆(2m − 1)/2m and Γm = (∆m − ∆m−1 )/4. Let ϕ : N → N2 be any bijection and let ϕi , i = 0, 1, be its two projections. Redeﬁne the mapping Φ : σ → N from the proof of Lemma IV.5.4 by letting Φ((xi )i )(n) = kn iﬀ

IV.5. Topological Ramsey Theory of Block Sequences in Banach Spaces

239

(a) B(X ϕ0 (kn ), 1/ϕ0 (kn )) ⊆ Xn . (b) e∗min supp xi (xi ) ≥ 1/ϕ1 (kn ) for every i in the set of the ﬁrst n-many j such that x2j+1 ∈ A1 The new condition (b) allows us to read some information out of a given [∞] block vector (xn )n ∈ BU (c0 ), such as, for example, which of its odd positions x2n+1 have the ﬁrst non zero coordinate positive (i.e., for which n do we have x2n+1 ∈ A1 ), and how large these coordinates are. It is clear that Φ is well deﬁned. For σ ∈ N<∞ , let [σ] = {a ∈ NN : a |σ| = σ}, and let Yσ = Φ−1 [σ]. Fix also a bijection ψ : N → N<∞ as in the proof of Lemma IV.5.4. Choose also a countable [<∞] ∆-net {sn }n of B1 (X) as in the proof of Lemma IV.5.4 with the additional property that {sn }n ∪ [(ei )k0 ][≤k] is ﬁnite for every k. Let X0 be the result of the application of Proposition IV.2.7 for the property of being large to X and the family {((Yσ )∆m +2Γm+1 )sn : σ ∈ N<∞ , n, m ∈ N}. Claim A. There is X1 = (xn ) ≤ X0 such that for every σ ∈ N<∞ and n ∈ N with min supp x|sn | −1 properties |sn | ≥ ψ −1 σ and sn ∈ [(ei )0 ], and every Y ≤ (xi )i≥|sn | , if (Yσ )∆|σ| (3Γ|σ|+1 , sn , X0 , φσ ) is very large for [Y ], then (Yσ )∆|σ| (3Γ|σ|+1 , sn , X0 , φσ ) is very large for [X1 ]. Proof of the claim. Set Zk = Yψ(k) (k ∈ N). Deﬁne a decreasing sequence Yi = (yji ) of inﬁnite normalized block subsequences of X0 such that for every i, every n such min supp y0i [<∞] that sn ∈ [(ek )0 ] , every k ≤ i, and every Y ≤ Yi , if (Zk )∆|ψ(k)| (3Γ|ψ(k)|+1 , sn , X0 , φψ(k) ) is very large for [Y ], then (Yσ )∆|ψ(k)| (3Γ|ψ(k)|+1 , sn , X0 , φψ(k) ) is very large for [Yi ]. It is not diﬃcult to show that the diagonal sequence X1 = (yii ) satisﬁes the conclusions. Fix X1 = (xn ) as in Claim A. Following the proof of the claim from the proof of Lemma IV.5.4, we get the following result. Claim B. For every σ ∈ N<∞ , every s ∈ [X][<∞] such that |s| ≥ ψ −1 σ and every Y ≤ (xi )i≥|sn | , if (Yσ )∆|σ| (Γ|σ|+1 , s, Y, φσ ) is very large for [Y ], then also (Yσ )∆|σ| (4Γ|σ|+1 , s, X1 , φσ ) is very large for [X1 ].

240 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Using again arguments analogous to those used in the proof of Lemma IV.5.4, we can show that for every Y ≤ X1 there are two sequences (nk ), (mk ) ∈ N ↑ such that mk ≥ ψ −1 (n0 , . . . , nk ) for every k, and such that (Yn0 ,...,nk )∆k+1 is large for [Y (m0 + · · · + mk ), X1 ].

(IV.18)

An argument analogous to the one used by the end of the proof of Lemma IV.5.4 shows that (IV.18) implies that there is some Y∞ ∈ X such that d(Y, Y∞ ) ≤ ∆. [∞] Moreover, by the new condition (b) on Φ, we have Y∞ ∈ BA (c0 ). This shows [∞] that [X1 ][∞] ⊆ (X ∩ BA (c0 ))∆ , as desired. Combining Lemmas IV.5.4 and IV.5.5 with Corollaries IV.4.11 and IV.4.13, we obtain the following result. Theorem IV.5.6 (Gowers). (a) For every Banach space E with a Schauder basis [∞] (en ), all analytic subsets of B1 (E) are strategically Ramsey. [∞] (b) Analytic subsets of B1 (c0 ) are approximately Ramsey. Exercise IV.5.7. Find a more direct proof of the previous theorem. (Hint: Given [∞] an analytic set X, ﬁx a continuous function f : N → B1 (E) such that f N = X, <∞ declare Yσ = f [σ] for σ ∈ N , and follow the proof of Lemma IV.5.4.)

IV.6

An Application to Rough Classiﬁcation of Banach Spaces

The most famous consequence of Theorem IV.5.6 is of course Gowers’ dichotomy presented above in the ﬁrst section of this chapter. This theorem however has some other interesting applications and the purpose of this section is to present a typical one. It involves a notion of minimality in the class of Banach spaces. Recall that an inﬁnite-dimensional Banach space E is minimal if every inﬁnitedimensional subspace of E has a further subspace isomorphic to E. Not every inﬁnite-dimensional Banach space has a minimal inﬁnite-dimensional subspace. A typical counterexample is the Tsirelson space. However, Tsirelson space has the following weaker minimality property so one can go on and investigate when a given Banach space contains a subspace with this weaker property. Deﬁnition IV.6.1. We say that two inﬁnite-dimensional Banach spaces E and F are totally incomparable if no inﬁnite-dimensional subspace of E is isomorphic to a subspace of F . A Banach space E is said to be quasi-minimal if it contains no pair of totally incomparable subspaces. Note that every hereditarily indecomposable Banach space is also an example of a quasi-minimal space. Since every inﬁnite-dimensional Banach space has an inﬁnite-dimensional subspace with a Schauder basis, in studying this notion we may restrict ourselves to Banach spaces E with a Schauder basis (en ).

