Some Results on Infinite DimensionalAsymptotic Structure of Banach Spaces

Thesis Submitted for the Degree “Doctor of Philosophy”

By

Roy Wagner

Submitted to the Senate of Tel-Aviv University

Month 1996

This work has been carried out underthe supervision of Professor V. Milman.

Acknowledgment

I would like to thank to my advisor, Prof. V. Milman for his invaluable guid-ance and intent supervision. I would also like to thank Professors B. Maurey, E.Odell and N. Tomczak-Jaegermann for their insightful conversation and patientinstruction. Finally, I would like to express deep gratitude to my excellent andsupportive teacher, Prof. E. Gluskin, who is responsible for my interest in the fieldof functional analysis.

Professors E. Odell and N. Tomczak-Jaegermann are co-authors of chapter 3;Professor V. Milman is co-author of chapter 4.

i

Abstract

This thesis presents some results in the newly developed field of infinite dimen-sional asymptotic structure of Banach spaces. the concept of infinite dimensionalasymptotic structure lies between local structure (finite dimensional subspaces) andglobal structure (infinite dimensional subspaces). It studies finite vector sequencesof increasing complexity in an attempt to gradually conduct local information intostronger contexts, sometime even an infinite dimensional context.

The first chapter of this thesis describes the combinatorial tools and conceptsof this theory, derived from earlier work, most notably [MMiTo] and [G5]. Thesecond chapter takes after [G5] and studies the dichotomy between unconditionalstructure in the asymptotic sense and proximity of subspaces (a strong version ofhereditary indecomposability). The third chapter studies spaces rich in copies of`n1 in the asymptotic sense. It quantifies this richness, analyzes it, and applies the

results to the study of Tsirelson-type spaces and infinite dimensional geometricquestions (stabilization of renormings). The last chapter defines and studies theasymptotic structure of operators. We use this structure to define new operatorideals, and to indicate an interesting direction in the study of Banach spaces withfew operators.

iii

Contents

Acknowledgment i

Abstract iii

Chapter 1. Preliminaries 11. Notation 12. Introduction 23. An intuitive introduction to asymptotic structure 54. The shallow game 65. The deep game 76. Asymptotic spaces 97. Asymptotic versions of operators 12

Chapter 2. Gowrers’ dichotomy for asymptotic structure 15

Chapter 3. Proximity to `1 and Distortion in Asymptotic `1 Spaces1 171. Introduction 172. Preliminaries 183. The Schreier families Sα 224. Asymptotic constants and ∆(X) 275. Examples–Tsirelson Spaces 356. Renormings of T , and spaces of bounded distortion 42

Chapter 4. Asymptotic Versions of Operators and Operator Ideals2 511. Preliminaries 512. Asymptotic versions of operator ideals 53

Bibliography 59

v

CHAPTER 1

Preliminaries

1. Notation

The theory of infinite dimensional asymptotic structure deals with the bodyvector sequences which are lacunary with respect to a basis. Therefore, in thisthesis we will always work in the category of Banach spaces with a basis (in fact,this restriction can be overridden, but since we can always pass to a subspace witha basis, more general concepts have no advantage in current applications).

X will denote an infinite dimensional space with a basis {ei}∞i=1. Unless statedotherwise, all the following spaces will be infinite dimensional.

We follow standard Banach-space theory notation, as outlined in [LTz]By Y ⊆ X we mean that Y is a closed infinite-dimensional linear subspace of

X. By S(X) = {x ∈ X : ‖x‖ = 1} we denote the unit sphere of X.If (ei) is a basic sequence and F ⊆ N, 〈ei〉i∈F is the linear span of {ei : i ∈ F}

and [ei]i∈F is the closure of 〈ei〉i∈F . The notation [X]n and [X]>n will standfor head ([ei]ni=1) and tail ([ei]∞i=n+1) subspaces respectively. Pn and P>n are thecoordinate-orthogonal projections on these subspaces respectively.

For F,G ⊆ N the notation F < G means that maxF < minG or either For G is empty. F < G are adjacent intervals of N if for some k ≤ m < n,F = [k, m] = {i ∈ N : k ≤ i ≤ m} and G = [m + 1, n].

If x ∈ 〈ei〉 and x =∑

aiei then supp(x) = {i : ai 6= 0} is the support ofx with respect to (ei) (w.r.t. (ei)). If the support of a vector is finite, we callx a block. For x, y ∈ 〈ei〉, we write x < y if supp(x) < supp(y). In this casewe say that x and y are consecutive. We denote S(X)n

< the collection of n-tuplesof consecutive normalized blocks in X, and S(X)∞< the collection of sequencesof consecutive normalized blocks in X (note that S(X)1< means normalized finitesupport vectors).

By (xi) ≺ (ei) we shall mean that (xi) is a block basis of (ei), that is a basismade of consecutive blocks. We say that Y is a block subspace of X, Y ≺ X, if Xhas a basis (xi) and Y = [yi]i∈N for some (yi) ≺ (xi).

Consider a Lipschitz function f on the unit sphere of a Banach space (in par-ticular a renorming). By existence of spectrum we mean existence of sections wherethe function (or norm) is almost constant. Exactly, for every ε > 0 there existsa subspace where the function ranges between C and C + ε for some constant C.C is then a spectrum point. When a spectrum does not exist, we say we havedistortion. The distortion is quantified by the range of fluctuation of the function,on subspaces where it is the smallest. We have λ-distortion if we have no spectrum,and for every ε in every subspace one can find two normalized vectors, x and y,such that |f(x)/f(y)| > λ− ε.

1

2 1. PRELIMINARIES

A norm is distortable if there exists an equivalent norm which is a distortion.A norm is λ distortable if there exists an equivalent norm which is a λ-distortion.A norm is boundedly-distortable if it is not λ-distortable for some λ > 1.

Two vector sequences, {xi}i and {yi}i, are C equivalent if for all {ai}i’s b <‖

∑i aixi‖

‖∑

i aiyi‖ < a with a/b ≤ C. If in a space X for every ε one can find sequences 1+ε

equivalent to a sequence {yi}i, then we say X contains almost isometric copies of{yi}i

Suppose B ⊆ A, where A is a metric space. Then Bδ denotes a delta extensionof B, that is all elements of A which are δ-close (or closer) to some element in B. IfB ⊆ A′ ⊆ An, then in the context of A′ Bδ1,...,δn is the collection of sequences b′ inA′, such that there exists b ∈ B, such that for every 1 ≤ i ≤ n the i-th coordinatesof b′ and b are δi close (or closer). In this case we also use the notation Bδ forBδ,...,δ.

2. Introduction

This thesis contains, primarily, results regarding the newly developed infinite di-mensional asymptotic theory of Banach spaces. Before describing the basic conceptsand tools of this theory, I would like to relate some background and motivation.

Infinite dimensional asymptotic theory studies the body of lacunary block se-quences of a Banach-space. By ’lacunary’ we may mean ’supported far along thebasis and widely spread out’ or ’contained in a fast decreasing chain of subspaces’(the former being identical to the latter if ’subspaces’ is replaced by ’tail-subspaces’).This theory has risen naturally from new developments in the theory of infinite di-mensional Banach-spaces. It comes from questions of infinite dimensional geometryconcerning existence of spectrum of Lipschitz functions and of distortion, (cf. [G1],[S]).

The ideology of infinite dimensional Banach-spaces in the 1960’s and early 70’sleaned towards proving strong stabilization theorems. The term stability is usedhere to indicate many symmetries in the space, many isomorphic subspaces. It canoften be described by existence of spectrum of some functions. For instance, if thefunctions

(1) fa(x, y) = ‖x + ay‖

have spectrum in every infinite dimensional subspace for every real a, thenthere are subspaces where all 2-dimensional subspaces are almost isometric.)

The most classical stability result dates back to Kolmogorov ([K], see also[Bo] and [Z]). This result claims that spaces with all finite equidimensional blocksubspaces isometric (uniformly isomorphic) are isometric (isomorphic) to `p, 1 ≤p ≤ ∞ or c0. Another result, which is of special importance in this context is theresult of James [J], stating that if a Banach space contains an isomorphic copy of`1 (c0), it contains almost isometric copies of `1 (c0).

Two crucial steps following this ideology in the finite dimensional direction areDvoretzky’s theorem ([Dv]) and Krivine’s theorem ([Kr]), the first stating thatan infinite dimensional Banach-space must contain arbitrarily good copies of `n

2

with arbitrarily high dimension, the second claims (in some versions) that everyinfinite dimensional Banach-space contains arbitrarily good copies of `n

p for some1 ≤ p ≤ ∞ with arbitrarily high dimension as block subspaces. The above four

2. INTRODUCTION 3

results suggest that Banach spaces are inherently stable, and may lead one to believethat all Banach spaces contained almost isometric copies of `p or c0.

In the finite dimensional case (local theory) a vast body of study proves thatthe ideology of inherent stability is correct (see [LMi] for a survey with exhaustivebibliography). The same ideology was discarded for the infinite dimensional casewith the introduction of a space not containing any isomorph of `p or c0, the first’unstable’ space, Tsirelson’s space in [T] (see also [FJo]). Much more recently, achain of constructions by Odell and Schlumprecht (with the involvement of Maureyand Tomczak) showed that James’ theorem can not be extended to `p’s in general([S], [OS2]; proofs that James’ theorem does not extend to other classes of spacescan be found in [MiTo], [To2], [M], [ArD], and chapter 3 of this thesis).

The situation is therefore as follows: when it comes to finite dimensional sub-spaces (local theory) stabilization results are extremely strong; in the infinite di-mensional case, however, such results are much less general. The natural directionis to find an intermediate level of stabilization, stronger than the local, yet moregeneral than infinite dimensional. This is the point where asymptotic theory ap-pears.

Let’s have a look at the dual of Tsirelson’s space (as introduced in [FJo]);being the exciting freak that it was, it has been exhaustively studied (see [CSh]for many results and a complete bibliography). As mentioned above it is not stablein the sense that it contains no `p or c0. However, it does have some importantinfinite-dimensional stability properties: first, it is reflexive and has an uncondi-tional basis; second, all block subspaces are uniformly equivalent to subsequencesof the basis; third, while it is not true that all subsequences of the basis are equiv-alent, a subsequence must be extremely lacunary in order to be non-equivalent tothe basis; fourth, the original Tsirelson space is minimal (every subspace containsan isomorph of the whole space). So it turns out that Tsirelson’s space is not asunstable as one might suspect. In fact, it has one more crucial stability property,which is not infinite dimensional, but is more than local: every sequence of n blocks,supported after the n-th element of the basis, is 2-equivalent to the unit vector basisof `n

1 .This new and hybrid type of stability appeared again, naturally, 20 years later.

A theorem of Milman from 1969 ([Mi]) states that if the result of James holds fora given Banach-space, that is if any isomorphic renorming of any infinite dimen-sional subspace has spectrum, then this space must contain `p or c0 as subspaces.(which, using [OS2], becomes `1 or c0). The result is achieved by showing thatunder this assumption functions similar to 1 have spectrum, meaning that all finitedimensional blocks subspaces in some subspace are almost isometric; adding theabove mentioned result of Kolmogorov completes the proof. An extension of thistheorem has been studied in [MiTo]. The same techniques were applied to a re-laxed hypothesis, that is assuming that every isomorphic renorming of any infinitedimensional subspace has only bounded distortion. It turned out that under thishypothesis there is a problem in a reiteration of a stabilization argument. Underthe relaxed hypothesis it was proved that some subspace must have a lot of `p’s init, in the same way that the dual of Tsirelson’s space has a lot of `1’s in it. We nowcall such spaces stable asymptotic `p spaces. So, as promised, we have a naturalstabilization concept, which is stronger than the local ones, and more common thaninfinite-dimensional stabilization.

4 1. PRELIMINARIES

Tsirelson’s space raises obvious queries in the direction of constructing evenless stable spaces. The construction of the first space known to be unboundedlydistortable in [S], was the corner stone used by Gowers and Maurey in [GM1] toconstruct a highly unstable Banach space: a space not containing any unconditionalbasic sequence, and moreover, hereditarily indecomposable, meaning that the an-gle between any two closed infinite dimensional subspaces is zero; from this lastproperty follows that every bounded linear operator on this space is a perturbationof a scalar operator by an operator which is not an isomorphism on any subspace.Otherwise put, a space with practically no significant symmetries. This construc-tion was later altered in numerous papers (among which are [G2] [G3] [G4], [H],[ArD], [GM2], [OS3] and [Fe]) to construct many other unstable spaces.

It turned out that the combination in [GM1] of not having an unconditionalbasic sequence and of being hereditarily indecomposable is no coincidence. Gowersproved in [G5] (see also [G6]) that the former implies containing the latter. Theproof, again, brings up concepts which have to do with asymptotic structure. Itis shown, using a Galvin-Prikry like argument, that if a space does not have anunconditional basic sequence, in other words does not have a subspace where everyfinite block sequence is stable (in the sense of unconditionality), means that the non-stable (badly-unconditional) finite sequences are not only present in every subspace,but in fact have to be quite prevalent in the space: one can find badly-unconditionalsequences in every list of subspaces (each vector from the sequence belongs tothe corresponding subspace in the list). Moreover, the construction of those non-stable sequences can not be obstructed by changing the next subspace on the listof subspaces even after all the preceding vectors are chosen. In other words, thechoice of first vectors in the sequence depends only on the first subspaces on thelist, and on the length of the sequence to be obtained.

As was shown later, this notion of prevalence is closely related to the conceptof asymptotic `p from [MiTo]. In fact, the combinatorial language introduced byGowers was then used to formally define the general concepts of asymptotic struc-ture in [MMiTo]. On one side it is a generalization of the concepts of asymptoticstructure from Tsirelson’s space and from [MiTo]. On the other side it is a con-cept which covers (a form of) Gowers’ notion of high prevalence. Thus, asymptoticstructure appears again in a natural way in the study of infinite dimensional geom-etry.

The papers [MMiTo] and [G5] are the starting point for this thesis.The first chapter introduces the asymptotic language. It derives ideas, nota-

tion and arguments from several papers, notably [G5], [MMiTo] and [C]. Thepresentation is tightened and formalized to suit the needs of this thesis.

The second chapter is an adaptation of [G5] to the language of asymptoticstructure. In [G5] the alternative to infinite dimensional stability (containing anunconditional basic sequence) is studied. In the second chapter of this thesis thesame techniques are used to study the alternative to the weaker asymptotic stabil-ity (the case where all block sequences which are lacunary enough are uniformlyunconditional). In the former case the alternative is containing a hereditarily inde-composable subspace. In the latter the alternative is a stronger uniform version ofthat situation.

The third chapter is a joint work with Prof. E. Odell from Austin universityand with prof. N. Tomczak-Jaegermann from Alberta university. It deals with

3. AN INTUITIVE INTRODUCTION TO ASYMPTOTIC STRUCTURE 5

quantifying the asymptotic stability of asymptotic-`1 spaces. This quantification isapplied to the study of distortability of those spaces.

The last chapter is joint work with my advisor, Prof. V. Milman. It appliesthe asymptotic language to operators. It is shown that this language is useful indescribing new operator ideals.

3. An intuitive introduction to asymptotic structure

We will begin with an informal discussion of asymptotic structure. In the fol-lowing sections of this chapter we will give detailed exposition of the combinatoriallanguage we use, formally define asymptotic structure, and prove basic existenceand stabilization results.

The main idea behind this theory is a stabilization at infinity of finite di-mensional objects (subspaces, restrictions of operators), which appear repeatedlyarbitrarily far and arbitrarily spread out with respect to the basis. Piping these sta-bilized objects together gives rise to an infinite dimensional notion: an asymptoticversion of a Banach-space X or of an operator acting on X.

To define this structure we first have to choose a frame of reference in the formof a family of subspaces, B(X). It is most convenient to choose B(X) such that theintersection of two subspaces from B(X) is in B(X). The family of tail subspacesis such a family; so is the family of finite codimensional subspaces, but here we willwork with the former. The construction proceeds as follows.

Fix n and ε. Consider the tail subspace [X]>N1 for some ’very large’ N1,and take a normalized vector in this tail subspace. Consider now a further tailsubspace [X]>N2 , with N2 ’very large’, depending on the choice of x1, and chooseagain any normalized vector, x2 in [X]>N2 . After n steps we have a sequence of nvectors, belonging to a chain of tail subspaces, each subspace chosen ’far enough’with respect to the previous vectors.

The span of a sequence in X, E = span{x1, . . . , xn}, is called ε-permissible ifwe can produce by the above process vectors {yi}n

i=1, which are (1 + ε)-equivalentto the basis of E, regardless of the choice of tail subspaces.

Now we can explain how far is ’far enough’. The choice of the tail subspaces[X]>Ni is such that no matter what normalized vectors are chosen inside them,they will always form ε-permissible spaces. The existence of such ’far enough’choices of tail subspaces is proved by a compactness argument: increasing the Ni’sdecreases the collection of vector sequences from the tail subspaces decreases, butarbitrary small extensions of the equivalence classes of sequences in these decreasingcollections cannot decrease to an empty set by compactness. Thus for large enoughNi’s we get sequences arbitrarily closely equivalent to permissible sequences.

We can now consider basic sequences which are 1+ε-equivalent to ε-permissiblesequences for every ε. These will be called n dimensional asymptotic spaces. Ourε-permissible sequences are (1 + ε)-realizations of asymptotic spaces in X.

Finally, a Banach-space whose every head-subspace is an asymptotic space ofX is called an asymptotic version of X.

The same construction can be made for an operator T as well. In this case, wewould like to stabilize not only the domain (which is an asymptotic space), but alsothe image and action of the operator. More precisely, our asymptotic sequenceswill now be sequences {xi}n

i=1, such that we can find arbitrarily far and arbitrarilyspread out sequences {yi}n

i=1 in our space, which are closely equivalent to {xi}ni=1,

6 1. PRELIMINARIES

and whose images under T are closely equivalent to {T (xi)}ni=1. Note that since we

view these operators as operators from [xi]ni=1 to the normalized [T (xi)]ni=1, theseoperators are always formally diagonal operators.

4. The shallow game

The shallow game is the game used in [MMiTo] to define asymptotic spaces

Definition 4.1. This is a game for two players. One is the subspace player,S, and the other is the vector player, V. The ’board’ of the game consists of aBanach-space with a basis, a natural number n, and two subsets of S(X)n

<: Φ andΣ. Player S begins, and they play n turns.

In the first turn player S chooses a tail subspace, [X]>m1 . Player V then choosesa normalized block in this subspace, x1 ∈ S([X]>m1). In the k-th turn, player Schooses a tail subspace [X]>mk

. Player V then chooses a normalized block, xk, suchthat xk ∈ S([X]>mk

) and xk > xk−1.V wins if the sequence (x1, . . . , xn) is in Φ.S wins if the sequence (x1, . . . , xn) is in Σ.Note that in this game it is not always true that one player wins and the other

loses. Furthermore, in some cases, we are only interested in the winning prospectsof one player, and therefore may ignore either Φ or Σ.

If V has a winning strategy in this game for Φ, that is a recipe for producingsequences in Φ considering any possible moves of S, we call Φ an asymptotic set oflength n. Formally this means:

∀[X]>m1 ∃x1∈ [X]>m1 ∀[X]>m2 ∃x2∈ [X]>m2 . . . ∀[X]>mn∃xn∈ [X]>mn

(2)such that (x1, . . . , xn) ∈ Φ

Note that this generalizes an earlier notion of an asymptotic set for n = 1 (inthis context of tail subspaces, rather than block subspaces; cf. [GM1]).

If S has a winning strategy in this game for the collection Σ, we call Σ anadmission set of length n. Formally this means:

∃[X]>m1 ∀x1∈ [X]>m1 ∃[X]>m2 ∀x2∈ [X]>m2 . . . ∃[X]>mn∀xn∈ [X]>mn

(3)such that (x1, . . . , xn) ∈ Σ

This terminology comes from admissibility criterions in the study of Tsirelson’sspace and its variants (all vector sequences beginning far enough and spread farenough).

A sequence of vectors produced by V while playing against the winning strategyof S is called a sequence of admissible vectors.

When the context is clear, we may omit the the length and simply write ’anasymptotic (admission) set’.

Remark 4.2.(1) Note that a collection containing an admission set is an admission set,

and that a collection containing an asymptotic set is an asymptotic set.(2) It is also useful to note that if V has a winning strategy for Φ, and S has a

winning strategy for Σ, then playing these strategies against each other will

5. THE DEEP GAME 7

necessarily produce sequences in Φ ∩ Σ. In fact, V has a winning strategyfor Φ ∩ Σ.

Indeed, suppose player V plays against a player S, and tries to producesequences in Φ ∩ Σ. To explain the strategy of player V we will use twoauxiliary players. The auxiliary player S ′ plays a winning strategy for Σ.The auxiliary player S ′′ plays the intersection of the tail subspaces chosenby S ′ and S at each turn. Player V plays a winning strategy for Φ choosinghis vectors as if its opponent were S ′′ rather than S. The moves of playerV are still legal in the game against player S. The sequence resulting fromthis strategy is always in Φ ∩ Σ. Thus the intersection of an admission setand an asymptotic set is asymptotic.

The following is simply formal negation of (2) and (3).

Lemma 4.3. Let Σ ⊆ S(X)n<. Then either Σ is an admission set, or Σc is an

asymptotic set. These options are mutually exclusive.

Tail subspaces of a Banach-space form a filter. This allows us to demonstratefilter (cofilter) behaviour for admission (asymptotic) sets.

Lemma 4.4.(1) Let Σ1, . . . , Σk ⊆ S(X)n

< be admission sets. Then⋂k

j=1 Σj is also an ad-mission set.

(2) Let Φ1, . . . , Φk ⊆ S(X)n<, such that

⋃kj=1 Φj is asymptotic. Then for some

1 ≤ j ≤ k Φj is asymptotic.

Proof.

(1) Suppose at any turn of the game player S has to choose [X]>m1 in orderto win for Σ1, [X]>m2 in order to win for Σ2, . . ., and [X]>mk

in order towin for Σk. If player S chooses [X]>m with m = max{m1,m2, . . . ,mk},the vector sequence chosen by player V will have to be in

⋂kj=1 Σj .

(2) Suppose for all 1 ≤ j ≤ k, Φj is not asymptotic. Then, by Lemma 4.3,Φc

j are all admission sets. By part 1 of the proof,⋂k

j=1 Φcj is an admission

set. Therefore, using Lemma 4.3 again,⋃k

j=1 Φj is not asymptotic, incontradiction.

�

5. The deep game

The deep game is the game used in [G5] for Gowers’ combinatorial lemma.

Definition 5.1. This game is defined exactly as the shallow game, except thatat each turn player S chooses any block subspace, rather than a tail subspace.

The terminology of asymptotic collections, admission collections and winningstrategies will be used for this game as well. Since a collection may be asymptotic forthe shallow game but not for the deep one (e.g., the collection Φ = {{emk

}nk=1;m1 < . . . < mn ∈ N}),

we will use the terminology deep/shallow-asymptotic collection (-admission set, -winning strategy, . . . ) where confusion may occur. Note however that a deep-asymptotic set is always shallow-asymptotic, and that a shallow-admission set isalways a deep-admission set.

8 1. PRELIMINARIES

In this section all notions refer to the deep game.Once more, formal negation yields:

Lemma 5.2. Let Σ ⊆⋃

n∈N S(X)n<. Then either Σ is an asymptotic set or Σc is

an admission set. These options are mutually exclusive.

In the deep game context, since the subspaces involved no longer form a filter,we no longer have ’filter-behaviour’.

Example 5.3. Let Y = Y1

⊕Y2. Let Σ1 = {y; y ∈ S(Y1)}, and Σ2 = {y; y ∈

S(Y2)}, Then both Σ1 and Σ2 are admission sets of length 1, but their intersectionis empty.

However, we do have a positive result which is crucial in this context. This isa restricted version of Gowers’ combinatorial Lemma from [G5], with roots in [C].

Theorem 5.4. Let Σ ⊆ S(X)n<. Let δ > 0. Then there exists a block subspace

Y in which (Σ)δ is asymptotic, or such that S(Y )n< ⊆ (Σc)δ.

A corollary to this result is a somewhat different form of filter behaviour ofadmission sets.

Lemma 5.5.(1) Let Σ1, . . . , Σk ⊆ S(X)n

< be admission sets in every block subspace. Letδ > 0. Then

⋂kj=1(Σj)δ is also an admission set; moreover the last set

contains S(Y )n< for some block subspace Y .

(2) Let Φ1, . . . , Φk ⊆ S(X)n<, such that

⋃kj=1 Φj is asymptotic. Let δ > 0.

Then, there exists a block subspace Y and 1 ≤ j ≤ k, such that (Φj)δ isasymptotic in Y .

Proof.

(1) The collection (((Σ1) δ2)c) δ

2is contained in Σc

1. Therefore it is not an as-ymptotic set. By Theorem 5.4 applied to (Σ1) δ

2and δ

2 , in some blocksubspace, Y1, all normalized block sequences belong to (Σ1)δ. repeat theargument inside Y1 for Σ2 instead of Σ1, and continue inductively to provethat

⋂kj=1(Σj)δ contains all normalized block sequences of length n in

some subspace, and is therefore an admission set.(2) Suppose for all 1 ≤ j ≤ k (Φj)δ is not asymptotic in any subspace. Then,

by Lemma 5.2, ((Φj)δ)c are all admission sets in every subspace. By part1 of the proof,

⋂kj=1(((Φj)δ)c)δ contains all normalized block sequences in

some block subspace. The last collection is contained in⋂k

j=1 Φcj . There-

fore, using Lemma 5.2 again,⋃k

j=1 Φj is not asymptotic, in contradiction.�

In the context of the deep game, it is useful to define the infinite and theunbounded games.

Definition 5.6. The infinite game is identical to the deep game, except thatΦ, Σ ⊆ S(X)∞< , and the game continues infinitely many turns. A player wins if theinfinite sequence generated is contained in the corresponding vector collection.

In the unbounded game Φ, Σ ⊆⋃

n∈N S(X)n<. The game continues infinitely

many turns. Player V wins if at some turn of the game, the finite sequence of

6. ASYMPTOTIC SPACES 9

vectors generated up to that turn is contained in Φ. Player S wins if at every turn,the sequence produced thus far is in Σ.

These games were studied and applied in [G5] and [G6]. In particular Gowerscombinatorial Lemma holds for the unbounded game in the same way as for thelength-n game.

Theorem 5.7. Let Σ ⊆⋃

n∈N S(X)n<. Let δ1 > 0, δ2 > 0, . . . and set ∆ =

(δ1, δ2, . . .). Then there exists a block subspace Y in which (Σ)∆ is asymptotic, orsuch that

⋃n∈N S(Y )n

< ⊆ (Σc)∆.

In the context of the infinite game, it is necessary to add a topological assump-tion on Σ (analyticity).

6. Asymptotic spaces

Definition 6.1. An n-dimensional Banach space F with a basis {fi}ni=1 is

called an asymptotic space of a Banach-space X, if for every ε > 0 the set of allsequences in S(X)n

<, which are (1 + ε)-equivalent to {fi}i, is an asymptotic set.A space X is an asymptotic version of X, if all spaces [X]n are asymptotic

spaces of X.We use {X}∞ to denote the collection of asymptotic versions of a space X,

and {X}n to denote the collection of its n-dimensional asymptotic spaces.

Obviously, the definition depends on the game we are using: deep or shallow.In this context we may consider another game, the fixed game, where player S isdoomed to repeat its choice of tail subspace for the first turn throughout the game.Note that in the fixed game, an admission set is simply a set that contains all blocksequences of a fixed length in some tail subspace.

For the shallow game an asymptotic space is a space which is approximatelyrepresented arbitrarily far and spread out along the basis. For the deep gameit is a space which is represented arbitrarily deep inside block subspaces (that is,represented by sequences with arbitrarily lacunary supports). For the fixed game anasymptotic space is simply a space which is approximately represented arbitrarilyfar along the basis. In the deep game context, the representation must still bepossible when going to any block subspaces. to point out the differences, considerthe following:

Example 6.2.(1) Let X be an `3 sum of copies of `102 . Then `102 is an asymptotic space for

the fixed game, but not for the shallow or deep games.(2) Let X be a space with a basis and a norm calculated as follows: Sum the `2

norm of the even coordinates and the `3 norm of the odd coordinates. Inthe shallow game both `n

2 and `n3 are asymptotic spaces for every n. In the

deep game however, there are no asymptotic spaces for X. Indeed, in thedeep game if the subspace player’s strategy is playing the even coordinatesubspace repeatedly throughout the game, the vector sequences producedwill all be `n

2 ’s. Therefore the only candidates for asymptotic spaces are`n2 ’s. However, if the subspace player plays repeatedly the odd coordinate

subspace, the vector player cannot produce `n2 ’s. Therefore there are no

deep asymptotic spaces in X.

10 1. PRELIMINARIES

Note that, in any context, the asymptotic spaces depend only on tail subspaces,that is, if two spaces have identical tails, they have the same asymptotic spaces.

We will now consider existence of asymptotic spaces. In the context of the fixedand shallow games, existence of asymptotic spaces is elementary, is was proved in[?] Spreading models make an obvious example of asymptotic spaces and versions.Existence of some special asymptotic versions was considered in [MMiTo]. Oneapproach of proving the following existence theorem is outlined in the intuitiveintroduction, and comes [MMiTo]. We will use a slightly different approach to theproof below.

Proposition 6.3. Let X be a space with a basis. Consider the shallow orfixed game. Fix ε > 0 and n ∈ N Let Σ be the collection of ε-perturbations (inthe equivalence sense) of asymptotic spaces of length n in S(X)n

<. Then Σ is anadmission set (for the respective game).

Proof. Suppose the proposition fails. Then Σc is an asymptotic set. Wewill show that it must contain an ε

2 approximation of some asymptotic space andconclude that Σ intersects its complement, a contradiction.

Take a finite covering of the Minkowski compactum of order n by cells of di-ameter smaller than ε

2 , {Ui}. Σc is the finite union of the sets: Φi = Σc ∩ Ui (withsome abuse of notation). By Lemma 4.4, one of these sets (to be noted Φi0) mustbe an asymptotic set (the Lemma 4.4 is obviously true for the fixed game).

Repeating the argument inductively inside the asymptotic Φi0 for arbitrarilysmall ε’s, and using compactness shows that there exists a space with a basis F ,such that for all δ > 0 the collection of δ perturbations of F inside Φi0 is asymptotic.Therefore F is an asymptotic space, and since Φi0 ⊆ Σ, and has diameter less thanε2 , our assertion is proved. �

We have seen in Example 6.2 that deep asymptotic versions need not exist.However, Lemma 5.4 implies:

Proposition 6.4. Let X be a space with a basis. Consider the deep game. Forevery ε > 0 and n ∈ N. There exists a block subspace where Σ, the collection ofsequences in S(X)n

< which are ε perturbations (in the equivalence sense) of spaceswhich are deep-asymptotic in some subspace, is an admission set for the deep gamein X.

Proof. Suppose the proposition fails. Then Σc is an asymptotic set in X. Wewill show that Σc must contain an ε

2 perturbation of some space which is deep-asymptotic in some subspace, thus intersecting its complement, a contradiction.

Take a finite covering of the Minkowski compactum of order n by cells of di-ameter smaller than ε

4 , {Ui}. Σc is the finite union of the sets: Φi = Σc ∩ Ui (withsome abuse of notation). By Lemma 5.5, an ε

8 -extension of one of these sets mustbe a deep-asymptotic set in some subspace.

Repeating the argument inductively in this subspace for the deep-asymptoticset (Φi0) ε

8with decreasing ε’s shows that there exists a space with a basis, F ,

such that for every n ∈ N the collection of δ-perturbations of F in (Φi) ε4

is deepasymptotic in some block subspace Xn, with Xn < Xn−1. Therefore F is anasymptotic space in the diagonal subspace. Thus (Φi) ε

4contains arbitrarily small

perturbations of some space which is asymptotic in some subspace, which leads toΣ containing an ε

2 -perturbation of a space which is asymptotic in some subspace,and we’re through. �

6. ASYMPTOTIC SPACES 11

The object of the following arguments is to show:

Theorem 6.5. There is a block subspace where the asymptotic spaces from thedeep, shallow and fixed games coincide, and where passing to a block subspace doesnot change them.

In particular, this means that every far enough block sequence of normalizedvectors is a small perturbation of a space which is deep asymptotic in every sub-space. This follows from the fact that in a block subspace as above the collection ofsmall perturbations of deep asymptotic spaces is the collection of small perturba-tions of fixed-game asymptotic spaces, which are a fixed-game admission set, andtherefore must contain all sequences supported far enough (with respect to theirlength).

The first part of the argument can be found in [C] and in [MMiTo].

Proposition 6.6. Consider the fixed, shallow or deep game. There exists asubspace where the collection of asymptotic spaces does not change when going to asubspace.

Proof. The proof is standard. Fix ε and n. Take a finite covering, {Ui}i∈I , ofthe Minkowski compactum of order n by cells of diameter smaller then ε. Choose asubspace where for every i ∈ I either no space from Ui is asymptotic in any furthersubspace, or in every further subspace, some space from Ui is asymptotic. Repeatfor ε’s going to zero, and diagonalize. Repeat for every n, and diagonalize. Callthe resulting subspace Y .

Now take a finite dimensional space with a basis F . Either for some ε no spacein an ε-neighbourhood of F is asymptotic in any subspace of Y , or for every ε, inevery subspace of Y some space in an ε neighbourhood of F is asymptotic, implyingthat F itself is asymptotic in every subspace.

�

The only remaining ingredient for Theorem 6.5 is:

Proposition 6.7. a space which is fixed-game asymptotic in every subspace,is deep-game asymptotic in some subspace.

Proof. This is an easy result of Theorem 5.4. If a space F is fixed-gameasymptotic in every subspace, then for every ε the collection Φε of sequences inS(X)n

<, which are ε-perturbations of F , intersects every subspace. Therefore, forall δ > 0,

((Φδ)c

)δ2, which contains Φ δ

2, cannot contain all normalized sequences

in some subspace. So by Theorem 5.4 applied to Φδ and δ2 , Φ 3δ

2has to be deep-

asymptotic in some subspace. Repeat for δ’s going to zero, diagonalize, and theproof is through. �

The above stabilization is wasteful, in the sense that one has to go to a subspaceto achieve it. The following concept of stabilization was conveyed to me by Odell([O]). The statement claims that the basis can be blocked into a finite dimensionaldecomposition such that every ’far enough skipped’ combination is almost a shallow-game asymptotic space.

Proposition 6.8. for all {εn}n∈N decreasing to zero there exist 0 = k0 < k1 <

. . . so that if Ei = 〈ej〉ki

j=ki−1+1 for i ∈ N, n ∈ N and xi ∈ 〈Ej〉li−1j=li−1+1 for some

12 1. PRELIMINARIES

sequence n − 1 ≤ l0 < l1 < . . . < ln then {xi}ni=1 is εn close to a shallow-game

asymptotic space.

Proof. Let k0 = 0. Let [X]>k1 be the first step of the subspace player in awinning strategy for sequences of length 2 which are ε2/2 close to shallow gameasymptotic spaces. Let [X]>k2 be the first step of the subspace player in a winningstrategy for sequences of length 3 which are ε3/2/ close to shallow game asymptoticspaces. Let Φ2 be a finite ε2/2-net in the unit sphere of [X]k2 ∩ [X]>k1 . For eachelement v of Φ2 Consider the next step of the subspace player in the strategy definingk1, assuming the vector player has just played the vector v. Take the intersectionof all those subspaces. Take the intersection between this subspace and the firstchoice of the subspace player in a winning strategy for sequences of length 4 whichare ε4/2 close to shallow game asymptotic spaces. This last subspace is [X]>k4 .

We will continue by induction. Suppose we have set ki for all i ≤ m, satisfyingthe following:

Set Ei = 〈ej〉ki

j=ki−1+1 for 1 ≤ i ≤ m. If xi ∈ 〈Ej〉li−1j=li−1+1 for 1 ≤ i ≤ n and for

some sequence n − 1 ≤ l0 < l1 < . . . < ln ≤ m + 1 and some n ≤ [m+12 ] + 1, then

xi is εl0+1/2 close to a sequence in a finite collection of sequences, Φm, in which:every sequence is the result of some number of leading turns in a game where thesubspace player plays a winning strategy for sequences of some length, l0, which areεl0/2 close to shallow game asymptotic spaces, and for which [X]>km

as a next stepis compatible with the strategy, as long as the game is not already over, and as longas the last vector in the sequence belongs to [X]km−1 .

We will now define km+1. fix l0. For each sequence from Φm beginning at[X]>kl0+1 with length not greater than l0+1, consider the next step of the subspaceplayer. Take the intersection of all these subspaces. Take the intersection betweenthe resulting tail subspace and the first step of the subspace player when it plays awinning strategy for sequences of length m + 2 which are εm+2/2 close to shallowgame asymptotic spaces. The last subspace is [X]>km+1 .

It remains only to extend the collection Φm to a finite Φm+1 satisfying theinductive hypothesis, which is done simply by playing one more step of the gamewith the appropriate strategy for every sequence which requires an extension. Theproperties of Φm, and the fact that [X]km+1 is finite dimensional (having finite netsas required) promise that this can be done to satisfy the inductive hypothesis.