IV.6. An Application to Rough Classiﬁcation of Banach Spaces

241

Proposition IV.6.2. Let E be a given Banach space with a Schauder basis (en ). (1) Let X and Y be two inﬁnite normalized block sequences of E. Then their closed linear spans are totally incomparable iﬀ there are no inﬁnite normalized block subsequences X ≤ X and Y ≤ Y such that the closed linear spans of X and Y are isomorphic. (2) The space E is quasi-minimal iﬀ for every pair of inﬁnite normalized block sequences X and Y of E there exist inﬁnite normalized block subsequences X ≤ X and Y ≤ Y such that the closed linear spans of X and Y are isomorphic. Proof. By Theorem IV.1.1 for every inﬁnite-dimensional closed linear subspace F of E there is an inﬁnite normalized block sequence X and a closed subspace F of F such that the closed linear span of X is isomorphic to F . When F is spanned by an inﬁnite normalized sequence Y the subspace F can be chosen to be spanned by an inﬁnite normalized block subsequence of Y . From these facts, the conclusions (1) and (2) are immediate. Let us say that two block subspaces E0 and E1 of E are disjointly supported if the support (relative the ﬁxed Schauder basis (en ) of E) of every vector of E0 is disjoint from the support of every vector of E1 . The following lemma hints towards an alternative of quasi-minimality which involves this notion. Lemma IV.6.3 (Casazza). Suppose that a given space E with a Schauder basis is isomorphic to a proper subspace of itself. Then E contains two equivalent inﬁnite block sequences (xn ) and (yn ) such that xn < yn < xn+1 for all n. Proof. Let (en )∞ n=0 be a ﬁxed Schauder basis of E. Let T : E → E be an isomorphism between E and its proper subspace. Perturbing T if necessary we may assume that T (en ) is a ﬁnitely supported vector for all n. Since the range of T is contained in some subspace of E of codimension one and since all such subspaces of E are isomorphic, we may assume that the range of T is contained in the closed linear span of e1 , e2 ,... ,i.e., the subspace that does not involve the basis vector e0 . For k ∈ N, let En denote the linear span of e0 , ..., ek . Note that by our assumption, dim(Ek ∩ T Ek ) < dim(T Ek ) for all k. Let x0 = e0 . Note that x0 < T (x0 ). Since T (x0 ) is ﬁnitely supported there is an integer k0 such that T (x0 ) ∈ Ek0 . Since dim(Ek0 ∩ T Ek0 ) < dim(T Ek0 ) there is a norm-one vector x1 ∈ Ek0 such that T (x1 ) > k0 and therefore x1 < T (x1 ). Clearly we can continue and produce an inﬁnite sequence (xn ) such that xn < T (xn ) < T (xn+1 ) for all n. By the Bessaga–Pelczynski argument (see Theorem IV.1.1 above), the sequence (xn ) can be reﬁned to a subsequence (xni ) which is equivalent to a block basic sequence (xni ) such that xni < T (xni ) for all i. This gives the conclusion of the lemma. Theorem IV.6.4 (Gowers). Let E be a Banach space with a Schauder basis (en ). Then either E has a quasi-minimal subspace or E has a block subspace F such

242 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces that no two disjointly supported subspaces of F are isomorphic, and therefore, every pair of two disjointly supported subspaces of F are totaly incomparable. Proof. Let us assume that that any block subspace F of E contains a pair of disjointly supported block subspaces that are isomorphic. For the notational convenience during the course of this proof, we shall identify the family N[∞] of inﬁnite subsets of N with the family of all inﬁnite increasing sequences σ = (σn )n of non-negative integers such that σ0 = 0. Given a pair of ﬁnite or inﬁnite normalized block sequences X = (xn )n and Y = (yn )n of E and a positive constant C, we write X ∼C Y if there is an isomorphism T from the closed linear span of X onto the closed linear span of Y such that T (xn ) = yn for all n and such that T . T −1 ≤ C. We write X ∼ Y whenever X ∼C Y for some constant C > 0. Consider the following set of normalized block sequences of E: [∞]

X = {(xn )n ∈ B1 (E) : ∃σ ∈ N[∞] ∃M ∈ R+ ∃(yn )n ∼M (zn )n σ

σ

n+1 [1] n+1 [1] ∀n yn ∈ [(x2k+1 )k=σ ] , zn ∈ [(x2k )k=σ ] }. n n

[∞]