The proof is now through. �

7. Asymptotic versions of operators

To define the concept of an asymptotic version of an operator we have to confineour attention to a space X with a shrinking basis. The game in this context willbe the shallow game.

Definition 7.1. Let T ∈ L(X). Define an asymptotic operator, U , of T tobe a formally diagonal operator between n-dimensional asymptotic spaces of X, Fn

and Gn, such that for every ε > 0, the following set is asymptotic:All sequences in S(X)n

<, which are (1 + ε)-equivalent to the basis of Fn, andwhose images under T are (1 + ε)-equivalent to the images under U of the basis ofFn.

An asymptotic version T of T is an operator between asymptotic versions of Xfor which every n-dimensional head is an asymptotic operator. A collection of such

7. ASYMPTOTIC VERSIONS OF OPERATORS 13

asymptotic sets of length n for every positive ε and every natural n will be referredto as: asymptotic sets approximating the asymptotic version T (arbitrarily well).

The set of all asymptotic versions of an operator T will be denoted {T }∞; theset of n-dimensional asymptotic operators will be denoted {T}n

Note that the asymptotic versions of the identity correspond simply to asymp-totic versions of the space.

The point in restricting to a shrinking basis is to promise that a sequenceof sufficiently spread out blocks will be mapped under T to a small perturbationof sufficiently spread out block, thus making sure (by Proposition 6.3) that theasymptotic version of T will have an asymptotic space as its range. This is also thereason that the fixed game is not suited for this context: ’far enough’ consecutiveblocks do not have to map to a small perturbation of consecutive blocks. Sincethere is no way in general to ’force’ images of consecutive blocks to belong (up toa small perturbation) to a prescribed block subspace, the deep game is not usefulin this context either.

Consider an operator T . An existence theorem parallel to Proposition 6.3 forthe shallow game (stating that sequences approximating arbitrarily well asymptoticoperators are an admission set) has a practically identical proof. The same is truefor Proposition 6.6 (stating that asymptotic operators do not change when passingto further subspaces of some subspace). We can now pass to a subspace where suf-ficiently far blocks will map under T into almost consecutive blocks. The argumentin Proposition 6.7 (stating that if U is a fixed-game asymptotic operator in every,then it is a deep-game, and in particular a shallow-game asymptotic operator insome subspace) shows we can pass to a subspace where sufficiently far blocks givearbitrarily good approximations of asymptotic operators. We thus conclude:

Theorem 7.2. For every operator T , there exists a subspace where for everybounded operator, the set of sequences approximating arbitrarily well some asymp-totic operator is a fixed-game admission collection. Furthermore, the collection ofasymptotic operators of T does not change when going to a subspace.

Another approach to existence of asymptotic operators, namely extracting sub-sets approximations an asymptotic operators from any collection of asymptotic setswith arbitrarily long lengths, is dealt with in chapter 4 (Theorem 1.3).

CHAPTER 2

Gowrers’ dichotomy for asymptotic structure

To be inserted.

15

CHAPTER 3

Proximity to `1 and Distortion in Asymptotic `1

Spaces1

1. Introduction

The first non-trivial example of what is now called an asymptotic `1 space wasdiscovered by Tsirelson [T]. This space and its variations were extensively studiedin many papers (see [CSh]). While the finite-dimensional asymptotic structure ofthese spaces is the same as that of `1, they do not contain an infinite-dimensionalsubspace isomorphic to `1, and thus their geometry is inherently different.

The idea of investigating the geometry of a Banach space by studying its as-ymptotic finite-dimensional subspaces arose naturally in recent studies related toproblems of distortion, i.e. the stabilization of equivalent norms on infinite dimen-sional subspaces of a given Banach space. These ideas were further developed andprecisely formulated in [MMiTo].

By a finite-dimensional asymptotic subspace of X we mean a subspace spannedby blocks of a given basis living sufficiently far along the basis. By an asymptotic`p space we mean a space all of whose asymptotic subspaces are `n

p , i.e. any nsuccessive normalized blocks of the basis {ei}∞i=1 supported after en are C-equivalentto the unit vector basis of `n

p .In this paper we introduce a concept which bridges the gap between this “first

order” structure of an asymptotic `1 space and the global structure of its infinite-dimensional subspaces. This concept employs a hierarchy of families of finite subsetsof N of increasing complexity, the Schreier classes (Sα)α<ω1 introduced in [AAr].For α < ω1 we define what it means for a normalized block basis to be Sα-admissiblewith respect to the basis (ei), and then measure the equivalence constant betweenall such blocks and the standard unit vector basis of `1, obtaining the parameterδα(ei). These constants increase when passing to block bases and this leads us todefine the ∆-spectrum of X, ∆(X), to be the set of all stabilized limits γ = (γα)of (δα(ei)) as (ei) ranges over all block bases of X.

We show that these concepts provide useful and efficient tools for studying theinfinite dimensional and asymptotic structure of asymptotic `1 spaces. Indeed, evensome first order asymptotic problems require a higher order analysis. The behaviorof the ∆-spectrum of X has deep implications in regard to the distortability of Xand its subspaces.

We now describe the contents of the paper in more detail.

1Joint work with Professors E. Odell and N. Tomczak-Jaegermann. To appear in Jour.Funct. Anal.

17

18 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES2

Section 2 reviews concepts and results concerning distortion and asymptotic `1spaces. We sketch the proof of the 2-distortability of Tsirelson’s space in Proposi-tion 2.7. This leads to a natural question as to whether the asymptotic structureof T can be distorted: can T be given an equivalent norm such that its asymptoticsubspaces are closer to `1? Without resorting to the higher order analysis developedin subsequent sections we only obtain a partial solution (the complete solution isthen provided in Section 5).

In Section 3 we define the Schreier families Sα and establish some facts abouttheir mutual relationship which are crucial for our later work.

Section 4 contains precise definitions of all the asymptotic `1 constants which weintroduce in this paper. We also define the spectrum ∆(X). Elements γ = (γα)α<ω1

of the spectrum satisfy γαγβ ≤ γα+β for all α, β < ω1 (Proposition 4.11). It followsthat γα = limn→∞ γ

1/nα·n exists for all α < ω1 and it is shown to equal δα(Y ) for

some subspace Y ⊆ X (Proposition 4.15). δα(Y ) is defined to be the largest ofδα((xi), | · |) as (xi) ranges over all block bases of Y and | · | over all equivalentnorms. The constants (δα(X))α<ω1 exhibit a remarkable regularity. They areconstantly one until α reaches the spectral index of X, I∆(X); and then decreasegeometrically to 0 as α reaches I∆(X) · ω (Theorem 4.23). An important tool inthis section is the renorming result of Theorem 4.20.

Section 5 contains the calculation of asymptotic constants for various asymp-totic `1 spaces. We consider T along with various other Tsirelson and mixedTsirelson spaces. These and other examples show that there is potentially con-siderable variety in the spectrum of X despite the regularity conditions imposedwhen considering all renormings. In addition it is shown that for γ ∈ ∆(X) anappropriate block basis in X admits a lower Tγ1 block Tsirelson estimate.

The central theme of Section 6 is the following problem: Does there exist anasymptotic `1 Banach space of bounded distortion? In particular, is Tsirelson’sspace of bounded distortion? We apply our work to obtain some partial resultsin this and related directions. We consider the consequences of assuming that anasymptotic `1 space is of bounded distortion. In particular the asymptotic constantsmust behave in a geometric fashion (Theorem 6.8, Corollary 6.9, Propositions 6.12and 6.13). Also, an asymptotic `1 space of bounded distortion bears a strikingresemblance to a subspace of a Tsirelson-type space T (Sα, θ) for some α < ω1 and0 < θ < 1 (Theorem 6.10). Furthermore we show that a renorming of Tsirelson’sspace T for which there exists γ in the spectrum with γ1 = 1/2 cannot distort Tby more than a fixed constant (Theorem 6.2).

2. Preliminaries

2.1. Distortion. If a Banach space (X, ‖ · ‖) is given an equivalent norm | · |we define the distortion of | · | by

Definition 2.1.

d(X, | · |) = infY

sup{|x||y|

: x, y ∈ S(Y, ‖ · ‖)}

,

where the infimum is taken over all infinite-dimensional subspaces Y of X.

Remark 2.2. If X has a basis, then a standard approximation argument easilyshows that in the above formula for d(X, | · |) it is sufficient to take the infimum over

2. PRELIMINARIES 19

all block subspaces Y ≺ X; and this is the form of the definition we shall alwaysuse.

The parameter d(X, | · |) measures how close | · | can be made to being a multipleof ‖ · ‖, by restricting to an infinite-dimensional subspace.

Definition 2.3. For λ > 1, (X, ‖ · ‖) is λ-distortable if there exists an equiva-lent norm | · | on X so that d(X, | · |) ≥ λ. X is distortable if it is λ-distortable forsome λ > 1. X is arbitrarily distortable if it is λ-distortable for all λ > 1.

Definition 2.4. A space (X, ‖ · ‖) is of D-bounded distortion if for all equiv-alent norms | · | on X and all Y ⊆ X, d(Y, | · |) ≤ D. A space X is of boundeddistortion if it is of D-bounded distortion for some D < ∞.

Let us mention a more geometric approach to distortion. A subset A ⊆ Xis called asymptotic if dist(A, Y ) = 0 for all infinite-dimensional subspaces Y ofX, i.e. for all Y and ε > 0 there is x ∈ A such that infy∈Y ‖x − y‖ < ε. Givenη > 0, consider the following property of X: there exist A,B ⊆ S(X) and A∗ inthe unit ball of X∗ such that: (i) A and B are asymptotic in X; (ii) for everyx ∈ A there is x∗ ∈ A∗ such that |x∗(x)| ≥ 1/2; (iii) for all y ∈ B and x∗ ∈ A∗,|x∗(y)| < η. It is well known and easy to see that if d(X, |·|) ≥ λ for some equivalentnorm | · | on X then in some Y ⊆ X there exist such asymptotic (in Y ) “almostbiorthogonal” sets, with η = 1/λ. Conversely, given sets A,B and A∗ as above, let|x| = ‖x‖+ (1/η) sup{|x∗(x)| : x∗ ∈ A∗} for x ∈ X. Then d(X, | · |) ≥ (1/2 + 1/4η).

A proof of the following simple proposition is left for the reader. Part b) wasshown in [To2].

Proposition 2.5. a): Let (X, ‖ · ‖) be of D-bounded distortion and let| · | be an equivalent norm on X. Then for all ε > 0 and Y ⊆ X thereexists Z ⊆ Y and c > 0 so that |z| ≤ c‖z‖ ≤ (D + ε)|z| for all z ∈ Z.

b): Every Banach space contains either an arbitrarily distortable subspaceor a subspace of bounded distortion.

Note that if X has a basis then one may replace Y ⊆ X and Z ⊆ Y , inDefinition 2.4 and Proposition 2.5, and the definition of an asymptotic set, byY ≺ X and Z ≺ Y , respectively.

It was shown in [OS1], [OS2] that every X contains either a distortable sub-space or a subspace isomorphic to `1 or c0 (both of which are not distortable [J]).Currently no examples of distortable spaces of bounded distortion are known. It isknown that such a space would for some 1 ≤ p ≤ ∞ necessarily contain an asymp-totic `p subspace (defined below for p = 1) with an unconditional basis and mustcontain `n

1 ’s uniformly ([MiTo], [M], [To2]).In light of these results it is natural to focus the search for a distortable space

of bounded distortion on asymptotic `1 spaces with an unconditional basis.

2.2. Asymptotic `1 Banach spaces. Several definitions of asymptotic `1spaces appear in the literature. We shall use the definition from [MiTo].

Definition 2.6. A space X with a basis (ei) is an asymptotic `1 space (w.r.t.(ei)) if there exists C such that for all n and all en ≤ x1 < · · · < xn,

‖n∑1

xi‖ ≥ (1/C)n∑1

‖xi‖ .

20 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES3

The infimum of all C’s as above is called the asymptotic `1 constant of X.

It should be noted that this definition depends on the choice of a basis: a spaceX may be asymptotic `1 with respect to one basis but not another. However whenthe basis is understood, the reference to it is often dropped.

In [MMiTo] a notion of asymptotic structure of an arbitrary Banach spacewas introduced; in as much as we shall not use it here, we omit the details. Thisled, in particular, to a more general concept of asymptotic `1 spaces; and spacessatisfying Definition 2.6 above were called there “stabilized asymptotic `1”. Severalconnections between the “MMT -asymptotic structure” of a space [MMiTo] andthe “stabilized asymptotic structure” of its subspaces can be proved; for instance,an MMT -asymptotic `1 space contains an asymptotic `1 space in the sense ofDefinition 2.6.

Before proceeding we shall briefly consider the prime example of an asymptotic`1 space not containing `1, namely Tsirelson’s space T [T]. Our discussion willmotivate our subsequent definitions. The space T is actually the dual of Tsirelson’soriginal space. It was described in [FJo] as follows.

Let c00 be the linear space of finitely supported sequences. T is the completionof (c00, ‖ · ‖) where ‖ · ‖ satisfies the implicit equation

‖x‖ = max

(‖x‖∞, sup

{12

n∑i=1

‖Eix‖ : n ∈ N and n ≤ E1 < · · · < En

}).

In this definition the Ei’s are finite subsets of N. Eix is the restriction of xto the set Ei. Thus if x = (x(j)) then Eix(j) = x(j) if j ∈ Ei and 0 otherwise.Of course it must be proved that such a norm exists. The unit vector basis (ei)forms a 1-unconditional basis for T and T is reflexive. If en ≤ x1 < · · · < xn w.r.t.(ei) then ‖

∑n1 xi‖ ≥ 1

2

∑n1 ‖xi‖ and so T is asymptotic `1 with constant less than

or equal to 2. The next proposition is the best that can currently be said aboutdistorting T . The proof, which we sketch, is illustrative.

Proposition 2.7. T is (2− ε)-distortable for all ε > 0.

Proof. (Sketch) Let ε > 0 and choose n so that 1/n < ε. Define for x ∈ T ,

|x| = sup{ n∑

i=1

‖Eix‖ : E1 < · · · < En

}.

Clearly, ‖x‖ ≤ |x| ≤ n‖x‖ for x ∈ T (in fact, for n ≤ x, |x| ≤ 2‖x‖). Let(xi) ≺ (ei)∞n . For any k > n some normalized sequence (yi)k

1 ≺ (xi)∞k is equivalentto the unit vector basis of `k

1 , with the equivalence constant as close to 1 as wewish. Thus if y = (1/k)

∑k1 yi, then ‖y‖ ≈ 1. Also if E1 < · · · < En then setting

I = {i : Ej ∩ supp(yi) 6= ∅ for at most one j} and J = {1, . . . , k} \ I we have that|J | ≤ n and

n∑1

‖Ejy‖ ≤ 1k

(∑i∈I

‖yi‖+∑i∈J

∑j

‖Ejyi‖)

≤ 1k

(∑i∈I

‖yi‖+∑i∈J

2‖yi‖)

≤ 1k

(k − |J |+ 2|J |) ≤ 1 +n

k.

2. PRELIMINARIES 21

Thus inf{|x| : ‖x‖ = 1, x ∈ 〈xi〉} = 1.Now let z = (2/n)

∑n1 zi ∈ 〈xi〉∞n where z1 < · · · < zn and each zi is an `ki

1 -average of the sort just considered. Here ki+1 is taken very large depending onmax supp(zi) and ε. Since ‖zi‖ ≈ 1, it follows that |z| ≥ (2/n)

∑‖zi‖ ≈ 2. Yet, if

m ≤ E1 < · · · < Em, and i0 is the smallest i such that em < max supp(zi), thenthe growth condition for ki implies that ki is much larger than m for i0 < i ≤ n.Hence by the argument above

12

m∑1

‖Ejz‖ =1n

m∑1

∥∥∥∥Ej

( n∑1

zi

)∥∥∥∥≤ 1

n

( m∑j=1

‖Ejzi0‖+n∑

i=i0+1

m∑j=1

‖Ejzi‖)

(4)

≤ 1n

(2‖zi0‖+

n∑i=i0+1

(1 + n/ki))

<∼ n + 1n

< 1 + ε .

By the definition of the norm we get ‖z‖ ≤ 1 + ε. This implies sup{|z| : ‖z‖ =1, z ∈ 〈xi〉} > 2/(1 + ε). �

Later we shall say that a sequence (yi)k1 is S1-admissible w.r.t. (xi) if xk ≤

y1 < · · · < yk. In the above proof we needed to consider an admissible sequence ofadmissible sequences; what we shall later call S2-admissible.

Inequality (4) obviously shows that the asymptotic `1 constant of T is greaterthan or equal, and hence equal, to 2. Furthermore, if X ≺ T , then X is anasymptotic `1 space with constant again equal to 2. In other words, passing to ablock basis of T does not improve the asymptotic `1 constant. Vitali Milman askedthe question what would happen if in addition we renormed? The above techniquegives that the constant cannot be improved too much.

Proposition 2.8. If |·| is any equivalent norm on X ≺ T then X is asymptotic`1 with constant at least

√2.

Proof. (Sketch) Let X ≺ T and consider an equivalent norm | · | on X sothat (X, | · |) is asymptotic `1 with constant θ. By multiplying | · | by a constantand passing to a block subspace of X if necessary we may assume that ‖ ·‖ ≥ | · | onX and for all Y ≺ X there exists y ∈ Y with ‖y‖ = 1 and |y| ≈ 1. Given n, choosez1 < z2 < · · · < zn w.r.t. X so that zi = (1/ki)

∑ki

1 zi,j where zi,1 < · · · < zi,kiin

X and ‖zi,j‖ = 1 ≈ |zi,j |. Here ki+1 is again large depending upon zi.Let z = (2/n)

∑n1 zi. Then as before we obtain |z| ≤ ‖z‖ <∼ 1 + (1/n). On the

other hand, |z| ≥ (2/nθ)∑n

1 |zi|>∼(2/nθ2)n = 2/θ2. Hence 2/θ2 <∼ 1. �

Remark 2.9. For any 0 < θ < 1 Tsirelson’s space Tθ is defined by the implicitequation analogous to the definition of T , in which the constant 1/2 is replaced byθ. The properties of T remain valid for Tθ as well, with appropriate modificationof the constants involved.

These results indicate that it could be of advantage to consider the `1-ness ofsequences which are S2-admissible with respect to a basis or even Sn-admissible.We do so in this paper and we shall obtain the best possible improvement of Propo-sition 2.8 in Theorem 5.2 (see also Remark 5.3). Of course the beautiful examplesof Argyros and Deliyanni [ArD] of arbitrarily distortable mixed Tsirelson spaces

22 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES4

(described below) also show the need for consideration of such notions when study-ing asymptotic `1 spaces. Our point here is that these are needed even to answerS1-admissibility questions.

3. The Schreier families Sα

Let F be a set of finite subsets of N. F is hereditary if whenever G ⊆ F ∈ Fthen G ∈ F . F is spreading if whenever F = (n1, · · · , nk) ∈ F , with n1 < · · · < nk

and m1 < · · · < mk satisfies mi ≥ ni for i ≤ k then (m1, . . . ,mk) ∈ F . F ispointwise closed if F is closed in the topology of pointwise convergence in 2N. Aset F of finite subsets of N having all three properties we call regular . If F and Gare regular we let

F [G] ={ n⋃

1

Gi : n ∈ N, G1 < · · · < Gn, Gi ∈ G for i ≤ n, (minGi)n1 ∈ F

}.

Note that this operation satisfies the natural associativity condition (F [G1])[G2] =F [G0], where G0 = G1[G2].

If N = (n1, n2, . . .) is a subsequence of N then F(N) = {(ni)i∈F : F ∈ F}. If Fis regular and M is a subsequence of N then, since F is spreading, F(M) ⊂ F(N).If F is regular and n ∈ N we define [F ]n by [F ]1 = F and [F ]n+1 = F

[[F ]n

].

Finally, if F is a finite set, |F | denotes the cardinality of F .

Definition 3.1. [AAr] The Schreier classes are defined by S0 = {{n} : n ∈N} ∪ {∅}, S1 = {F ⊆ N : minF ≥ |F |} ∪ {∅}; for α < ω1, Sα+1 = S1[Sα], and if αis a limit ordinal we choose αn ↑ α and set

Sα = {F : for some n ∈ N, F ∈ Sαnand F ≥ n} .

It should be noted that the definition of the Sα’s for α ≥ ω depends upon thechoices made at limit ordinals but this particular choice is unimportant for ourpurposes. Each Sα is a regular class of sets. It is easy to see that S1 ⊆ S2 ⊆ · · ·and Sn[Sm] = Sm+n, for n, m ∈ N, but this fails for higher ordinals. However wedo have

Proposition 3.2. a): Let α < β < ω1. Then there exists n ∈ N so thatif n ≤ F ∈ Sα then F ∈ Sβ.

b): For all α, β < ω1 there exists a subsequence N of N so that Sα[Sβ ](N) ⊆Sβ+α.

c): For all α, β < ω1 there exists a subsequence M of N so that Sβ+α(M) ⊆Sα[Sβ ].

We start with an easy formal observation.

Lemma 3.3. Let Fand G be sets of finite subsets of N and let G be spreading.Assume that there exists a subsequence N of N so that F(N) ⊆ G. Then for allsubsequences L of N there exists a subsequence L′ of L with F(L′) ⊆ G.

Proof. Let L = (li). Let N = (ni) such that F(N) ⊆ G. Since G is spreading,any L′ = (l′i) ⊆ (li) such that l′i ≥ ni for all i satisfies the conclusion (for instanceone can take L′ = (lni

)). �

3. THE SCHREIER FAMILIES Sα 23

Proof of Proposition 3.2. a) We proceed by induction on β. If β = γ + 1 thenα ≤ γ and so we may choose n so that if n ≤ F ∈ Sα then F ∈ Sγ ⊆ Sβ . If β isa limit ordinal and βn ↑ β is the sequence used in defining Sβ , choose n0 so thatα < βn0 . Choose n ≥ n0 so that if n ≤ F ∈ Sα then F ∈ Sβn0

. Thus also F ∈ Sβ .b) We induct on α. Since S0[Sβ ] = Sβ , the assertion is clear for α = 0. If

α = γ + 1, then Sα[Sβ ] = S1[Sγ [Sβ ]] and Sβ+α = S1[Sβ+γ ]. Thus we can take Nto satisfy Sγ [Sβ ](N) ⊆ Sβ+γ .

If α is a limit ordinal we argue as follows. First, by Lemma 3.3, the inductivehypothesis implies that for every α′ < α and every subsequence L of N there existsa subsequence N of L with Sα′ [Sβ ](N) ⊆ Sβ+α′ . Let αn ↑ α and γn ↑ β + α be thesequences of ordinals used to define Sα and Sβ+α, respectively.

Choose subsequences of N, L1 ⊇ L2 ⊇ · · · so that Sαk[Sβ ](Lk) ⊆ Sβ+αk

. IfLk = (`k

i )∞i=1 we let L be the diagonal L = (`k) = (`kk)∞k=1. It follows that if

F ∈ Sαk[Sβ ](L) and F ≥ `k then F ∈ Sαk

[Sβ ](Lk) and so F ∈ Sβ+αk. For each

k choose n(k) so that β + αk < γn(k). Using a) choose j(1) < j(2) < · · · so thatif j(k) ≤ F ∈ Sβ+αk

then F ∈ Sγn(k) . Let N = (n(k))∞k=1 be a subsequence of Nwith n(k) ≥ `k ∨ j(k)∨ n(k) for all k. Then if F ∈ Sα[Sβ ](N) there exists k so thatnk ≤ F ∈ Sαk

[Sβ ](N) and so `k ≤ F ∈ Sβ+αkand j(k) ≤ F ∈ Sγn(k) , whence since

n(k) ≤ F we have F ∈ Sβ+α.c) As in b) we induct on α. The cases α = 0 and α = γ + 1 are trivial.

Thus assume that α is a limit ordinal. Let αr ↑ α and γr ↑ β + α be the sequencesdefining Sα and Sβ+α respectively. We may write (γr) = (γ1, . . . , γn0−1, β+γn0 , β+γn0+1, . . .) where γi < β if i < n0. By a) there exists m0 so that if m0 ≤ F ∈⋃n0−1

1 Sγithen F ∈ Sβ . We shall take later M = (mi)∞1 where m1 ≥ m0. By

the inductive hypothesis and Lemma 3.3. choose sequences Ln0 ⊇ Ln0+1 ⊇ · · · sothat Sβ+γk

(Lk) ⊆ Sγk[Sβ ] for k ≥ n0 and m0 ≤ Ln0 . If Lk = (`k

i )i set L = (`k)where `k = `k

k for k ≥ n0, and m0 ≤ `1 < · · · < `k−1 < `k < · · · . Thus if k ≥ n0,`k ≤ F ∈ Sβ+γk

(L) implies that F ∈ Sγk[Sβ ]. Also for k < n0, `k ≤ F ∈ Sγk

implies that F ∈ Sβ . For k ≥ n0 choose m(k) so that β + γk < β + αm(k). Bya) there exists n(k) so that n(k) ≤ F ∈ Sγk

[Sβ ] implies that F ∈ Sαm(k) [Sβ ] forall k ≥ n0. Finally we choose M = (m(k)) where m(k) = `k for k < n0 andm(k) ≥ `k ∨ m(k) ∨ n(k) for k ≥ n0. Thus if F ∈ Sβ+α(M) then F ∈ Sβ or elsethere exists k ≥ n0 with m(k) ≤ F ∈ Sβ+γk

(M). Hence `k ≤ F ∈ Sγk[Sβ ] and so

n(k) ≤ F ∈ Sαm(k) [Sβ ]. Since F ≥ m(k) we get that F ∈ Sα[Sβ ]. 2

Corollary 3.4. For all α < ω1 and n ∈ N there exist subsequences M and Nof N satisfying [Sα]n(N) ⊆ Sα·n and Sα·n(M) ⊆ [Sα]n.

Proof. This is easily established by induction on n using Proposition 3.2. Forexample, if [Sα]n(P ) ⊆ Sα·n and Sα[Sα·n](L) ⊆ Sα·(n+1), let N = (pli) (here P =(pi) and L = (li)). Then [Sα]n(N) ⊆ Sα·n(L) and so [Sα]n+1(N) = Sα[Sα]n(N) ⊆Sα[Sα·n](L) ⊆ Sα·(n+1). �

Remark 3.5. The Schreier family Sα has been used in [AAr] to construct aninteresting subspace Sα of C(ωωα

) as follows. Sα is the completion of c00 under thenorm

‖x‖ = sup{|∑i∈E

x(i)|`1 : E ∈ Sα} .

24 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES5

The unit vector basis is an unconditional basis for Sα. The space Sα does notembed into C(ωωβ

) for any β < α.

The next important proposition is a slight generalization of a result in [ArD]and is a descendent of results in [B].

Proposition 3.6. Let β < α < ω1, ε > 0 and let M be a subsequence ofN. Then there exists a finite set F ⊆ M and (aj)j∈F ⊆ R+ so that F ∈ Sα(M),∑

j∈F aj = 1 and if G ⊆ F with G ∈ Sβ then∑

j∈G aj < ε.

Proof. We proceed by induction on α. The result is clear for α = 1. LetM = (mi). We choose 1/k < ε, F ⊆ M , F > mk, |F | = k and let aj = 1/k ifj ∈ F .

If α is a limit ordinal let αn ↑ α be the sequence used to define Sα. Choose n sothat β < αn. Applying the induction hypothesis to β, αn and {m ∈ M : m ≥ mn}yields the result.

If α = γ + 1 we may assume (by Proposition 3.2) that β = γ. If γ is a limitordinal let γn ↑ γ be the sequence used to define Sγ . Choose k so that 1/k < ε/2.Choose sets Fi ⊆ M with mk ≤ F1 < · · · < Fk along with scalars (aj)j∈

⋃k1 Fi

⊆ R+

and n1 < · · · < nk satisfying the following:

1):∑

j∈Fiaj = 1 for i ≤ k

2): Fi ∈ Sγni(M) and mni

< Fi for 1 ≤ i ≤ k

3):∑

j∈G aj < 1/2i if G ⊆ Fi+1 with G ∈ Sγ`whenever ` ≤ max Fi for

1 ≤ i < k.

Let F =⋃k

1 Fi. Then F ⊆ M and F ∈ Sγ+1(M). For j ∈ F set bj = k−1aj .Then (bj)j∈F ⊆ R+,

∑j∈F bj = 1 and if G ∈ Sγ , G ⊆ F then

∑j∈G bj < ε. Indeed

there exists n with n ≤ G ∈ Sγn. Thus if i0 = min{i : G∩Fi 6= ∅} then n ≤ max Fi0

and so by 3),

∑j∈G

bj =k∑

i=i0

( ∑j∈Fi∩G

bj

)≤ 1

k

(1 +

12i0

+ · · ·+ 12k

)< ε .

If γ = η + 1 we again choose 1/k < ε/2 and sets mk ≤ F1 < · · · < Fk,Fi ∈ Sγ(M), along with (aj)j∈Fi ⊆ R+,

∑j∈Fi

aj = 1 so that if G ∈ Sη then∑j∈G∩Fi+1

aj < (1/2i max(Fi)) for 1 ≤ i < k. As above we set F =⋃k

1 Fi and letbj = k−1aj if j ∈ Fi. Thus F ∈ Sα(M) and if G ∈ Sγ , G ⊆ F , write G =

⋃ps=1 Gs

where p ≤ G1 < · · · < Gp and Gi ∈ Sη for each i. Then if i0 = min{i : G∩Fi 6= ∅},since max(Fi0) ≥ p,

∑j∈G

bj ≤1k

+k∑

i=i0+1

12i

p

k

1max(Fi)

≤ 1k

+1k

k∑i=i0+1

2−i < ε .

�

Definition 3.7. Let ε > 0 and β < α < ω1. If (ei) is a normalized basicsequence, M is a subsequence of N and F and (ai)i∈F are as in Proposition 3.6,we call x =

∑i∈F aiei an (α, β, ε)-average of (ei)i∈M . If (xi) is a normalized block

basis of (ei) and F and (ai)i∈F are as in Proposition 3.6 for M = (min supp(xi)),we call x =

∑i∈F aixi an (α, β, ε)-average of (xi) w.r.t. (ei).

3. THE SCHREIER FAMILIES Sα 25

The Schreier families are large within the set of all classes of pointwise closedsubsets of [N]<ω. Our next two propositions show that they are in a sense the largestamong all regular classes of a given complexity. To make this concept precise weconsider the index I(F) defined as follows. Let D(F) = {F ∈ F : there exist(Fn) ⊆ F with 1Fn

→ 1F pointwise and Fn 6= F for all n}, Dα+1(F) = D(Dα(F))and Dα(F) =

⋂β<α Dβ(F) when α is a limit ordinal. Then

I(F) = inf{α < ω1 : Dα(F) = {∅}} .

F is a countable compact metric space in the topology of pointwise convergenceand so I(F) must be countable, see e.g. [Ku], p. 261-262.

Remark 3.8. The Cantor-Bendixson index of F (under the topology of point-wise convergence) is I(F) + 1. This is because ∅ corresponds to the 0 functionand one needs one more derivative to get ∅ : DI(F)+1(F) = ∅, which defines theCantor-Bendixson index.

Now we have ([AAr])

Proposition 3.9. For α < ω1, I(Sα) = ωα.

Proof. We induct on α. The result is clear for α = 0. If the proposition holdsfor α it can be easily seen that for n ∈ N, Dωα·n(Sα+1) = {F : there exists k ∈ N,k > n, with F =

⋃k−n1 Fi, k ≤ F1 < · · · < Fk−n, and Fi ∈ Sα for i ≤ k−n}. Hence

I(Sα+1) = ωα+1. The case where α is a limit ordinal is also easily handled. �

Proposition 3.10. If F is a regular set of finite subsets of N with I(F) ≤ ωα

then there exists a subsequence M of N with F(M) ⊆ Sα.

This proposition is a special case of more complicated statements (Proposi-tion 3.12 and Remark 3.13) below. First let us recall (see e.g., [Mo]) that everyordinal β < ω1 can be uniquely written in Cantor normal form as

β = ωα1 · n1 + ωα2 · n2 + · · ·+ ωαj · nj

where (ni)j1 ⊆ N and ω1 > α1 > · · · > αj ≥ 0.

Definition 3.11. If (αi)j1 are countable ordinals and (ni)

j1 ⊆ N, by ((Sα1)

n1 , . . . , (Sαj)nj )

we denote the class of subsets of N that can be written in the form

E11 ∪ · · · ∪ E1

n1∪ E2

1 ∪ · · · ∪ E2n2∪ · · · ∪ Ej

1 ∪ · · · ∪ Ejnj

where E11 < E1

2 < · · · < Ejnj

and Eki ∈ Sαk

for all k ≤ j and i ≤ nk.

Proposition 3.12. Let F be a regular set of finite subsets of N with

I(F) = ωα1 · n1 + · · ·+ ωαk−1 · nk−1 + ωαk · nk,

in Cantor normal form. Then there exists a subsequence M of N so that we haveF(M) ⊆ ((Sα1)

n1 , . . . , (Sαk)nk).

Remark 3.13. The conclusion of the proposition holds even if I(F) < ωα1 ·n1+· · ·+ ωαk · nk. Indeed this follows from the fact that if α < β and F(N) ⊆ Sα, andN = (ni), then there exists r ∈ N so that F((ni)i≥r) ⊆ Sβ (by Proposition 3.2(a)).

26 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES6

Proof of Proposition 3.12. We induct on I(F). If I(F) = 1 then F contains onlysingletons {n} and so F(N) ⊆ ((S0)1).

Assume the proposition holds for all classes with index ≤ β, and let I(F) =β + 1. Let β = ωα1 · n1 + · · ·+ ωαk−1 · nk−1 + nk (nk ≥ 0). Then D(F) is a regularclass of sets with I(D(F)) = β and so there exists M with

D(F)(M) ⊆ ((Sα1)n1 , . . . , (Sαk−1)

nk−1 , (S0)nk)

where (S0)0 ≡ ∅). Note that if F ∈ F \D(F) then if k = maxF and G = F \ {k}we have that G ∈ D(F). Indeed since F is regular, for n ≥ k, G ∪ {n} ∈ F andG ∪ {n} → F . It follows that F(M) ⊆ ((Sα1)

n1 , . . . , (S0)nk+1).If β is a limit ordinal and the proposition holds for regular classes with index

< β we proceed as follows. Let β = ωα1 · n1 + · · · + ωαk · nk where αk > 0 andnk > 0. For j ∈ N set Fj = {F ∈ F : F > j and {j} ∪ F ∈ F}. Then clearlyeach Fj is regular. Also I(Fj) < β for if I(Fj) = β then Dβ(Fj) = {∅}. By thedefinition of Fj this implies that {j} ∈ Dβ(F) which contradicts I(F) = β. Foreach j set

Gj = {{j} ∪ F : F ∈ Fj} ∪ Fj .

Each Gj is regular, I(Gj) ≤ I(Fj) + 1 < β and F =⋃

j Gj .If αk = (αk − 1) + 1 (i.e., is a successor ordinal), choose pj ↑ ∞ so that

I(Gj) < ωα1 · n1 + · · ·+ ωαk · (nk − 1) + ωαk−1 · pj .

By Remark 3.13 and the inductive hypothesis, there exists Mj ⊆ N with

Gj(Mj) ⊆ ((Sα1)n1 , . . . , (Sαk

)nk−1 , (Sαk−1)pj ) .

Without loss of generality we can choose the Mj ’s so that M1 ⊇ M2 ⊇ · · · . LetMj = (mj

i )∞i=1. Let M = (m1

p1,m2

p2, . . .) ≡ (mi). If ∅ 6= F ∈ F then F ∈ Gj for

j = minF . Since (mj ,mj+1, . . .) ⊆ (mjj ,m

jj+1, . . .) and Gj is spreading, we have

that (mi)i∈F ∈ ((Sα1)n1 , . . . , (Sαk

)nk−1, (Sαk−1)pj ). Also mj = mjpj≥ pj and so

(mi)i∈F ∈ ((Sα1)n1 , . . . , (Sαk

)nk) .

If αk is a limit ordinal, let ηj ↑ αk be the sequence of ordinals defining Sαk.

Choose `(j) ↑ ∞ so that

I(Gj) ≤ ωα1 · n1 + · · ·+ ωαk · (nk − 1) + ωη`(j) .

As above we choose M1 ⊇ M2 ⊇ · · · so that

Gj(Mj) ⊆ ((Sα1)n1 , . . . , (Sαk

)nk−1, (Sη`(j))) .

As in Remark 3.13 we may choose rj ↑ ∞ so that when considering Gj((mji )i≥rj

) thelast set we get is not only in Sη`(j) but also in Sαk

. Thus if M = (mj) = (mjrj

)∞j=1

it follows that F(M) ⊆ ((Sα1)n1 , . . . , (Sαk

)nk). 2

Corollary 3.14. Let F be a pointwise closed class of finite subsets of N. Thenthere exist α < ω1 and a subsequence M of N so that F(M) ⊆ Sα.