Note that X is an analytic subset of B1 (E) and therefore strategically Ramsey by Theorem IV.5.6. Using our assumption that every block subspace of E contains two disjointly supported block subspaces that are isomorphic, it is not diﬃcult to show that X is large. Let C be the basis constant of (en ) and chose a summable 2 ∆ ∈ R∞ + , let d be its sum and let K = (1 + 2Cd) . Then X ∼K Y for every pair [∞] X, Y ∈ B1 (E) such that d(X, Y ) ≤ ∆. Since X is large, by Theorem IV.5.6, there is some Y such that (X)∆ is strategically large for [Y ][∞] . We claim that the closed linear span of Y is quasi-minimal. We shall use the criterion of quasiminimality given in Proposition IV.6.2. Consider a pair of inﬁnite block sequences Z0 , Z1 ≤ Y . Let Z be an inﬁnite block subsequence of Y given by the strategy of player II in the game G(X)∆ (Y ), when player I plays alternatively between Z0 ∈ X such that d(Z, Z) ≤ ∆. Then and Z1 . Then Z ∈ (X)∆ . So, there exist Z Z ∼K Z and so we let T denote an operator witnessing this equivalence. Chose = (w = ( and a a sequence (σn )n ∈ N[∞] , a block sequences W n ), U un ) ≤ Z, positive constant M witnessing the membership Z ∈ X. Note that, in particular, ∼M U . Let W, U ≤ Z be inﬁnite block sequences given we have the equivalence W ) and U = T (U ). Then W ∼ W and U ∼ U since Z ∼ Z. Note also by W = T (W that wn ∈ Z1 and un ∈ Z0 for all n. But (wn )n ∼ (w n )n ∼ ( un )n ∼ (un )n , and we are done. Corollary IV.6.5 (Gowers). Every inﬁnite-dimensional Banach space E has an inﬁnite-dimensional subspace F such that F is either quasi-minimal or no subspace of F is isomorphic to any proper subspace of itself. We ﬁnish this section by mentioning the following interesting result whose proof involves several ideas exposited above including Gowers’ result about quasiminimal spaces.

IV.7. An Analytic Set whose Complement is not Approximately Ramsey

243

Theorem IV.6.6 (Rosendal). Let E be an inﬁnite-dimensional Banach space. Then E contains either a minimal subspace or continuum many pairwise incomparable subspaces.

IV.7

An Analytic Set whose Complement is not Approximately Ramsey

It is well-known that using the axiom of choice one is likely to ﬁnd a set that fails to have a given regularity property. The classical such use is the construction of a set of reals that is not Lebesgue measurable. After the discovery of G¨ odel’s universe of constructible sets it has been realized that the corresponding pathological sets can be constructed even on a very low level of Luzin’s hierarchy of projective sets provided of course one is willing to conﬁne oneself to the constructible subuniverse. All these constructions are based on a particularly nice well-ordering of the reals. Deﬁnition IV.7.1. A Σ12 -good well ordering of NN is a Σ12 -relation <w ⊆ NN × NN that well-orders NN in order type ω1 in such a way that the relation R ⊆ NN × NN deﬁned by letting (x, y) ∈ R iﬀ {z ∈ NN : z <w y} = {(x)i : i ∈ N}5 is also Σ12 . Recall the classical fact essentially due to G¨odel (see [40]) that in the constructible subuniverse L there is a Σ12 -good well-ordering of NN . The point of this deﬁnition is that most of the known constructions of irregular sets (such as for example a non Lebesgue measurable set) that use a well-ordering of R would give us a Σ12 irregular set provided we start with a Σ12 -good well-ordering. We shall now see that this is indeed also the case with a natural approach of constructing a set of normalized block-sequences that is not strategically Ramsey by diagonalizing over a well-ordering of all strategies for one of the players. There is however a slight problem with this approach since clearly there are more than continuum many strategies. It is for this reason that we need to reformulate the game GX (Y ) so that the set of strategies is in a natural way equinumerous with NN allowing us to transfer the Σ12 -good well ordering. [∞]

[∞]

Deﬁnition IV.7.2. Given Y ∈ B1 (E) and X ⊆ B1 , let GfX (Y ) be the inﬁnite perfect-information game between two players I and II played as follows: • I plays a norm-one vector y0 of the linear span of Y . • II responds by playing either a norm-one vector z0 in the linear span of {y0 }, or by playing z0 = 0 meaning that II chooses not to play any vector at the moment. 5 For

x ∈ NN and i ∈ N we let (x)i be deﬁned by (x)i (j) = x(2i (2j + 1)).

244 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces • I plays another norm-one vector z1 of the linear span of Y . • II responds by playing either a norm-one vector z1 in the linear span of {y0 , y1 }, or by playing z1 = 0 meaning that II chooses not to play any vector at the moment, and so on. • At stage k player I plays a norm-one vector yk in the linear span of Y and II responds either by ﬁnding a maximal place l < k where he played a vector and then choosing a norm-one vector zk in the linear span of {yl+1 , ..., yk }, or by playing zk = 0. We require that, if in a given interval [l, m] of integers the player II chooses never to play a vector, then the vectors yk (l ≤ k ≤ m) picked by I must form a block subsequence of Y. We also require that if zl and zk are vectors played by II and if l < k then zl < zk . After inﬁnitely many steps, we let (zki )i be the increasing enumeration of vectors played by II and we say that II wins the play if this block subsequence of Y belongs to X. Otherwise (i.e., if II does not produce an inﬁnite sequence, or if (zki )i ∈ / X) we say that I wins the inﬁnite play of the game. Thus, while at the ﬁrst step of the new game GfX (Y ) the player I cannot play an inﬁnite block subsequence Y0 = (yn0 ) of Y as he could in GX (Y ), he can play instead y00 , y10 , . . . , yk00 until the ﬁrst place k0 when II plays a norm-one vector zk0 belonging to the linear span of {y00 , y10 , . . . , yk00 } (and therefore to the linear span of Y0 ). After that, while I cannot play an inﬁnite block subsequence Y1 = (yn1 ) of Y as he could have done this in the game GX (Y ), he can instead play vectors y01 , y11 , . . . , yk11 as long as II allows him to do this , i.e., until the place k1 when II plays a vector, and so on. In other words, while the two games GX (Y ) and GfX (Y ) look diﬀerent, they are really just two diﬀerent formulations of the same game, and so we have the following result. Proposition IV.7.3. The player I has a winning strategy in the game GfX (Y ) iﬀ the player I has a winning strategy in the game GX (Y ). Similarly, the player II has a winning strategy in the game GfX (Y ) iﬀ the player II has a winning strategy in the game GX (Y ). The point of replacing the game GX (Y ) by the new game GfX (Y ) is that the sets of strategies for any of the players in the new game can naturally be injected into NN , so well-orderings of NN naturally transfer to well-orderings of the sets of strategies. So we are in a situation to prove the following result. Theorem IV.7.4 (Bagaria-Lopez Abad, Gowers). If there is a Σ12 -good well ordering of NN , then for every inﬁnite dimensional Banach space E there is a Σ12 -subset X [∞] of B1 (E) which is not strategically Ramsey. Under the same assumption, there [∞] is also a Σ12 -subset X of B1 (c0 ) which is not approximately Ramsey. Proof. We shall in fact prove a stronger result: For every ∆ = (δn )n ∈ R∞ + such that δn < 1 for all n there is a Σ12 set of normalized block sequences X such that