Proof. Let R be the regular hull of F ; that is, R = {G : there exists F =(n1, . . . , nk) ∈ F with G ⊆ (mi)k

1 for some m1 < · · · < mk with mi ≥ ni for i ≤ k}.Clearly, R is hereditary and spreading. We check that it is also pointwise closed,

and hence the corollary follows from Proposition 3.10. Let Gn → G pointwise forsome (Gn) ⊆ R. If |G| < ∞ then G is an initial segment of Gn for large n and soG ∈ R. It remains to note that |G| = ∞ is impossible. If G = (n1, n2, . . .) then for

4. ASYMPTOTIC CONSTANTS AND ∆(X) 27

all k, (n1, . . . , nk) is a subset of some spreading of some set Fk ∈ F . In particular|{n ∈ Fk : n ≤ nj}| ≥ j for 1 ≤ j ≤ k. Thus any limit point of (Fk)∞k=1 is infinitewhich contradicts the hypotheses that F is pointwise closed and consists of finitesets. �

For some other interesting properties of the Schreier classes we refer the readerto [ArMeTs] and [AnO].

4. Asymptotic constants and ∆(X)

Asymptotic constants considered in this paper will be determined by the Schreierfamilies Sα; nevertheless it should be noted that they can be introduced for a verygeneral class of families of finite subsets of N.

Definition 4.1. If F is a regular set of finite subsets of N, a sequence of setsE1 < · · · < Ek is F-admissible if (min(Ei))k

i=1 ∈ F . If (xi) is a basic sequencein a Banach space and (yi)k

1 ≺ (xi), then (yi)k1 is F-admissible (w.r.t. (xi)) if

(supp(yi))k1 is F-admissible, where supp(yi) is taken w.r.t. (xi). We use a short

form α-admissible to mean Sα-admissible.

The next definition was first introduced in [To1] for asymptotic `p spaces with1 ≤ p < ∞.

Definition 4.2. Let F be a regular set of finite subsets of N. For a basicsequence (xi) in a Banach space X we define δF (xi) to be the supremum of δ ≥ 0such that whenever (yi)k

1 ≺ (xi) is F-admissible w.r.t. (xi) then

‖k∑

i=1

yi‖ ≥ δk∑

i=1

‖yi‖ .

If X is a Banach space with a basis (ei) we write δF (X) for δF (ei). For α < ω1,we set δα(xi) = δSα(xi) and δα(X) = δSα(X).

Remark 4.3. Note that δF (xi) is equal to the supremum of all δ′ ≥ 0 suchthat ‖y‖ ≥ δ′

∑‖Eiy‖, for all y ∈ 〈xi〉 and all adjacent F-admissible intervals

E1 < · · · < Ek such that⋃

Ei ⊇ supp(y). Here the support of y and restrictionsEiy are understood to be w.r.t. (xi). Indeed, clearly sup δ′ ≥ δF (xi). Conversely,given (yi)k

1 ≺ (xi) F-admissible we set y =∑

yi and we let (E1, . . . , Ek) be adjacentintervals such that Ei ⊇ supp(yi) and minEi = min supp(yi) for all i.

In as much as distortion problems involve passing to block subspaces andrenormings, it is natural to make two more definitions.

Definition 4.4. Let F be a regular set of finite subsets of N and let (ei) be abasis for X.

δF (X) = δF (ei) = sup{δF (xi) : (xi) ≺ (ei)} and

δF (X) = δF (ei) = sup{δF ((ei), | · |) : | · | is an equivalent norm on X} .

We write δSα(X) = δα(X) and δSα

(X) = δα(X).

The asymptotic constants provide a measurement of closeness of block sub-spaces of X to `1. Clearly X is asymptotic `1 w.r.t. (ei) if and only if δ1(X) > 0.The asymptotic `1 constant of X is then equal to δ1(X)−1. On the other hand wealso have

28 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES7

Proposition 4.5. X contains a subspace isomorphic to `1 if and only if δα(X) >0 for all α < ω1.

Proof. This follows from Bourgain’s `1 index of a Banach space X whichwe recall now. For 0 < c < 1, T (X, c) is the tree of all finite normalized se-quences (xi)k

1 ⊆ X satisfying ‖∑k

1 aixi‖ ≥ c∑k

1 |ai| for (ai)k1 ⊆ R. The order

on the tree is (xi)k1 ≤ (yi)n

1 if k ≤ n and xi = yi for i ≤ k. For ordinalsβ < ω1 we define Dβ(T (X, c)) inductively by D1(T (X, c)) = {(xi)k

1 ∈ T (X, c) :(xi)k

1 is not maximal}. Dβ+1(T (X, c)) = D1(Dβ(T (X, c))) and Dβ(T (X, c)) =⋂γ<β Dγ(T (X, c)) if β is a limit ordinal. The index I(X) is defined by I(X) =

sup0<c<1 inf{β : Dβ(T (X, c)) = ∅}, where the infimum is set equal to ω1 if no suchβ exists. Bourgain showed that for a separable space X, I(X) < ω1 if and only ifX does not contain a subspace isomorphic to `1 [Bou].

Now observe that if F is a regular set of finite subsets of N then D(F) = {F ∈F : F ∪ {k} ∈ F for some F < k}. It follows that if δF (xi) > 0 for some basicsequence (xi) in X then I(X) ≥ I(F). Hence by Proposition 3.9, if δα(X) > 0 forevery α < ω1 then I(X) = ω1, hence X contains a subspace isomorphic to `1. Theconverse implication is obvious. �

Other facts about Bourgain’s `1 index can be found in [JuO].The next lemma collects some simple observations about the asymptotic con-

stants.

Lemma 4.6. Let (ei) be a basis for X and let (xi) ≺ (ei). Let F and G beregular classes of finite subsets of N.

a): δF (ei) ≤ δF (xi) and δF (xi) ≤ δF (ei);b): δF (ei) ≤ δF (ei) ≤ δF (ei);c): infn δn(ei) > 0 iff (ei/‖ei‖) is equivalent to the unit vector basis of `1;d): δF (ei) = sup(xi)≺(ei) sup{δF ((xi), |·|) : |·| is an equivalent norm on [xi]i∈N};e): δF [G](xi) ≥ δF (xi)δG(xi).

Proof. a) and b) are immediate; the first part of a) uses that F(M) ⊆ F .c) follows from the fact that

⋃∞n=1 Sn contains all finite subsets of {2, 3, . . .}. d)

is true because if Y ⊆ X and | · | is an equivalent norm on Y then | · | can beextended to an equivalent norm on X. For e) notice that if (yi)k

i is F [G]-admissiblew.r.t. (xi), then it can be blocked in a F-admissible way into successive blockseach of which consists of G-admissible vectors (w.r.t. (xi)). This directly impliesthe inequality. �

The most important situation for the study of the constants δα is when thewhole sequence (δα)α<ω1 is stabilized on a nested sequence of block subspaces.This leads to the concept of the ∆-spectrum of X to be all possible stabilized limitsof δα’s of block bases. We formalize it in the following definition.

Definition 4.7. Let X be a Banach space and let γ = (γα)α<ω1 ⊆ R. Wesay that a basic sequence (xi) in X ∆-stabilizes γ if there exist εn ↓ 0 so thatfor every α < ω1 there exists m ∈ N so that for all n ≥ m if (yi) ≺ (xi)∞n then|δα(yi)− γα| < εn.

Let X have a basis (ei). The ∆-spectrum of X, ∆(X), is defined to be the setof all γ’s so that there exists (xi) ≺ (ei) such that (xi) ∆-stabilizes γ. By ∆(X)

4. ASYMPTOTIC CONSTANTS AND ∆(X) 29

we denote the set of all γ’s so that (xi) ∆-stabilizes γ for some (xi) ≺ (ei), undersome equivalent norm | · | on [xi]i∈N.

Remark 4.8. It is important to note that the asymptotic constants δα(yi)considered here and appearing in the definition of the spectrum ∆(X) refer to theadmissibility with respect to the block basis (yi) itself. It is sometimes convenient,however, to consider asymptotic constants that keep a reference level for admis-sibility fixed when passing to block bases. Precisely, if (ei) is a basis in X and(xi) ≺ (ei), we define δF ((xi), (ei)) as the supremum of δ ≥ 0 such that when-ever (yi)k

1 ≺ (xi) is F-admissible w.r.t. (ei) then ‖∑k

1 yi‖ ≥ δ∑k

1 ‖yi‖. Clearly,δF (ei) ≤ δF ((xi), (ei)) ≤ δF (xi). We can then define the spectrum ∆(X, (ei)) byreplacing δα(yi) by δSα

((yi), (ei)), in Definition 4.7 above. Let us also note that ithas been proved in [AnO] that these two concepts of spectrum actually coincideand ∆(X, (ei)) = ∆(X).

Remark 4.9. The definition of Sα for α ≥ ω0 depended upon certain choicesmade at limit ordinals. It follows that the constants δα(ei) also depend upon theparticular choice of Sα. However ∆(X) is independent of the choice of each Sα.Indeed, this follows from a consequence of Propositions 3.9 and 3.10. If Sα and Sα

are two choices for the Schreier class then there exist subsequences of N, M andN such that Sα(N) ⊆ Sα and Sα(M) ⊆ Sα. We also deduce that the constants δα

and δα are independent of the particular choice of Sα.

The following stabilization argument shows that ∆(X) is always non-empty.

Proposition 4.10. Let X be a Banach space with a basis (ei). Then thereexists γ = (γα)α<ω1 and (xi) ≺ (ei) so that (xi) ∆-stabilizes γ. In particular,∆(X) 6= ∅.

Proof. Fix εn ↓ 0. If [ei]i∈N contains `1, then, since `1 is not distortable, wecan choose a normalized sequence (xi) ≺ (ei) with ‖

∑∞n aixi‖ ≥ (1− εn)

∑|ai| for

all (ai); thus the proposition follows with γα = 1 for all α.If [ei]i∈N does not contain `1 then by Proposition 4.5, δα(ei) > 0 for at most

countably many α’s.Fix an arbitrary α < ω1. It follows from Lemma 4.6 that if (yi) ≺ (ei) then

δα((yi)∞n ) = δα(yi) for all n. Since δα(yi) ≤ δα(zi) whenever (yi) ≺ (zi), by astandard argument we can stabilize δα. That is, given (wi) ≺ (ei) we can find(zi) ≺ (wi) so that

γα ≡ δα(zi) = δα(yi) for all (yi) ≺ (zi) .

(To do this, construct (wi) � (z(1)i ) � (z(2)

i ) � . . . such that δα(z(k+1)i ) ≤ inf{δα(yi) :

(yi) ≺ (z(k)i )}+ 2−k, for every k, and set zi = z

(i)i for all i.)

Now choose by induction (zi) � (x(1)i ) � (x(2)

i ) � . . . such that

|δα(x(n+1)i )− δα(x(n)

i )| = |δα(x(n+1)i )− γα| ≤ εn for all n ,

and let xi = x(i)i for all i. Then |δα((xi)∞n ) − γα| < εn for all n. If (yi) ≺ (xi)∞n

then δα((xi)∞n ) ≤ δα(yi) ≤ δα(yi) = γα.Then using this and a diagonal argument for the countably many α’s so that

δα(ei) > 0 we obtain the proposition. �

Our next proposition collects some basic facts about the ∆-spectrum.

30 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES8

Proposition 4.11. Let X have a basis (ei).a): ∆(X) 6= ∅ and if γ ∈ ∆(X) then γα ∈ [0, 1] for α < ω1.b): X contains `1 iff there exists γ ∈ ∆(X) with γ1 = 1.c): If γ ∈ ∆(X) then γα ≥ γβ if α ≤ β < ω1.d): If γ ∈ ∆(X) and α, β < ω1, then γαγβ ≤ γβ+α.e): If γ ∈ ∆(X) and α < ω1, n ∈ N then γα·n ≥ (γα)n.f): If γ ∈ ∆(X) then γ is a continuous function of α.g): δα(X) = sup{γα : γ ∈ ∆(X)}.

Proof. We have already seen the non-trivial part of a) and one implication inb). Next, e) follows immediately from d) while f) and g) follow from the relevantdefinitions, using c) to get f).

To complete b) note that if γ1 = 1 then γα = 1, for all α < ω1 (for α = β + 1this follows from d) and for α a limit ordinal—from f)). Thus by Proposition 4.5,X contains `1.

c) Let γ ∈ ∆(X) and α ≤ β < ω1. For n ∈ N let Nn = (n, n + 1, . . .).Let (xi) stabilize γ. Given m ∈ N choose n ≥ m by Proposition 3.2 so thatSα(Nn) ⊆ Sβ(Nm). It follows that δα((xi)∞n ) ≥ δβ((xi)∞m ). Letting m →∞ we getγα ≥ γβ .

d) Let (yi) be basic. By Proposition 3.2 there exists M with Sβ+α(M) ⊆ Sα[Sβ ].It follows that δβ+α((yi)M ) ≥ δSα[Sβ ](yi). By Lemma 4.6 we see that δSα[Sβ ](yi) ≥δα(yi)δβ(yi). Thus δβ+α((yi)M ) ≥ δα(yi)δβ(yi). Using this for (yi) = (xi)i∈Nn

where (xi) stabilizes γ, we obtain that γβ+α ≥ γαγβ . �

Remark 4.12. It is often useful to note that the constants δn satisfy conditionsc) and d) for natural numbers. If m,n ∈ N and m ≤ n then δm(xi) ≥ δn(xi) andδm+n(xi) ≥ δm(xi)δn(xi), hence also δmn(xi) ≥ (δn(xi))m (because Sm ⊆ Sn andSn[Sm] = Sm+n).

It is well known that the supermultiplicativity property d) of sequences γ ∈∆(X) formally implies a “sub-power-type” behavior of γ, which we shall find usefulin various situations. This depends on an elementary lemma. For two sequences(bn), (cn) ⊆ (0, 1] we shall write cn � bn to denote that limn bn/cn = ∞.

Lemma 4.13. Let (bn) ⊆ (0, 1] satisfy bn+m ≥ bnbm for all n, m ∈ N. Thenlimn b

1/nn exists and equals supn b

1/nn . Moreover, for every 0 < ξ < limn b

1/nn we

have ξn � bn.

Proof. Let an = log(b−1n ). Then an ≥ 0 and an+m ≤ an + am for all n, m.

It suffices to prove that an/n → a ≡ infm{am/m}. Given ε > 0 choose k with|ak/k − a| < ε. For n > k, an/n − a < an/n − ak/k + ε. Setting n = pk + r,0 ≤ r < k and using apk

≤ pak we obtain

an

n− ak

k+ ε ≤ apk + ar

n− ak

k+ ε ≤ pak

pk + r+

ar

n− ak

k+ ε

≤ pak

pk− ak

k+

ar

n+ ε =

ar

n+ ε .

The first part of the lemma follows. The moreover part can be easily proved bycontradiction. �

We have an immediate corollary.

4. ASYMPTOTIC CONSTANTS AND ∆(X) 31

Corollary 4.14. Setting γα = limn(γα·n)1/n for α < ω1 we have that forevery 0 < ξ < γα, ξn � γα·n ≤ γ n

α , for all α < ω1 and n ∈ N.Setting δ = limn(δn(xi))1/n, for a basic sequence (xi), we have that for every

0 < ξ < δ, ξn � δn(xi) ≤ δn for all n ∈ N.

There is an interesting connection between the constants δα(X) which allow forrenormings of a given space X, and the supermultiplicative behavior of γ ∈ ∆(X),in particular of γα, which involved the original norm only.

Proposition 4.15. Let X have a basis (ei) and let γ ∈ ∆(X). Then thereexists (yi) ≺ (ei) so that (yi) ∆-stabilizes γ and so that for all α < ω1, δα(yi) =limn(γα·n)1/n ≡ γα.

The argument is based on the following renorming result which we shall useagain.

Proposition 4.16. Let Y be a Banach space with a bimonotone basis (yi). Letα < ω1 and n ∈ N. Then there exists an equivalent bimonotone norm ||| · ||| on Y

with δα((yi), ||| · |||) ≥(δ[Sα]n(yi)

)1/n

.

Proof. Denote the original norm on Y by | · | and set θ = δ[Sα]n(yi). For0 ≤ j ≤ n define a norm | · |j on Y by

|y|j = sup{

θj∑1

|Eiy| : (Eiy)`1 is [Sα]j-admissible w.r.t. (yi)

and E1 < · · · < E` are adjacent intervals}

.

Here we take [Sα]0 = S0 so that |y|0 = |y|. For 0 ≤ j ≤ n we have |y|j ≥ θj |y| and|y| ≥ θn−j |y|j . The former inequality follows trivially from the definition of | · |jand the latter from the fact that any [Sα]j-admissible family is [Sα]n-admissibleand the definition of θ.

Set |||y||| = 1n

∑n−10 |y|j for y ∈ Y . Then ||| · ||| is an equivalent norm on Y .

Let (xs)r1 be α-admissible w.r.t. (yi). First observe that |

∑r1 xs| ≥ θ

∑r1 |xs|n−1.

Indeed, arbitrary [Sα]n−1-admissible decompositions for each xs can be put togetherto give a [Sα]n-admissible decomposition for

∑r1 xs, thus the estimate follows from

the definition of | · |n−1 and the fact that δ[Sα]n(yi) = θn. To be more precise, for1 ≤ s ≤ r choose adjacent intervals of integers Es

1 < · · · < Esk(s) so that (Es

j )k(s)1 is

[Sα]n−1 admissible and

|xs|n−1 = θn−1

k(s)∑j=1

|Esj xs| .

Let F sj = Es

j if j < k(s) and F sk(s) = [minEs

k(s),minEs+11 ) if s < r and F r

k(r) =Er

k(r). Then F1 < · · · < F 1k(1) < · · · < F r

k(r) are [Sα]n-admissible adjacent intervals

32 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES9

of N and so

∣∣∣ r∑l=1

xl

∣∣∣ ≥ θnr∑

s=1

k(s)∑j=1

∣∣F sj

( r∑l=1

xl

)∣∣≥ θ

r∑s=1

θn−1

k(s)∑j=1

|Esj (xs)| = θ

r∑s=1

|xs|n−1 ,

(since∣∣F s

k(s)

(∑rl=1 xl

)∣∣ = |F sk(s)(xs + xs+1)| ≥ |Es

k(s)(xs)| if s < r, using that thenorm is monotone).

Similarly, |∑r

1 xs|j+1 ≥ θ∑r

1 |xs|j for j = 1, 2, . . . , n − 2, by the definitions of| · |j+1 and | · |j . Thus

∣∣∣∣∣∣∣∣∣ r∑1

xs

∣∣∣∣∣∣∣∣∣ = 1n

n−1∑j=0

∣∣∣ r∑s=1

xs

∣∣∣j≥ θ

n

r∑s=1

|xs|n−1 +1n

(n−2∑j=0

θr∑

s=1

|xs|j)

.