IV.7. An Analytic Set whose Complement is not Approximately Ramsey

245

for every block subsequence Y, the player I has no winning strategy in the game GfX (Y ) and the player II has no winning strategy in the game Gf(X)∆ (Y ). Fix a Σ12 -good well-ordering <w of NN . We have already observed that the [∞] union of B1 (E) and the two sets SI = {(Y, σ) : σ is a strategy of I in GfY (Y ) for some Y} and SII = {(Y, σ) : σ is a strategy of I in GfY (Y ) for some Y} can naturally be injected into NN , so the well-ordering <w induces the corresponding well-ordering <s on the union. Recursively on <s we chose two assignments {ZI (Y, σ) : (Y, σ) ∈ SI } and {ZII (Y, σ) : (Y, σ) ∈ SII } such that: (1) If (Y, σ) ∈ SI , i.e., σ is a strategy for the player I in the game GfY (Y ) for some set Y of inﬁnite block sequences, then ZI (Y, σ) is the <s -least block subsequence Z = (zn ) of Y with the following two properties: (1.a) There is an inﬁnite play of GfY (Y ) in which I plays according to σ and the player II produces the inﬁnite block sequence Z = (zn ), i.e., zn ∈ [y0n , . . . , yknn ], where (k0 )

yin

(kn−1 )

(i)

! ! ! = σ(0, . . . , 0, z0 , . . . , 0, . . . , 0, zn−1 , 0, . . . , 0).

(1.b) d(ZI (Y , σ ), X) ≤ ∆ and d(ZII (Y , σ ), X) ≤ ∆ for every (Y , σ ) <s (Y, σ). (2) If (Y, σ) ∈ SII , i.e., if σ is a strategy for player II in the game GfY (Y ) for some set Y of block sequences, then ZII (Y, σ) is the <s -least inﬁnite block sequence Z = (zn ) with the following two properties: (2.a) There is an inﬁnite run of GfY (Y ) in which player II plays according to σ and at the end produces the block sequence Z = (zn ). (2.b) d(ZII (Y , σ ), Z) ≤ ∆ and d(ZI (Y , σ ), X) ≤ ∆ for every (Y , σ ) <s (Y, σ). Since every (Y, σ) has only countably many predecessors, a simple diagonalisation procedure shows that there are always block sequences satisfying (1) or (2) depending whether (Y, σ) ∈ SI or (Y, σ) ∈ SII , respectively. Let X = {ZI (Y, σ) : (Y, σ) ∈ SI }. The fact that <s is obtained in a natural way from <w which is a Σ12 -good wellordering guarantees that X is a Σ12 set. Consider a strategy σ for player I in the game GfX (Y ) for some inﬁnite block sequence Y . Then by (1) the Player II can play against σ and win by producing ZI (Y, σ) ∈ X. So, the strategy σ is not winning for I in the game GfX (Y ).

246 Chapter IV. Approximate and Strategic Ramsey Theory of Banach Spaces Suppose now that σ is a strategy for II in the game Gf(X)∆ (Y ) for some inﬁnite block sequence Y . Then, by deﬁnition of ZII (Y, σ) in case (2), there is a run of the game in which II uses σ and produces the set ZII (Y, σ). We show that Z = ZII (Y, σ) is not in X∆ . Consider an arbitrary Z ∈ X and choose (Y , σ ) ∈ SI such that Z = ZI (Y , S ). If (Y , σ ) <s (Y, σ) then Z was chosen so that d(Z, Z ) ≤ ∆. Otherwise, (Y, σ) <s (Y , σ ), and hence Z was chosen so that d(Z , Z) ≤ ∆. Since Z was an arbitrary member of X this shows that the set ZII (Y, S) does not belong to X∆ . This proves that σ is not a winning strategy for player II in the game Gf(X)∆ (Y ). In conclusion, we have just produced a Σ12 set X of inﬁnite normalized block sequences of the Banach space E such that there is no inﬁnite normalized block sequence Y such that the player I has a winning strategy in the game Gf(X) (Y ) [∞]