Thus |||∑r

1 xs||| ≥ θ∑r

1 |||xs|||. �

Proposition 4.17. Let X be an asymptotic `1 space and let γ ∈ ∆(X). If(ei) ≺ X ∆-stabilizes γ then for all εi ↓ 0 there exists (xi) ≺ (ei) and an equivalentnorm | · | on [xi] satisfying

a): For all n and x ∈ 〈xi〉∞n we have

‖x|| ≤ |x| ≤ (2 + εn)‖x‖ .

b): (xi) is bimonotone for | · |.c): (xi) ∆-stabilizes γ ∈ ∆(X, | · |) with γα ≥ γα for all α < ω1.

Proof. We may assume that [ei] does not contain `1. Thus by Rosenthal’stheorem [R] there exists (xi) ≺ (ei) which is normalized and weakly null. Bypassing to a subsequence of (xi) we may assume that for all n < m and (ai)m

1 ⊆ R,‖∑n

1 aixi‖ ≤ (1 + εn)‖∑m

1 aixi‖, where εn = εn/2.Define the norm | · | for x ∈ X by

|x| = sup{‖Ex‖ : E is an interval} .

Passing to a block basis of (xi) we may assume that (xi) ∆-stabilizes some γ ∈∆(X, | · |). For x =

∑mi=n aixi with |x| = ‖Fx‖ we have

‖x‖ ≤ |x| ≤ ‖max F∑i=n

aixi‖+ ‖min F−1∑

i=n

aixi‖ ≤ 2(1 + εn)‖x‖ = (2 + εn)‖x‖ .

Thus a) holds and b) is immediate. It remains to check c). Fix α < ω1 and m ∈ N.Let xm < y1 < · · · < y` (w.r.t. (xi)∞m ) where (yi)`

1 is α-admissible w.r.t. (xi)∞mand hence w.r.t. (ei)∞m . Choose intervals E1 < · · · < E` such that |yi| = ‖Eiyi‖ fori ≤ ` and Ei ⊆ [min supp(yi),max supp(yi)]. Define (Fi)`

1 to be adjacent intervalsso that minFi = minEi. Thus Fi = [minEi,minEi+1) ⊆ N for i < ` and F` = E`.

4. ASYMPTOTIC CONSTANTS AND ∆(X) 33

Let F =⋃`

1 Fi. Then, by Remark 4.3,∣∣∣∑1

yi

∣∣∣ ≥∥∥∥F (

∑1

yi)∥∥∥ ≥ δα((ei)∞m )

∑j=1

‖Fj(∑1

yi)‖

≥ δα((ei)∞m )(1 + εm)−1∑j=1

‖Fjyj‖ = δα((ei)∞m )(1 + εm)−1∑j=1

|yj | .

It follows that δα((xi)∞m , | · |) ≥ δα((ei)∞m )(1 + εm)−1. Letting m → ∞ we obtainγα ≥ γα. �

Remark 4.18. It is worth noting the following. Let (ei) be a basic sequencein X ∆-stabilizing γ ∈ ∆(X). Then there exists (xi) ≺ (ei) and an equivalentmonotone norm | · | on [xi] so that (xi) ∆-stabilizes γ ∈ ∆(X, | · |). Furthermore∣∣∣|x| − ‖x‖

∣∣∣ < εn for x ∈ 〈xi〉∞n and some εn ↓ 0. Assuming as we may that [ei]does not contain `1, this is accomplished by taking (xi) to be a suitable weakly nullblock basis of (ei) and setting |

∑aixi| = supn ‖

∑n1 aixi‖.

A similar argument yields

Proposition 4.19. Let F be a regular set of finite subsets of N and let (ei)be a basis for X. Given ε > 0 and εi ↓ 0 there exists an equivalent norm | · |on some block subspace [xi] ⊆ X satisfying a) and b) of Proposition 4.17 andδF ((xi), | · |) ≥ δF (ei)− ε.

As a corollary to these propositions we obtain

Theorem 4.20. Let Y be a Banach space with a basis (yi). Let α < ω1,n ∈ N, ε > 0 and θn = δ[Sα]n(yi). Then there exists an equivalent norm ||| · ||| onX = [xi] ≺ Y with δα((xi), ||| · |||) ≥ θ − ε.

Proof of Proposition 4.15. Let (xi) ≺ (ei) ∆-stabilize γ (for the original norm‖ · ‖). We may assume that X ′ = [xi] does not contain `1. It follows that thereexists α0 < ω1 so that δβ(X ′) = 0 = γβ for all β > α0. Also from Lemma 4.6,δα(zi) ≤ δα(wi) if (zi) ≺ (wi) ≺ (ei); moreover, δα((zi)∞n ) = δα(zi) for all n ∈ N.We can therefore stabilize the δα’s (as in the proof of Proposition 4.10) to find(yi) ≺ (xi) so that for all α ≤ α0, δα(yi) = δα(zi) if (zi) ≺ (yi). Of course (yi) still∆-stabilizes γ. We shall prove that δα(yi) = limn(γα·n)1/n.

Note that if | · | is an equivalent norm on [yi] and γ ∈ ∆((yi), | · |) thenlimn(γα·n)1/n = limn(γα·n)1/n. Indeed if (zi) ≺ (yi) ∆-stabilizes γ in | · | then since(zi) ∆-stabilizes γ in ‖ · ‖ and the norms are equivalent, we obtain cγβ ≤ γβ ≤ dγβ

for all β < ω1 and for some constants c, d > 0. Thus

γα ≤ supn

(γα·n)1/n = limn

(γα·n)1/n .

By Proposition 4.11 we obtain that δα(yi) ≤ limn(γα·n)1/n.Fix θ < limn(γα·n)1/n. Thus there exists n0 with θn0 < γα·n0 . Choose (zi) ≺

(yi) with θn0 < δα·n0(zi). By Corollary 3.4 there exists M so that [Sα]n0(M) ⊆Sα·n0 , which yields δα·n0(zi) ≤ δ[Sα]n0 ((zi)M ). So letting (wi) = (zi)M we haveδ[Sα]n0 (wi) > θn0 . By Theorem 4.20 there exists an equivalent norm ||| · ||| on [w′

i]N,

34 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES10

for some (w′i) ≺ (wi) with δα((w′

i), ||| · |||) > θ. The reverse inequality, δα(yi) ≥limn(γα·n)1/n, follows. 2

As we will see in later sections, some further regularity properties of sequencesγ ∈ ∆(X) are closely related to distortion properties of the space X, and theymay or may not hold in general. In contrast, the sequences (δα) which allowfor renorming display a complete power type behavior. In fact, we will give acomprehensive description of behavior of such sequences in Theorem 4.23 below.

In the result that follows we shall be particularly interested in part c).

Proposition 4.21. Let X have a basis (ei). Let α < ω1 and n ∈ N.

a): δ[Sα]n(X) = (δα(X))n

b): δ[Sα]n(X) = δα·n(X)c): δα·n(X) = (δα(X))n

Proof. c) will follow from a) and b).a) Since for any equivalent norm |·| on X we have δ[Sα]n((yi), |·|) ≥ (δα((yi), | · |))n

(Lemma 4.6, e)), the inequality δ[Sα]n(X) ≥ (δα(X))n follows from g) of proposi-tion 4.11. To see the reverse inequality let | · | be an equivalent norm on X, andlet (yi) ≺ (ei) and θ > 0 satisfy δ[Sα]n((yi), | · |) > θn. By Theorem 4.20 there exist(xi) ≺ (yi) and an equivalent norm ||| · ||| on [xi]i∈N such that δα((xi), ||| · |||) > θ.This completes the proof.

b) As we have shown earlier, whenever (yi) ≺ (ei) and | · | is an equivalentnorm, by Corollary 3.4 there exists a subsequence M such that δ[Sα]n((yi)M , | · |) ≥δα·n((yi), | · |). It follows that δ[Sα]n(X) ≥ δα·n(X). The reverse inequality followsby choosing N with Sα·n(N) ⊆ [Sα]n. �

Let us introduce the following natural and convenient definition.

Definition 4.22. Let X be an asymptotic `1 space. The spectral index of X,I∆(X), is defined to be

I∆(X) = inf{α < ω : δα(X) < 1} .

Theorem 4.23. If X is an asymptotic `1 space not containing `1, then I∆(X) =ωα for some α < ω1. If I∆(X) = α0 and δα0(X) = θ then δα0·n+β(X) = θn for alln ∈ N and β < α0. Finally, δβ(X) = 0 for all α0 · ω ≤ β < ω1.

Proof. For the proof of the first statement, it suffices to show that if β <I∆(X) then for all n ∈ N, β·n < I∆(X) ([Mo], Thm. 15.5). But by Proposition 4.21,δβ·n(X) = (δβ(X))n = 1, so β · n < I∆(X).

Now let α0 = ωα for some α and assume that δα0(X) = θ for some 0 < θ < 1.Fix β < α0. We first show that for any ε > 0 we can find (yi) ≺ X and anequivalent norm ||| · ||| on [yi]N with δβ((yi), ||| · |||) > 1− ε and δα0((yi), ||| · |||) > θ− ε.Indeed, let θ′ = θ − ε and choose by Proposition 4.17 (xi) ≺ X and an equivalentbimonotone norm | · | on X so that δα0((xi), | · |) > θ′. Given m ∈ N we can choosea subsequence N of N so that Sα0

[[Sβ ]j

](N) ⊆ Sβ·j+α0 = Sα0 for j = 0, 1, . . . ,m;

this follows from Proposition 3.2, Corollary 3.4 and the fact that β ·m + ωα = ωα.Let (yi) = (xi)N and am ≡ δ[Sβ ]m((yi), | · |). Note that since [Sβ ]m(N) ⊆ Sα0 then

5. EXAMPLES–TSIRELSON SPACES 35

am ≥ θ′ and so a ≥ (θ′)1/m. For y ∈ [yi]N and 0 ≤ j ≤ m set

|y|j = sup{aj∑1

|Eiy| : (Eiy)`1 is [Sβ ]j-admissible w.r.t. (yi)

and E1 < · · · < Ek are adjacent intervals} .

It can be checked by a straightforward calculation, using the choice of N andthat (yi) is monotone for |·|, that δα0((xi), |·|j) ≥ δα0((yi), |·|) > θ′ for j = 0, . . . ,m.For y ∈ [yi]N set |||y||| = 1

m

∑m−1j=0 |y|j . Then δα0((yi), ||| · |||) > θ′ and from the proof

of Proposition 4.16, δβ((yi), ||| · |||) ≥ a > (θ′)1/m. Taking m such that (θ′)1/m ≥ 1−εwe get what we wanted.

Now by Proposition 3.2 there exists a subsequence M of N with Sα0+β(M) ⊆Sβ [Sα0 ]. It follows that

δα0+β((yi)M , ||| · |||) > (1− ε)θ′ = (1− ε)(θ − ε) .

Hence δα0+β(X) = θ.The case of general n is proved similarly, replacing α0 by α0 · n above and

recalling (Proposition 4.21) that δα0·n(X) = (δα0(X))n. The last statement isobvious. �

5. Examples–Tsirelson Spaces

Our primary source of examples of asymptotic `1 spaces with various behaviorsof asymptotic constants is the class of mixed Tsirelson spaces introduced by Argyrosand Deliyanni in [ArD].

Definition 5.1. Let I ⊆ N and for n ∈ I let Fn be a regular family of finitesubsets of N. Let (θn)n∈I ⊆ (0, 1) satisfy supn∈I θn < 1. The mixed Tsirelson spaceT (Fn, θn)n∈I is the completion of c00 under the implicit norm

‖x‖ = max

(‖x‖∞, sup

n∈Isup{

θn

k∑i=1

‖Eix‖ : (Ei)ki=1 is Fn-admissible

}).

It is shown in [ArD] that such a norm exists. It is also proved that if I isfinite or if θn → 0, then T (Fn, θn)n∈I is a reflexive Banach space, in which thestandard unit vectors (ei) form a 1-unconditional basis. In [ArD] it is provedthat for an appropriate choice of θn and Fn the space T (Fn, θn)n∈N is arbitrarilydistortable. Deliyanni and Kutzarova [DKut] proved a result that illustrates thepossible complexity these spaces can possess. They proved that a mixed Tsirelsonspace may uniformly contain `n

∞’s in all subspaces. Notice that the Tsirelson spaceT satisfies T = T (Sn, 2−n)n∈N = T (S1, 2−1). For 0 < θ < 1 we denote the θ-Tsirelson space by Tθ = T (S1, θ).

Theorem 5.2. Let (ei) denote the unit vector basis for T .

a): If (xi) ≺ (ei) then for all n, δn(xi) = 2−n and δn(xi) = 2−n.b): For all γ ∈ ∆(T ), γn = 2−n for n ∈ N and γα = 0 for α ≥ ω.c): For all γ ∈ ∆(T ), γn ≤ 2−n for n ∈ N.d): I∆(X) = 1 for all X ≺ T .

Remark 5.3. Condition a) immediately implies that for an arbitrary equivalentnorm | · | on T and (xi) ≺ (ei), we have δ1((xi), | · |) ≤ 1/2. Since the asymptotic

36 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES11

`1 constant is equal to δ−11 , this improves the constant in Proposition 2.8 from

√2

to 2.

Remark 5.4. For Tθ we have δn(Tθ) = δn(Tθ) = θn for n ∈ N; and all otherequalities and inequalities from Theorem 5.2 hold with appropriate modifications.Also, clearly, I∆(Tθ) = 1.

Proof of Theorem 5.2. a) By definition of the norm ‖ · ‖ for T , δn(ei) ≥ 2−n andso if (xi) ≺ (ei) then δn(xi) ≥ 2−n as well.

We next show that there exists C < ∞ so that δm(xi) ≤ C2−m for all m. Thiswill yield the equality for δn. Indeed if for some n, δn(xi) = A/2n where A > 1 thensince δnk(xi) ≥ (δn(xi))k (Remark 4.12), we would have that C2−nk ≥ δnk(xi) ≥Ak2−nk for all k, which is impossible.

First we consider the case (xi) = (ei)i∈M where M is a subsequence of N. Letε > 0, n ∈ N and let x =

∑i∈F aiei be an (n, n − 1, ε)-average of (ei)i∈M (see

Proposition 3.6 and Notation 3.7). Thus ‖x‖ ≥ 2−n. Iterating the definition of thenorm in T yields that ‖x‖ =

∑ni=1 2−i

∑j∈Fi

aj where (Fi)ni=1 partitions F into

sets with Fi ∈ Si for i ≤ n. Thus if ε < 2−n,

‖x‖ = ‖∑i∈F

aiei‖ ≤n−1∑i=1

2−iε + 2−n∑j∈Fn

aj ≤ 2/2n = 2/2n∑i∈F

‖aiei‖ .

Hence δn((ei)M ) ≤ 2/2n.If (xi) is normalized with (xi) ≺ (ei) then by [CJoTz] (see also [CSh]), there

exists a subsequence M such that (xi) is D-equivalent to (ei)i∈M , where D isan absolute constant (we let mi = min supp(xi), and then M = (mi)). Thusδn(xi) ≤ Dδn((ei)M ) ≤ 2D/2n.

To get the equality for δn we first observe that for any equivalent norm | · | onT there is a constant C ′ (depending on | · |) such that δn((xi), | · |) ≤ C ′δn(xi), andthen we follow the previous argument.

b) is immediate from the first part of a); and c) and d) follow from the secondpart of a). 2

Remark 5.5. For the subsequence M = (mi) above one could take any mi ∈supp(xi) for all i. In the space Tθ, any normalized block basis is D-equivalent to(ei)M as well, with the equivalence constant D = cθ−1, where c is an absoluteconstant. The choice of a subsequence M is the same as indicated above (forθ = 1/2).

The next example illustrates Theorem 4.23.

Example 5.6. Let α < ω1 and let X = T (Sωα , θ). Thena): δωα·n(X) = θn for n ∈ Nb): I∆(X) = ωα

Proof. a) Let (xi) ≺ X be a normalized block basis that ∆-stabilizes γ ∈∆(X). Let n ∈ N and let ε > 0. Choose N by Corollary 3.4 so that [Sωα ]n ⊇Sωα·n(N) and also [Sωα ]n−1(N) ⊆ Sωα·(n−1). Choose x =

∑F aixi to be an (ωα ·

n, ωα · (n − 1), ε) average of (xi)N w.r.t. (ei), the unit vector basis of X. Clearly‖x‖ ≥ θn. As in T , ‖x‖ is calculated by a tree of sets where the first level of sets isSωα -admissible, the second level is [Sωα ]2-admissible and so on.

5. EXAMPLES–TSIRELSON SPACES 37

If we stop this tree after n−1 levels, discarding sets which stopped before thenand shrinking those sets which split the support of some xi we obtain for some(Eix)`

1 being ωα · (n− 1)-admissible,

‖x‖ ≤ θn−1∑1

‖Eix‖+ ε .

The next level of splitting may indeed split the supports of some of the xj ’s. How-ever since those xj ’s have not yet been split the contribution of ajxj to the nextlevel of sets is at most ajθ

−1. Thus we obtain

‖x‖ ≤ θn(∑

ajθ−1)

+ ε = θn−1 + ε .

It follows that γωα·n ≤ θn−1 = 1θ (θn).

Thus, just as in the case of T , γωα·n = θn. Indeed, if γωα·n0 > θn0 then

γωα·n0k ≥ (γωα·n0)k >

1θθn0k

for large enough k (Proposition 4.11), which is a contradiction.Similarly if γ ∈ ∆(X) then for some C, γωα·n ≤ Cθn and so γωα·n ≤ θn for all

n. This yields that δωα·n(X) = θn.b) The argument in Proposition 4.23(b) yields this result: for β < ωα and ε > 0

there exists (xi) and ||| · ||| with δβ((xi), ||| · |||) > 1− ε. �

Before we pass to further examples, let us note a fundamental and useful con-nection between the spectrum ∆(X) and a lower estimate for the norm on someblock subspace.

Proposition 5.7. Let X be an asymptotic `1 space and let (zi) ≺ X be anormalized bimonotone block basis ∆-stabilizing some γ ∈ ∆(X) with 0 < γ1 < 1.Let (ei) be the unit vector basis of Tγ1 ≡ T (S1, γ1). Then for all ε > 0 there existsa subsequence (xi) of (zi) satisfying for all (ai) ⊆ R

‖∑

aixi‖ ≥ (1− ε)‖∑

aiei‖Tγ1.