(and so in particular [Y ][∞] ∩X is nonempty for every Y ∈ B1 (E)) and also there is no such Y for which the player II has a winning strategy in Gf(X)∆ (Y ) (and so [∞]

in particular [Y ][∞] X∆ for every Y ∈ B1 (E)). It follows that X is neither approximately nor strategically Ramsey. Combining this and Theorem IV.4.14 we obtain the following result. Theorem IV.7.5 (Lopez–Abad). If there is a Σ12 -good well ordering of NN , then for every inﬁnite-dimensional Banach space E there is a coanalytic subset X of [∞] B1 (E) which is not strategically Ramsey. Under the same assumption, there is [∞] also a coanalytic subset of B1 (c0 ) which is not approximately Ramsey. So, in particular, the classes of approximately and strategically Ramsey sets are in general not closed under the operation of complementation, and so that these notion behave quite diﬀerently from the classical notion of Ramsey sets. Going to G¨ odel’s constructible subuniverse to obtain this non-complementarity result is in some sense necessary as it is shown in [9] (see also [49]) that under some large cardinal assumptions all projective (or more generally ‘reasonably’ deﬁnable) families of inﬁnite normalized block sequences of an arbitrary Banach space E with a Schauder basis (en ) are strategically Ramsey. We ﬁnish this section and this chapter with the following problem that asks for yet another picture of the class of approximate and strategic Ramsey sets diﬀerent from the one given by G¨odel’s constructible universe. Problem IV.7.6 (Gowers). Does the axiom of determinacy imply that all sets of inﬁnite normalized block sequences are strategically Ramsey? Remark IV.7.7. The material of this chapter is based on sources [35], [36], [56], [9], [10], and [49] where some alternative proofs of these results can be found. The reader is referred to these sources also for a wealth of additional information about the approximate and strategic Ramsey theory that might lie outside the scope of the present notes.

Bibliography [1] A. Alexiewicz, On sequences of operations. II. Studia Math. 11 (1950), 200– 236. [2] D. E. Alspach, S. A. Argyros, Complexity of weakly null sequences. Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44 pp. [3] D. E. Alspach, E. Odell, Averaging weakly null sequences. Functional analysis (Austin, TX, 1986–87), 126–144, Lecture Notes in Math., 1332, Springer, Berlin, 1988. [4] S. A. Argyros, I. Deliyanni, Examples of asymptotic l1 Banach spaces. Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. [5] S. A. Argyros, I. Gasparis, Unconditional structures of weakly null sequences. Trans. Amer. Math. Soc. 353 (2001), no. 5, 2019–2058 [6] S. A. Argyros, G. Godefroy, H. P. Rosenthal, Descriptive set theory and Banach spaces. Handbook of the geometry of Banach spaces, Vol. 2, 1007–1069, North-Holland, Amsterdam, 2003. [7] S. A. Argyros, S. Mercourakis, A. Tsarpalias, Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107 (1998), 157–193. [8] A. D. Arvantakis, Weakly null sequences with an unconditional subsequence. Preprint 2004. [9] J. Bagaria, J. Lopez-Abad, Weakly Ramsey sets in Banach spaces. Advances in Mathematics 160 (2001), 133–174. [10] J. Bagaria, J. Lopez-Abad, Determinacy and weakly Ramsey sets in Banach spaces. Trans. Amer. Math. Soc. 354 (2002), no. 4, 1327–1349 [11] B. Beauzamy, Banach-Saks properties and spreading models. Math. Scand. 44 (1979), no. 2, 357–384. [12] S. Bellenot, The Banach space T and the fast growing hierarchy from logic, Israel J. Math. 47 (1984), 305–313.

248

Bibliography

[13] Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. A.M.S. Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000. [14] C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math., 42 (1958), 151–164. [15] A. Borichev, R. Deville, E. Matheron, Strongly sequentially continuous functions. Quaest. Math. 24 (2001), no. 4, 535–548. [16] A. Brunel, L. Sucheston, On B-convex Banach spaces, Math. System Theory, 7 (1974), 294–299. [17] J. Burzyk, C. Kli´s, Z. Lipecki, On metrizable abelian groups with a completeness-type property. Colloq. Math. 49 (1984), no. 1, 33–39. [18] P. G. Casazza, T. J. Shura, Tsirelson’s space. Lecture Notes in Mathematics, 1363. Springer-Verlag, Berlin, 1989. [19] R. Deville, E. Matheron, Pyramidal vectors and smooth functions on Banach spaces. Proc. Amer. Math. Soc. 128 (2000), no. 12, 3601–3608 [20] J. Diestel, Sequences and series in Banach spaces. Graduate Texts in Mathematics, 92. Springer-Verlag, New York, 1984. [21] A. Dvoretzky, Some results on convex bodies and Banach spaces. 1961 Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), 123–160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford. [22] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic, 39 (1974), 163–165. [23] J. Elton, Thesis, Yale University, New Haven, CT. [24] J. Elton, E. Odell, The unit ball of every inﬁnite-dimensional normed linear space contains a (1 + ε)-separated sequence. Colloq. Math. 44 (1981), no. 1, 105–109. [25] P. Erd¨ os and M. Magidor, A note on regular methods of summability and the Banach-Saks property, Proc. Amer. Math. Soc., 59 (1970), 232–234. [26] P. Erd¨ os and R. Rado, A combinatorial theorem, J. London Math. Soc., 25 (1950), 249–255. [27] J. Farahat, Espaces de Banach contenant l1 , d’apr`es H. P. Rosenthal. Espaces Lp , applications radoniﬁantes et g´eom´etrie des espaces de Banach, Exp. No. ´ 26, 6 pp. Centre de Math., Ecole Polytech., Paris, 1974. [28] V. Farmaki, Systems of Ramsey families. Atti Sem. Mat. Fis. Univ. Modena 50 (2002), no. 2, 363–379. [29] V. Ferenczi and C. Rosendal, On the number of non-isomorphic subspaces of a Banach space, Studia Math., to appear.

Bibliography

249

[30] T. Figiel , R. Frankiewicz , R. Komorowski and C. Ryll-Nardzewski, On hereditarily indecomposable Banach spaces Ann. Pure Appl. Logic 126 (2004), 293–299. [31] F. Galvin, A generalization of Ramsey’s Theorem, Notices Amer. Math. Soc., 15 (1968), Abstract 68T-368. [32] F. Galvin, K. Prikry, Borel sets and Ramsey’s Theorem, J. Symbolic Logic, 38 (1973), 193–198. [33] I. Gasparis, E. Odell, and B. Wahl, Weakly null sequences in the Banach space C(K). ArXiv 2004. [34] W. T. Gowers, Lipschitz Function on Classical Spaces, European J. Combinatorics, 13 (1992), 141–151. [35] W. T. Gowers, An Inﬁnite Ramsey Theorem and Some Banach-Space Dichotomies, Ann. Math. 156 (2002), 797–833. [36] W. T. Gowers, Ramsey methods in Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, 1071–1097, North-Holland, Amsterdam, 2003. [37] S. Guerre-Delabri`ere, Classical sequences in Banach spaces. Monographs and Textbooks in Pure and Applied Mathematics, 166. Marcel Dekker, Inc., New York, 1992. [38] N. Hindman, Finite sums from sequences within cells on a partition of N, J. Combinatorial Theory(A), 17 (1974), 1–11. [39] W. Just, A. R. D. Mathias, K. Prikry, P. Simon, On the existence of large p-ideals. J. Symbolic Logic 55 (1990), no. 2, 457–465. [40] A. Kanamori, The higher inﬁnite. Large cardinals in set theory from their beginnings. Second edition. Springer Monographs in Mathematics. SpringerVerlag, Berlin, 2003. [41] V. Kanellopoulos, A proof of Gowers’ c0 theorem, Proc. Amer. Math. Soc., to appear. [42] A.S. Kechris Classical descriptive set theory, Springer-Verlag 1995. [43] O. Klein and O. Spinas, Canonical form of Borel functions on the Milliken’s space, preprint 2002. [44] J. L. Krivine, Sous-espaces de dimension ﬁnie des espaces de Banach r´eticul´es. Ann. of Math. (2) 104 (1976), no. 1, 1–29. [45] R. Laver, On Fra¨ıss´e’s order type conjecture, Ann. Math. 93 (1971), 89–111. [46] R. Laver, Well-quasi-orderings and sets of ﬁnite sequences. Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 1–10.

250

Bibliography

[47] R. Laver, Better-quasi-orderings and a class of trees. Studies in foundations and combinatorics, 31–48, Adv. in Math. Suppl. Stud., 1, Academic Press, New York-London, 1978. [48] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Springer-Verlag, 1977. [49] J. Lopez-Abad, Coding into Ramsey sets, CRM Preprint 557. [50] J. Lopez-Abad, Canonical equivalence relations on nets of P Sc0 , CRM Prepint 558. [51] J. Lopez-Abad, A. Manoussakis, On Tsirelson type spaces. In preparation 2004. [52] J. Lopez-Abad, S. Todorcevic, Pre-compact families of ﬁnite sets and unconditional basic sequences. Preprint 2004. [53] E. Matheron, A useful lemma concerning subseries convergence. Bull. Austral. Math. Soc. 63 (2001), no. 2, 273–277. [54] A. R. D. Mathias, A remark on rare ﬁlters. Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975. [55] A. R. D. Mathias, Happy families. Ann. Math. Logic 12 (1977), no. 1, 59–111. [56] B. Maurey, A note on Gowers’ dichotomy theorem, Convex Geom. Analysis, MSRI Publications, Vol 34 (1998), 149–157. [57] B. Maurey, Type, cotype and K-convexity. Handbook of the geometry of Banach spaces, Vol. 2, 1299–1332, North-Holland, Amsterdam, 2003. [58] B. Maurey, H.P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math., 61 (1977), 77–98. [59] S. Mazur, W. Orlicz, Sur les espaces m´etriques lin´eaires. II. Studia Math. 13, (1953). 137–179. [60] K. Milliken, Ramsey’s theorem with Sums and Unions, J. Combinatorial Theory (A), 18 (1975), 276–290. [61] V. D. Milman, G. Schechtman, Asymptotic theory of ﬁnite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. [62] E. C. Milner, Basic wqo- and bqo-theory. Graphs and order (Banﬀ, Alta., 1984), 487–502, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 147, Reidel, Dordrecht, 1985. [63] C. St. J. A. Nash-Williams, On well quasi-ordering transﬁnite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39.

Bibliography

251

[64] C. St. J. A. Nash-Williams, On better-quasi ordering transﬁnite sequences, Proc. Cambridge Philos. Soc. 64 (1968), 273–290. [65] C. St. J. A. Nash-Williams, On well quasi-ordering inﬁnite trees, Proc. Cambridge Philos. Soc. 64 (1968). [66] E. Odell, Applications of Ramsey theorems to Banach space theory. Notes in Banach spaces, 379–404, Univ. Texas Press, Austin, Tex., 1980. [67] E. Odell, On Schreier unconditional sequences. Banach spaces (M´erida, 1992), 197–201, Contemp. Math., 144, Amer. Math. Soc., Providence, RI, 1993. [68] E. Odell, On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic. Analysis and logic (Mons, 1997), 189–267, London Math. Soc. Lecture Note Ser., 262, Cambridge Univ. Press, Cambridge, 2002. [69] E. Odell, T. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281. [70] E. Odell, N. Tomczak-Jaegermann, R. Wagner, Proximity to l1 and distortion in asymptotic l1 spaces. J. Funct. Anal. 150 (1997), no. 1, 101–145. [71] J. Pawlikowski, Parametrized Ellentuck theorem. Topology Appl. 37 (1990), no. 1, 65–73. [72] M. Pouzet, Sur les pr´emeillieurordres, Ann. Inst. Fourier (Grenoble) 22 (1972), 1–19. [73] H. J. Pr¨ omel; B. Voigt, Canonical forms of Borel-measurable mappings ∆ : [ω]ω → R. J. Combin. Theory Ser. A 40 (1985), no. 2, 409–417. [74] V. Pt´ ak, A combinatorial theorem on system of inequalities and its applications to analysis, Czech. Math. J., 84 (1959), 629–630. [75] P. Pudlak and V. R¨ odl, Partition theorems for systems of ﬁnite subsets of integers, Discrete Math., 39 (1982), 67–73. [76] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 30 (1929), 264–286. [77] C. Rosendal, Incomparable, non isomorphic and minimal Banach spaces, preprint 2004. [78] H. P. Rosenthal, A characterization of Banach spaces containing l1 . Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. [79] H. P. Rosenthal, On a theorem of J. L. Krivine concerning block ﬁnite representability of lp in general Banach spaces. J. Funct. Anal. 28 (1978), no. 2, 197–225. [80] H. P. Rosenthal, A characterization of Banach spaces containing c0 . J. Amer. Math. Soc. 7 (1994), no. 3, 707–748.