Proof. We shall prove the proposition in the case where γ1 = 1/2 (and soTγ1 = T ). We shall describe below the argument in a general case, but the readeris advised to first test the special case when δ1(zi) = 1/2 (when εn = 0 for all nand the mi’s can be omitted.) Choose integers mi ↑ ∞ so that

∑∞1 2−mi < ε and

then choose εn ↓ 0 to satisfy, for all k ∈ N,k∏1

(12− εn(i)

)> (1− ε)2−k whenever (n(i))k

i=1 ⊆ N

satisfy for every j, |{i : n(i) = j}| ≤ mj .(5)

Let (xi) be a subsequence of (zi) which satisfies: for all n, if xn ≤ y1 < · · · < yn

w.r.t. (xi) then ‖∑n

1 yi‖ > ( 12 − εn)

∑n1 ‖yi‖. Such a sequence exists since (zi)

∆-stabilizes γ with γ1 = 1/2.Let x =

∑`1 aixi and assume that ‖

∑`1 aiei‖T = 1. We shall show that ‖x‖ >

(1 − ε)2. If ‖∑

aiei‖T = |aj | for some j then ‖x‖ = 1. Otherwise for some 1-admissible family of sets, ‖

∑aiei‖T = 1

2

∑nj=1 ‖Ej(

∑aiei)‖T . Accordingly we

38 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES12

have that (here is where the bimonotone assumption is used)

‖x‖ >

(12− εi

) n∑j=1

‖Ejx‖

where i = min(suppE1x). We then repeat the step above for each Ejx. Ultimatelywe obtain for some J ⊆ N,

1 = ‖∑

aiei‖T =∑i∈J

2−`(i)|ai|

where `(i) = the number of splittings before we stop at |ai|. We follow the sametree of splittings in getting a lower estimate for ‖x‖ with one additional proviso.Each splitting of Ex in 〈xi〉 will introduce a factor of ( 1

2 − εn) for some n. Agiven factor (1

2 − εn) may be repeated a number of times. If any ( 12 − εn) is

repeated mn times we shall discard the corresponding set ‖Ex‖ at that instant.By virtue of (5) we thus obtain that ‖x‖ ≥

∑i∈I(1 − ε)2−`(i)|ai| where I ⊆ J

and aixi belonged to a discarded set for i ∈ J \ I. However the contributionof the discarded sets to ‖

∑aiei‖T is at most

∑∞n=1 2−mn < ε since from our

construction for any given n (where ( 12 − εn) is repeated mn times) we will discard

at most one set, something of the form 2−k‖Ex‖T where k ≥ mn. It follows that‖x‖ > (1− ε)(‖x‖T − ε) = (1− ε)2. �

The proof also yields the following block result.

Corollary 5.8. Let (zi) be a bimonotone basic sequence in a Banach spaceX which ∆-stabilizes γ ∈ ∆(X) where 0 < γ1 < 1. Let (ei) be the unit vector basisof Tγ1 . Then for all ε > 0 there exists a subsequence (xi) of (zi) satisfying for all(yj)k

1 ≺ (xi) if mj = min(supp(yi)) w.r.t. (xi) then∥∥∥ k∑1

yi

∥∥∥ ≥ (1− ε)∥∥∥ k∑

1

‖yj‖emj

∥∥∥Tγ

.

Remark 5.9. We can remove the bimonotone assumption on the norm if wehave that for some εn ↓ 0, ‖y0 +

∑m1 yi‖ ≥ (γ1 − εn)

∑m1 ‖yi‖, whenever zn ≤ y0 ≤

zm < y1 < · · · < ym. Without either this assumption or the bimonotone propertywe obtain a slightly weaker result.

Theorem 5.10. Let X be an asymptotic `1 space and let (zi) ≺ X be a basicsequence ∆-stabilizing some γ ∈ ∆(X), with 0 < γ1 < 1. Then for all ε > 0 thereexists a normalized (xi) ≺ (zi) satisfying for all (ai) ⊆ R

‖∑

aixi‖ ≥12(1− ε)‖

∑aiei‖Tγ1

.

Moreover if (yi)k1 ≺ (xi) with mj = min(supp(yi)) w.r.t. (xi) then one has

‖k∑1

yi‖ ≥12(1− ε)

∥∥∥ k∑1

‖yi‖emi

∥∥∥Tγ1

.

Proof. By Proposition 4.17 there exists a ‖ · ‖-normalized (xi) ≺ (zi) and abimonotone norm | · | on [xi] with ‖x‖ ≤ |x| ≤ (2 + ε)‖x‖ for x ∈ [xi] and such that(xi) ∆-stabilizes γ ∈ ∆(X, |·|) with γ1 ≥ γ1. We may thus assume that (xi) satisfies

5. EXAMPLES–TSIRELSON SPACES 39

the conclusion of Corollary 5.8 for | · | and ε′ such that (1− ε′)/(2 + ε′) = 12 (1− ε).

Thus if (yi)k1 is as in the statement of the theorem,∥∥∥ k∑

1

yi

∥∥∥ ≥ 12 + ε′

∣∣∣ k∑1

yi

∣∣∣ ≥ 1− ε′

2 + ε′

∥∥∥ k∑1

|yi|emi

∥∥∥Tγ1

≥ 12(1− ε)

∥∥∥ k∑1

‖yi‖emi

∥∥∥Tγ1

.

�

The following can be proved by an argument similar to that in Proposition 5.7.

Proposition 5.11. Let X be an asymptotic `1 space and let (zi) ≺ X bea normalized bimonotone block basis ∆-stabilizing γ ∈ ∆(X). Let α < ω1 with0 < γα < 1 and let ε > 0. Then there exists a subsequence (xi) of (zi) satisfyingthe following: if (yi)k

1 ≺ (zi) with min(supp(yi)) = mi (w.r.t. (xi)) then∥∥∥ k∑1

yi

∥∥∥ ≥ (1− ε)∥∥∥ k∑

1

‖yi‖emi

∥∥∥T (Sα,γα)

.

The next example is a space X for which the sequences of asymptotic constants(δα(X)) and (δα(X)) are “essentially” the same as for Tsirelson’s space T ; still, Xand T have no common subspaces–no subspace of X is isomorphic to a subspace ofT . It is worth noting that X also has the property that the sequence δ = (δα(X))does not belong to ∆(X).

Example 5.12. Let 0 < c < 1 and let X = T (Sn, c2−n)n∈N. Then

a): δn(X) = 2−n for all n

b): For all γ ∈ ∆(X), γn < 2−n for all n.c): No subspace of X embeds isomorphically into T .

Before verifying these assertions we first require some observations.The norm of x ∈ X, if not equal to ‖x‖∞, is computed by a tree of sets, the

first level being (Ei)`1 where for some j, (Ei)`

1 is j-admissible and

‖x‖ =c

2j

∑i=1

‖Eix‖ .

For each i, if ‖Eix‖ does not equal ‖Eix‖∞, then we split ‖Eix‖ into a second levelof sets mi-admissible for some mi, and so on. If every set keeps splitting then afterk steps we obtain an expression of the form

(6) ckr∑

s=1

2−n(s)‖Fsx‖ .

Of course some sets may stop splitting, in which case if we carry on for k-steps,we only obtain a lower estimate for ‖x‖. Consider the case where (xi) ≺ X andx ∈ 〈xi〉. We set ‖x‖Tk,(xi) to be the largest of the expressions of the form (6)obtained by splitting k-times (a k level tree of sets, where (Fs)r

1 is the kth-level),subject to the additional constraint that for all i and s, Fs does not split xi. ThusFsxi is either xi or 0.

Lemma 5.13. Let (xi) ≺ X, ε > 0 and k ∈ N. Then there exists x ∈ 〈xi〉 with‖x‖ = 1 such that ‖x‖Tk,(xi) > 1− ε.

40 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES13

Proof. Assume without loss of generality that ‖xi‖ = 1 for i ∈ N. We call x ∈[xi] an (n, ε)-normalized average (of (xi) w.r.t. (ei)) if x =

∑i∈F aixi/‖

∑i∈F aixi‖,

where∑

i∈F aixi is an (n, n− 1, cε/2n)-average of (xi) w.r.t. (ei). Thus (xi)i∈F isn-admissible w.r.t. (ei) and if G ⊆ F satisfies (xi)i∈G is (n − 1)-admissible then∑

G ai < cε/2n. Also∑

i∈F ai = 1 and ai > 0 for i ∈ F . (We can always find suchvectors by Proposition 3.6.) Note that if (xi)i∈G is (n−1)-admissible and if we writex in the form x =

∑i∈F bixi (for some bi > 0), then

∑G bi < (cε/2n)(2n/c) = ε

(since ‖∑

i∈F aixi‖ ≥ c/2n).We first indicate how to find x satisfying ‖x‖ = 1 and ‖x‖T1,(xi) > 1 − ε. Let

εi = 2−(i+1)ε so that∑∞

1 εi = ε/2. Let

‖ · ‖n = sup{

c2−n∑j=1

‖Ejx‖ : (Ejx)`j=1 is n-admissible

}.

and observe that for all x, limn ‖x‖n = 0. Let n1 = 1 and choose (y1i ) ≺ (xi) and

nj ↑ ∞ by induction so that each y1j is an (nj , εj)-normalized average of (xi) and

for all j, ‖∑j

i=1 y1i ‖m < εj+1 if m ≥ nj+1. Then we choose y2 to be an (n, ε/2)-

normalized average of (y1i ) where n ∈ N is not important but we may assume that

y2 =∑

F biy1i where n < nmin F .

We have 1 = ‖y2‖ and so by the definition of the norm in X, there exists j suchthat 1 = ‖y2‖j = c/2j

∑`s=1 ‖Es(y2)‖ where (Esy

2)`1 is j-admissible. We claim that

by somewhat altering the Es’s we can ensure, by losing no more than ε, that thesets Es do not split any of the xi’s. Indeed if 1 ≤ j < n, then G = {i ∈ F : Es

splits y1i for some s} ∈ Sj . Since j < n,

∑s∈G bs < ε/2 and thus by shrinking the

offending sets Es to avoid splitting yi’s we obtain the desired sets. If n ≤ j < nmin F

then if we fix i ∈ F and consider Gi = {r : Es splits one or more of the xr’s in thesupport of yi} we get that, by similarly shrinking the offending Es’s so as to notsplit such an xr, and letting Es be the new sets, that

c

2j

∑‖Esy

2‖ > 1−∑i∈F

biεi > 1− ε .

Finally if F = (k1, . . . , kr) and nkp≤ j < nkp+1 then∥∥∥ ∑

i∈F,i<kp

biyi

∥∥∥j

< εkp and bkp < ε/2

so we first discard the Es’s which intersect supp(∑

i≤kpbiyi). Then arguing as

above we shrink the remaining Es’s so as to not split any xi. We obtainc

2j

∑‖Esy

2‖ ≥ 1− εkp− ε/2−

∑i∈F,i>kp

biεi > 1− ε .

This proves the lemma in the case k = 1. For the general case we continue asabove letting (y2

i ) be (n2i , εi)-normalized averages of (y1

i ), etc. If x = yk+11 then x

satisfies the lemma for k. We omit the tedious calculations. �

Proof of the assertions in Example 5.12. By Proposition 4.16, since δn(X) ≥ c2−n,we have δ1(X) ≥ 2−1. If there exists γ ∈ ∆(X) with γ1 ≥ 2−1 then by Theorem 5.10

5. EXAMPLES–TSIRELSON SPACES 41

there exists (xi) ≺ (ei) and d > 0 so that for all (yj)`1 ≺ (xi) if mj = min(supp(yj))

w.r.t. xi, then

(7)∥∥∥∑

1

yj

∥∥∥ ≥ d∥∥∥∑

1

‖yj‖emj

∥∥∥T

.

Fix an arbitrary k. By Lemma 5.13 there exists x ∈ 〈xi〉 with ‖x‖ = 1and ‖x‖Tk,(xi) > 1/2. Thus there exists a k-level tree of sets whose final levelis (E1, . . . , Er) so that ck

∑rs=1 2−n(s)‖Esx‖ > 1/2. Following the same partition

scheme in T and using (7) for ys = Esx we get (with ms = min(supp(Esx))),

d−1 = d−1‖x‖ ≥∥∥∥ r∑

s=1

‖Esx‖ems

∥∥∥T≥

r∑s=1

2−n(s)‖Esx‖ >12(c−k) .

Since c < 1, this is impossible for large enough k. This proves b) for n = 1 andthat δ1(X) = 2−1. Then Proposition 4.21 yields δn(X) = 2−n for all n.

The remainder of b) easily follows from the proof of Proposition 4.16. Indeedassume that some γ ∈ ∆(X) satisfies γn = 2−n, for some n > 1. By Proposition 4.17there is (yi) ≺ X and an equivalent bimonotone norm | · | on [yi] such that (yi)∆-stabilizes γ ∈ ∆(X, | · |) and γn = 2−n. By passing to a subsequence we mayassume that for some sequence εn ↓ 0, for all m,∣∣∣ k∑

1

xi

∣∣∣ ≥ 2−n(1− εm)k∑1

|xi|

if (xi)k1 ≺ (yi)∞m and (xi) is n-admissible w.r.t. (yi)∞1 . Let ||| · ||| be the norm

constructed in the proof of Proposition 4.16 for α = 1 and θ = 1/2. If yr ≤ x1 <· · · < xr then ∣∣∣ r∑

1

xs

∣∣∣ ≥ (1/2)(1− εr)r∑1

|xs|n−1 .

The remaining estimates remain true and, as in the proof of Proposition 4.16, weobtain

|||r∑1

xs||| ≥ (1/2)(1− εr)r∑1

|||xs||| .

Thus γ1 = 1/2 which is impossible.If c) were not true, then, by Theorem 5.2 b), a subspace Y of X isomorphic to

a subspace of T would admit a renorming for which γ1(Y ) = 1/2, in contradictionto b). 2

Remark 5.14. The above example X yields the following. There exists (xi) ≺(ei) and a sequence of equivalent norms ||| · |||j so that for all k on [xi]∞k , ‖x‖ ≥|||x|||j ≥ c2‖x‖ if j ≥ k and furthermore δ1(||| · |||j , (xi)) > 1

2 − εj for some εj → 0.Yet γ1 < 1

2 for all γ ∈ ∆(X). To see this one needs only choose (xi) so that on [xi]∞k ,‖x‖ = sup`≥k ‖x‖`. This can be accomplished by taking each xj to be an iteratedj+1-normalized average of (ei) (as in lemma 5.13). Then set |||x|||j = (1/j)

∑j1 ‖x‖i.

Since ‖x‖ ≥ ‖x‖i ≥ c‖x‖j ≥ c2‖x‖ on 〈xs〉∞j , ‖x‖ ≥ |||x|||j ≥ c2‖x‖.

We mention one other example, taken from [AnO]. First suppose that X =T (Sn, θn)n∈N where 1 > supn θn and limn→∞ θn = 0. We shall call (θn) regular if

42 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES14

for all n, m ∈ N, θn+m ≥ θnθm. It is easy to verify that every such X has a regularrepresentation, i.e., for some regular sequence (θn) we have X = T (Sn, θn)N. Thuslimn θ

1/nn exists by Lemma 4.13.

Example 5.15. Let X = T (Sn, θn)N where 1 > supn θn, θn → 0 and (θn) isregular. Let θ = limn θ

1/nn . Then

a): For all Y ≺ X we have δ1(Y ) = θ.b): For all Y ≺ X and for all n ∈ N, δn(Y ) = θn and δω = 0.

c): For all Y ≺ X, I∆(Y ) ={

ω if θ = 11 if θ < 1

d): For all Y ≺ X and j ∈ N we have δj(Y ) ≤ θj supn≥j θnθ−n ∨ θj/θ1. Inparticular, if θnθ−n → 0 then X is arbitrarily distortable.

6. Renormings of T , and spaces of bounded distortion

Definition 6.1. The distortion constant of a space X is defined by

D(X) = sup|·|∼‖·‖

d(X, | · |) .

So X is distortable iff D(X) > 1. Similarly, X is arbitrarily distortable iffD(X) = ∞. Finally, X is of bounded distortion iff there is D < ∞ such thatD(Y ) ≤ D for every subspace Y ⊆ X.

As we saw in Proposition 2.7, Tsirelson’s space T satisfies D(T ) ≥ 2. Similarlyone can show that D(Tθ) ≥ θ−1. However, not much more is known about distortingT . It is unknown if T is arbitrarily distortable, or at least whether it contains anarbitrarily distortable subspace; and, if not, what is D(T ) or at least a reasonableupper estimate for it. The interest in these questions lies in the fact that, as alreadymentioned, no examples are yet known of distortable spaces which are of boundeddistortion.

¿From techniques developed earlier in this paper we easily get some informa-tion on asymptotic constants of equivalent norms on Tsirelson space. This shouldbe compared with Theorem 5.2 where the constants for the original norm wereestablished.

Surprisingly, it is not known if there exists (xi) ≺ T and an equivalent norm | · |on [xi] with δ1((xi), | · |) < 1/2. Our next result shows that the class of equivalentnorms for which δ1 = 1/2 cannot arbitrarily distort T .

Theorem 6.2. There exists an absolute constant D with the following property.Let X ≺ T and let | · | be an equivalent norm on X such that for some γ ∈ ∆(X, | · |),γ1 = 1/2. Then d(X, | · |) ≤ D.

Proof. Let (zi) be a basic sequence in X ∆-stabilizing γ under | · | whereγ1 = 1/2. Let ε > 0. By passing to a block basis of (zi) and multiplying | · | by aconstant if necessary we may assume that ‖ · ‖T ≥ | · | on [zi] and for all (wi) ≺ (zi)there exists w ∈ 〈wi〉 with 1 + ε > ‖w‖T ≥ |w| = 1. Choose a normalized blockbasis (wi) of (zi) satisfying 1 + ε ≥ ‖wi‖T ≥ |wi| = 1 for all i. Theorem 5.10 allowsus to also assume that

|∑

aiwi| ≥ (1/2− ε)‖∑

aiei‖T .

There exists an absolute constant D1 so that (wi/‖wi‖T ) is D1-equivalent to (emi)

in ‖ · ‖T , where mi = min supp(wi) w.r.t. (ei), for each i [CJoTz]. Thus we have,

6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION 43

for all (ai) ⊆ R,

(8) (1 + ε)D1‖∑

aiemi‖T ≥ ‖

∑aiwi‖T ≥ |

∑aiwi| ≥ (1/2− ε)‖

∑aiei‖T .

Consider the subsequence (pi) of N defined by induction by p1 = 1 and pi+1 =mpi

, for i ≥ 1. There is a universal constant D2 so that (epi) is D2-equivalent to

(epi+1) in ‖ · ‖T [CJoTz]. Also, on the subspace [wpi] we have, by (8),

(1 + ε)D1‖∑

aiepi+1‖T ≥ |∑

aiwpi| ≥ (1/2− ε)‖

∑aiepi

‖T .