252

Bibliography

[81] H. P. Rosenthal, The Banach spaces C(K). Handbook of the geometry of Banach spaces, Vol. 2, 1547–1602, North-Holland, Amsterdam, 2003. [82] T. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math., 76 (1991), 81–95. [83] J. Schreier, Ein Gegenbeispiel zur Theorie der Schwachen Konvergenz, Studia Math., 2 (1930), 58–62. [84] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic, 35 (1970), 60–64. [85] S. G. Simpson, Bqo-theory and Fra¨ıs´e’s conjecture, in Recursive aspects of descriptive set theory, R. Mansﬁeld and G. Weikkamp, Oxford Univ. Press. 1985. [86] J. Stern, A Ramsey theorem for trees, with an application to Banach spaces, Israel. J. Math. 29 (1978), 179–188. [87] A. D. Taylor, A canonical Partition Relation for ﬁnite Subsets of ω, J. Combinatorial Theory (A), 21 (1970), 137–146. [88] S. Todorcevic, Topics in topology. Lecture Notes in Mathematics, 1652. Springer-Verlag, Berlin, 1997. [89] S. Todorcevic, An Introduction to Ramsey Spaces, in preparation. [90] N. Tomczak-Jaegermann, Banach spaces of type p have arbitrarily distortable subspaces. Geom. Funct. Anal. 6 (1996), no. 6, 1074–1082 [91] B. S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl., 8 (1974), 81–95.

Index FIN[<∞] , 201 FIN[∞] , 199 FIN[k] , 199 (0, j)-dependent sequence, 87 (1, j)-dependent sequence, 76 (θ, C, j) exact pair, 36 (eα )α<κ , 7 2j + 1-special sequence, 74 C-equivalent basic sequences, 131 C-spreading basic sequence, 131 C − ck0 vector, 44 C − k1 average, 33 Q-array bad, 181 minimal bad, 181 Q-tree, 145 W (M, θ), 9 W (M, θ), 9 Wκ [G], 22 Wκ [G], 23 ∆-net, 223 α-extrauniform family, 160 α-uniform family, 153 FIN, 197 γFIN, 197 λ-distortable Banach space, 36 M-admissible ﬁnite family, 9 ran x, 7 -function, 72 152, 156 B, c00 (κ), 7 n2j−1 -special sequence, 40 p-closed, 73 1-subsymmetric transﬁnite basis, 99 accepts M accepts s, 154 Y accepts s, 203 Z accepts (x, y), 219

admissible ﬁnite family, 9 antichain, 106 maximal, 106 regular, 106 arbitrarily distortable, 36 arbitrarily distortable space, 209 array bad, 144 minimal bad, 144 assignment, 108 canonical, 109 asymptotic pair, 229 discrete, 230 separated, 230 set, 210, 229 asymptotic 1 space, 134 asymptotically equivalent, 94 ﬁner, 94 asymptotically spreading basic sequence, 131 automatic continuity principle, 193 average, 33 uniformly bounded, 44 bad Q-array, 181 minimal, 181 bad array, 144 minimal, 144 Banach space λ-distortable, 36 arbitrarily distortable, 36, 209 distortable, 209 ﬁnitely interval representable, 83 ﬁnitely representable, 135 HI, 39 quasi-reﬂexive, 57 Banach–Saks property, 188

254 barrier, 147, 202 dominates, 156 ﬁnite rank, 174 Schreier, 152 basic inequality, 27 basic sequence C-spreading, 131 C-equivalent, 131 spreading model, 130 asymptotically spreading, 131 unconditional, 130 basis Schauder, 130 bimonotone transﬁnite , 99 boundedly complete transﬁnite, 102 constant, 130 transﬁnite, 99 basis constant unconditional, 130 better-quasi-ordered set (b.q.o.), 146 block sequence, 132, 197 subsequence, 132 block subsequence, transﬁnite, 100 canonical assignment, 109 decomposition, 84 canonical equivalence relation, 166 canonical projections, 130 Cantor–Bendixson derivative, 14 index of the family M (i(M)), 9 catcher, 106 Ces`aro-summable sequence, 188 codes the pair, 230 coding function, 40 collection of ﬁnite convex means, 162 combinatorial forcing, 148, 149, 154, 219 convergent, 180 critical set, 93

Index decided, ϕ is, 200 decides M decides s, 154 Y decides s, 203 Z decides (x, y), 219 dense below, 148 dense-open-set assignment, 148 dependent sequence, 76 diagonal step operator, 90 discrete, asymptotic pair, 230 disjointly supported vectors, 219 distortable Banach space, 15, 36 distortable, space, 209 dominates, barrier, 156 Ellentuck topology, 185 equation, 200 equivalence relation canonical, 166 exact pair, 36 extensible, strongly, 149 extensor, 148 false, ϕ is, 200 family α-uniform on Y, 204 hereditary, 157 pre-compact, 157 Schreier, 152 spreading, 153 family F ⊆ FIN[<∞] Ramsey, 202 Sperner, 202 thin, 202 family of ﬁnite subsets of N α-extrauniform, 160 α-uniform, 153 pre-compact, 157 Ramsey, 148 Sperner, 148 spreading, 159 thin, 148 ﬁltration, 108 ﬁnite convex means, MFIN , 162

Index ﬁnite family M-admissible , 9 ﬁnite rank, 174 ﬁnitely interval representable, 83 ﬁnitely representable Banach space, 135 front, 147 function oscillation stable, 210 functional special, 176 ground norm, 21 set, 21 Hereditarily Indecomposable Banach space, 39 hereditarily indecomposable space, 218 hereditary family, 157 indecomposable limit ordinal, 152 independence, 179 inextensible, 149 James-like space, 81 large for, 222 limit ordinal, indecomposable, 152 Maurey–Rosenthal space, XM R , 176 maximal antichain, 106 regular antichain, 106 regular array, 106 Mazur map, 209 Mazur–Orlicz complete topological group, 195 unconditionally complete topological group, 195 Milliken’s topology, 206 minimal bad Q-array, 181 array, 144 mixed Tsirelson norm, 15 space, 15

255 Nikodym boundedness principle, 194 norm ground, 21 mixed Tsirelson, 15 Tsirelson, 8 open set, 148 operator diagonal step, 90 strictly singular, 100 oscillation stable function, 210 pair asymptotic, 229 parametrized perfect-set theorem, 207 partial ranking, 181 pre-compact family, 157 projection, trivial, 39 property Banach–Saks, 188 property, tree-like, 72 quasi-order (q.o.), 143 well-founded , 144 quasi-reﬂexive Banach space, 57 Ramsey family F ⊆ FIN[<∞] , 202 family of ﬁnite subsets of N, 148 set in FIN[∞] , 206 set in N[∞] , 177 strategically, 222 ultraﬁlter on N, 128 Ramsey-null set in FIN[∞] , 206 in N[∞] , 185 ranking, partial, 181 rapidly increasing sequence (R.I.S.), 26 reﬂexive extension, 35 regular antichain, 106 array, 106 regular method of summability, 188

256 rejects M rejects s, 154 Y rejects s, 203 Z rejects (x, y), 219 relevant node, 114 restriction, 157, 176 rooted tree, 145 scattered linearly ordered set, 184 Schauder basis, 130 canonical projections of a, 130 Schlumprecht space, 15 Schreier barrier, 152 family, 152 space, 134 Schreier family, 9, 10 Schur’s 1 -theorem, 193 separated, asymptotic pair, 230 sequence n2j−1 -special, 40 block, 132, 197 Ces`aro-summable, 188 codes a pair, 230 dependent, 76 independence, 179 special, 176 unconditional basic, 169 sequence of pairs of disjoint subsets convergent, 180 series, unconditionally convergent, 192 set asymptotic, 210, 229 better-quasi-ordered (b.q.o.), 146 critical, 93 ground, 21 open, 148 quasi-ordered (q.o.), 143 Ramsey, 177, 206 Ramsey-null, 185, 206 scattered, 184 well-quasi-ordered (w.q.o.), 143 shrinking transﬁnite basis, 102 small subset, 203

Index space ε-H.I., 218 arbitrarily distortable, 209 asymptotic 1 , 134 hereditarily indecomposable, 218 James-like, 81 Maurey–Rosenthal, XM R , 176 Schlumprecht , 15 Schreier, 134 Tsirelson, 8 Tsirelson , 134 special functional, 72, 176 sequence, 40, 176 sequence of vectors, 176 Sperner family, 148, 202 splitter, 106 spreading family, 153, 159 model, 130 step function, 90 strategically large for, 222 Ramsey, 222 strictly singular operator, 22 tree complete extension, 26 extension, 22 strongly extensible, 149 subsemigroup, 197 subsequence block, 132 subset small, 203 summability, regular method, 188 thin family F ⊆ FIN[<∞] , 202 thin family of ﬁnite subsets of N, 148 topological group Mazur–Orlicz complete, 195 unconditionally Mazur–Orlicz complete, 195

Index topology Ellentuck, 185 metric, 206 Milliken’s , 206 totally incomparable, Banach spaces, 92 trace, 157 transﬁnite basis, 99 1-subsymmetric, 99 boundedly complete, 102 shrinking, 102 unconditional , 99 transﬁnite block subsequence, 100 tree, 145 complete extension, 26 analysis, 9, 23 labeled by Q, 145 rooted, 145 tree-like property, 72 trivial projection, 39 true, ϕ is, 200 Tsirelson mixed space, 15 norm, 8 space, 8 Tsirelson space, 134 type 0, 23 I, 23 II, 23 ultraﬁlter as a quantiﬁer, 127 Ramsey, 128 unconditional basic sequence , 130 basis constant, 130 unconditional transﬁnite basis, 99 unconditionally convergent series, 192 uniform boundedness principle, 192 uniformly bounded averages, 44

257 vectors disjointly supported, 219 very large for, 222 weight, 23 well-founded quasiordered set, 144 well-quasi-ordered set(w.q.o.), 143