Thus the conclusion follows with D = 2D1D2. �

A natural question in light of the above results is whether one can quantify thedistortion d(X, | · |) of an equivalent norm | · | on X ≺ T in terms of ∆(X, | · |).

Problem 6.3. Let | · | be an equivalent norm on T and let (xi) ≺ T (∆, | · |)-stabilize γ. Thus for some c > 0, c2−n ≤ γn ≤ 2−n for all n. Does there exist afunction f(c) so that d(X, | · |) ≤ f(c)?

We shall give a suggestive partial answer to a weaker problem. First we notethe following proposition.

Proposition 6.4. For n ∈ N define the equivalent norm ‖ · ‖n on T by ‖x‖n =sup{2−n

∑`1 ‖Eix‖ : (Eix)`

1 is n-admissible}. Given X ≺ T and εn ↓ 0 there exists

(xi) ≺ X so that for all n if x ∈ 〈xi〉∞n then∣∣∣‖x‖ − ‖x‖n

∣∣∣< εn‖x‖.

Proof. First note that if

‖ · ‖Sn= sup

{∑i∈E

|x(i)| : E ∈ Sn

}then for all x ∈ T we have ‖x‖n ≤ ‖x‖ ≤ ‖x‖n+‖x‖Sn . Indeed, if ‖x‖ 6= ‖x‖∞ then‖x‖ = x∗(x) for some functional x∗ (with ‖x∗‖ = 1) determined by the successiveiterations of the implicit equation of the norm in T ; in particular, x∗(ei) = ±2−n(i)

for all i. We may write x∗ = y∗ + z∗ where z∗(ei) = ±2−n(i) if n(i) ≤ n and0 otherwise. Thus, since the support of z∗ is n-admissible, |z∗(x)| ≤ (1/2)‖x‖Sn

and |y∗(x)| ≤ ‖x‖n. Furthermore, ‖x‖Sn≤ 2n‖x‖. Since the Schreier space Sn is

isomorphic to a subspace of C(ωωn

) (Remark 3.5), , it is c0-saturated, i.e., everyinfinite-dimensional subspace contains a copy of c0, and thus ‖ · ‖Sn

cannot beequivalent to ‖ · ‖ on any infinite-dimensional subspace of T . In particular wecan chose (xi) ≺ X so that for all x ∈ 〈xi〉∞n , ‖x‖Sn

≤ εn‖x‖. The conclusionfollows. �

Problem 6.5. Let | · | be an equivalent norm on X = [xi] ≺ T . Let (yi) ≺ (xi),C < ∞ and suppose that for all n, if y ∈ [yi]∞n then C−1|y|n ≤ |y| ≤ C|y|n, where|y|n = sup{2−n

∑`1 |Eiy| : (Eiy)`

1 is n-admissible w.r.t. (xi)}. Does there exist afunction F (C) so that d(Y, | · |) ≤ F (C)?

Proposition 6.6. Let (yi) ≺ (xi) ≺ T and let | · | be an equivalent norm on[xi]. Suppose that for all n and y ∈ [yi]∞n , C−1|y|n ≤ |y| ≤ C|y|n (where | · |n isdefined as above). Then for all ε > 0 there exists n0 and an equivalent norm ||| · |||on [yi]∞n0

such that C−1|||y||| ≤ |y| ≤ C|||y||| for y ∈ [yi]∞n0and δ1((yi)∞n0

, ||| · |||) > 12 −ε.

44 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES15

Proof. Choose n0 so that C2/n0 < ε. On [yi]∞n0define |||y||| = 1

n0

∑n01 |y|j .

Clearly the inequality between the norms hold. Let p ∈ N and let (zi)p1 ≺ [yi]∞n0

satisfy yn0+p ≤ z1 < · < zp. Let z =∑p

1 zi. Then (see the proof of Proposition 4.16)|z|j+1 ≥ 1

2

∑pi=1 |zi|j for j = 1, . . . , n0 − 1. Hence

|||z||| ≥ 1n0

n0−1∑j=1

12

p∑i=1

|zi|j =12

p∑i=1

|||zi||| −1

2n0

p∑i=1

|zi|n0 .

Now |zi|n0 ≤ C|zi| ≤ C2|||zi||| and so

|||z||| ≥ 12

p∑i=1

|||zi|||(

1− C2

2n0

)> (1− ε)

12

p∑i=1

|||zi||| ,

completing the proof. �

Finally, let us recall the following known [CJoTz] property of T . There existsan absolute constant D1 so that if x1 < y1 < x2 < y2 < · · · are normalized in Tthen (xi) is D1-equivalent to (yi). It turns out that equivalent norms on T thatsatisfy this property (with a fixed constant) cannot arbitrarily distort T . The result,in fact, holds in any space having this subsequence property.

Proposition 6.7. There exists a function f(D) satisfying the following. If | · |is an equivalent norm on [xi]N ≺ T so that (yi) is D-equivalent to (zi) whenevery1 < z1 < y2 < · · · is a normalized block basis of (xi), then d(X, | · |) ≤ f(D).

Proof. By passing to a block basis of (xi) and scaling the norm | · | we mayassume that there exists d > 1 so that for all x ∈ [xi], d−1‖x‖ ≤ |x| ≤ ‖x‖;furthermore, in any block subspace Y of (xi) there exist y, z ∈ Y with |y| = |z| = 1and ‖y‖ ≤ 2 and ‖z‖ > d/2. Choose a | · |-normalized block basis of (xi), y1 < z1 <y2 < · · · with ‖zi‖ > d/2 and ‖yi‖ ≤ 2 for all i. There exists w =

∑aizi satisfying

|w| = 1 and ‖w‖ < 2. Since (zi) and (yi) are D-equivalent for | · |, |∑

aiyi| > D−1.Also (zi/‖zi‖T ) and (yi/‖yi‖T ) are D1-equivalent in T . Thus

‖∑

aiyi‖T ≤ 2D1‖∑

aizi/‖zi‖T ‖T ≤ 4D1/d ‖∑

aizi‖T ≤ 8D1/d .

Thus D−1 ≤ 8D1/d and so d ≤ 8D1D ≡ f(D). �

We now turn to some results about spaces of bounded distortion.

Theorem 6.8. Let X be an asymptotic `1 space. Let γ ∈ ∆(X) and let (yi) ≺X ∆-stabilize γ. If Y = [yi] is of D-bounded distortion then for any α < ω1 andn, m ∈ N,

a): D−1(δα(Y ))n ≤ γα·n ≤ (δα(Y ))n

b): γα·nγα·m ≤ γα·(n+m) ≤ D2γα·nγα·m.

Proof. a) Let γ = ( γα) ∈ ∆(Y ). Choose an equivalent norm | · | on Y and(wi) ≺ (yi) which (∆, | · |)-stabilizes γ. Let ε > 0. By passing to a block basis of(wi) and scaling | · | we may suppose that

|w| ≤ ‖w‖ ≤ (D + ε)|w| for w ∈ [wi] .

6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION 45

Let α < ω1 and n ∈ N. We may assume that δα·n((wi), | · |) > γα·n − ε. Thusif (xs)r

1 is α · n-admissible w.r.t. (wi),∥∥∥ r∑1

xs

∥∥∥ ≥ ∣∣∣ r∑1

xs

∣∣∣ ≥ ( γα·n − ε)r∑1

|xs| ≥γα·n − ε

D + ε

r∑1

‖xs‖ .

It follows that γα·n ≥ γα·n/D and so γα·n ≤ Dγα·n. Passing to the supremumover all γα·n and using Proposition 4.11 g), we get δα·n(Y ) ≤ Dγα·n. Hence byProposition 4.21,

D−1(δα(Y ))n = D−1δα·n(Y ) ≤ γα·n ≤ δα·n(Y ) = (δα(Y ))n .

b) Using part a) and Proposition 4.11 d),

γα·nγα·m ≤ γα·(n+m) ≤ (δα(Y ))n+m

= (δα(Y ))n(δα(Y ))m ≤ D2γα·nγα·m ,

completing the proof. �

Combining the proposition with Theorem 4.23 we get a complete description,up to equivalence, of sequences γ from ∆(X), in spaces of D-bounded distortion.We leave the details to the reader.

Recall the notation γα = limk(γα·k)1/k, for α < ω1 (Corollary 4.14). If Y ≺ X∆-stabilizes γ, we may write γα(Y ) to emphasize the subspace Y . By Proposi-tion 4.15, γα(Y ) = δα(Y ). Therefore, by Proposition 2.5, we have an importantsufficient condition for an asymptotic `1 space to contain an arbitrary distortablesubspace.

Corollary 6.9. Let X be an asymptotic `1 space. Let γ ∈ ∆(X) and let (yi) ≺X ∆-stabilize γ. If there exists α < ω1 such that γα > 0 and limn γα·nγα(Y )−n = 0,then Y contains an arbitrarily distortable subspace.

Let us present an alternative approach to Corollary 6.9, taken from [To1], whichis of independent interest. It is based on a construction of certain asymptotic setsin a general asymptotic `1 space.

An alternative proof of Corollary 6.9. (Sketch) Let γ ∈ ∆(X), let Y = [yi] ≺ X∆-stabilize γ and let (y∗i ) be the biorthogonal functionals in Y ∗. Suppose thatY is of D-bounded distortion. Fix an arbitrary α < ω1. We shall show that(1/3D)(γα(Y ))n ≤ γα·(n−1). By Proposition 4.15, this is slightly weaker thanTheorem 6.8, but sufficient to imply Corollary 6.9.

Fix n ∈ N. First we shall show that for all ε > 0, all normalized blocks(xi) ≺ (yi), and all 0 < λ < 1, there is an (α · n, α · (n − 1), ε) average x of (xi)w.r.t. (yi) such that ‖x‖ ≥ λ(γα(Y ))n ≡ λ′.

This is done by blocking, in the spirit of James [J]. Fix m sufficiently large andpick N ⊆ N such that [Sα·n]m(N) ⊆ Sα·(n m) (Corollary 3.4) and that λγα·(n m) ≤δα·(n m)((xi)N ) (this is possible by the Definition 4.7 of the ∆-spectrum). Pick(z(1)

i ) ≺ (xi)N such that for all i, z(1)i is an (α · n, α · (n − 1), ε) average of (xi)N

w.r.t. (yi). If for all i, ‖z(1)i ‖ < λ′, then pick (z(2)

i ) ≺ (z(1)i ) such that for all i,

z(2)i is an (α · n, α · (n− 1), ε) average of (z(1)

i /‖z(1)i ‖) w.r.t. (yi). And keep going.

Assume that after m steps we still had that ‖z(k)i ‖ < λ′ for all i and all k ≤ m.

Write z(m)1 =

∑j∈N bjxj ; then bj ≥ 0 and let J be the set of all j ∈ N such that

46 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES16

bj > 0. It is easily seen that (xj)j∈J is [Sα·n]m(N)-admissible w.r.t. (yi), hencealso (α · (n m))–admissible w.r.t. (yi). Moreover, our assumption on the norms ofthe z

(k)i ’s easily yields that

∑bj > (1/λ′)m−1. Thus

(1/λ′)m−1λγα·(n m) ≤ λγα·(n m)

∑j∈J

‖bjxj‖

≤ δα·(n m)((xi)N )∑j∈J

‖bjxj‖ ≤ ‖∑j∈J

bjxj‖ = ‖z(m)1 ‖ < λ′ .

It follows that (λγα·(n m))1m < λ′, hence (γα·(n m))1/n m < λ1/n−1/mnγα(Y ), a con-

tradiction, if m is large enough.Now we shall define asymptotic sets A,B ⊆ S(Y ) and a set A∗ in the unit ball

of Y ∗ such that A∗ 2-norms A and the action of A∗ on B is small. By passing toa tail subspace of Y if necessary, we may assume without loss of generality that34γα·(n−1) ≤ 7

8δα·(n−1)(Y ). Fix ε > 0, quite small as determined at the end of thisproof. Let A∗ consist of all functionals in Y ∗ of the form y∗ = 3

4γα·(n−1)

∑k∈K w∗

k,where (w∗

k)K ≺ (y∗i ) is (α · (n− 1))-admissible (w.r.t. (y∗i )); and let A consist of ally ∈ S(Y ) that are 2-normed by A∗. The set A is asymptotic by the definition of the∆-stabilization. Since Y ∆-stabilizes γ, it is not difficult to see that A is asymptoticin Y and that functionals from A∗ have the norm not exceeding 1. Then B consistsof all vectors of the form x/‖x‖, where x is an (α · n, α · (n − 1), ε) average w.r.t.(yi) of some normalized (xi) ≺ (yi), such that ‖x‖ ≥ (1− ε)(γα(Y ))n. By the firstpart of this proof, B is asymptotic in Y . We will show that if y∗ ∈ A∗ and z ∈ B,then |y∗(z)| ≤ 3

4 γα(Y )−n(γα·(n−1) + 76ε)/(1− ε) ≡ η.

This is a direct consequence of the following estimate. If x is an (α · n, α · (n−1), ε) average as above, and if (Ek) ∈ Sα·(n−1), and Ekx denotes the restriction ofx whose support w.r.t. (yi) is Ek; then

∑‖Ekx‖ ≤ 1 + 7ε/6γα·(n−1). To see this,

write x in the form x =∑

i∈F aixi where (xi)i∈F is α · n-admissible w.r.t. (yi) andif J ⊆ F satisfies (xi)J is α · (n− 1)-admissible then

∑G ai < ε. Also

∑i∈F ai = 1

and ai > 0 for i ∈ F . Set I = {i : Ek ∩ supp(xi) 6= ∅ for at most one k} andJ = F \ I; and for i ∈ J let Ki = {k : Ek ∩ supp(xi) 6= ∅}. Then it can be checkedthat (xi)J is α · (n− 1)-admissible, hence∑

k

‖Ekx‖ ≤∑i∈I

ai‖xi‖+∑i∈J

ai

∑k∈Ki

‖Ekxi‖ ≤ 1+ε/δα·(n−1)(Y ) ≤ 1+7ε/6γα·(n−1) .

Now, if y∗ = 34γα·(n−1)

∑k∈K w∗

k ∈ A∗ then letting Ek = supp(wk) for all k we get|y∗(z)| ≤ η, as required.

As mentioned in Section 2, Y is (1/2+1/4η)-distortable. Hence the assumptionof D-bounded distortion implies 1/2 + 1/4η ≤ D. Substituting the definition of ηand taking ε > 0 sufficiently small we get the inequality (1/3D)(γα(Y ))n ≤ γα·(n−1),as promised. 2

As we remarked earlier, the assumption of bounded distortion implies the ex-istence of certain subspaces with a nice structure ([MiTo], [M], [To2]). We wouldlike to identify more such regular subspaces in the class of asymptotic `1 spaces ofbounded distortion.

6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION 47

Recall (Proposition 6.4) that in Tsirelson’s space T = Tθ, for all εn ↓ 0 thereexists (xi) ≺ T so that for all n and all x ∈ 〈xi〉∞n we have

(1 + εn)−1‖x‖T ≤ sup{θn∑1

‖Eix‖T : (Ei)`1 is n-admissible} ≤ ‖x‖T .

In any asymptotic `1 space with bounded distortion one can find a block basisthat displays an isomorphic version of this phenomenon.

Theorem 6.10. Let X be an asymptotic `1 space of D-bounded distortion notcontaining `1. There exist (wi) ≺ X, α = ωβ0 , 0 < θ < 1, and (zi) ≺ (wi) suchthat for every k ∈ N we have, for z ∈ [zi]∞k ,

(1/4 D) sup1≤n≤k

sup{

θn∑

‖Eiz‖ : (Ei) is α · n-admissible}≤ ‖z‖

≤ 4 D inf1≤n≤k

sup{

θn∑

‖Eiz‖ : (Ei) is α · n-admissible}

.

(Here, for an interval E of N and z =∑

aiwi ∈ [wi], Ez denotes the restrictionw.r.t. (wi), i.e., Ez =

∑i∈E aiwi.)

Proof. By Proposition 4.5, δβ(X) > 0 for at most countably many β’s; writethis set as (βm). For an arbitrary β < ω1, it follows from Lemma 4.6 that if(yi) ≺ (ei) then δβ((yi)∞n ) = δβ(yi) for all n; and that δβ(zi) ≤ δβ(yi) whenever(zi) ≺ (yi). Letting, for example, f(yi) =

∑2−mδβm

(yi), by a standard inductionargument, similar to that in Proposition 4.10, we can stabilize f(yi). That is, wecan find (yi) ≺ X such that f(zi) = f(yi) for all (zi) ≺ (yi). Since δβ(X) = 0implies δβ(zi) = 0 for all (zi) ≺ X, the stabilization of f implies that we have, forall (zi) ≺ (yi),

δβ(zi) = δβ(yi) for all β < ω1 .

Let α = I∆(yi); by Theorem 4.23, α = ωβ0 for some β0 < ω1. Let θ = δα(yi).Then δα·n(yi) = θn for n ∈ N, by Proposition 4.21. By an inductive constructionfollowed by a diagonal argument, using Proposition 4.17, we can find (wi) ≺ (yi)and equivalent bimonotone norms | · |n on [wi]∞n such that for all (zi) ≺ (wi)∞n andn ∈ N,

(9) δα([zi]∞n , | · |n) ≥ 2−1/nθ.

Notice that (9) is preserved if the norms involved are multiplied by constants.Therefore by scaling and the assumption of bounded distortion we may additionallyensure that ‖w‖ ≤ |w|n ≤ 2 D‖w‖ for w ∈ [wi]∞n and all n ∈ N.

Now, given any α-admissible family of intervals (Fi)k1 of N, let (Gi)k

1 be adjacentintervals such that min Fi = minGi for i < k and let Gk = Fk. Since the norms| · |n are bimonotone, |Fiw|n ≤ |Giw|n for w ∈ [wi]∞n and all n ∈ N. In particular,by Remark 4.3 for n ∈ N and w ∈ [wi]∞n we get

|w|n ≥ δα

([wi]∞n , | · |n

) k∑i=1

|Giw|n ≥ δα

([wi]∞n , | · |n

) k∑i=1

|Fiw|n .

48 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES17

Using this and the assumption (9) on δα’s we easily get, for n ∈ N and w ∈[wi]∞n ,

2 D‖w‖ ≥ |w|n ≥ sup{

δnα

k∑i=1

|Eiw|n : (Ei) is [Sα]n-admissible}

≥ (1/2) sup{

θnk∑

i=1

‖Eiw‖ : (Ei) is [Sα]n-admissible}

,

where we have abbreviated δα

([wi]∞n , |·|n

)to δα. Finally, using Corollary 3.4 and a

diagonal argument, construct a subsequence M = (mi) of N such that setting Mn =(mi)∞n we get Sα·n(Mn) ⊆ [Sα]n for all n. Thus for w ∈ [wi]i∈Mn

replacing thesupremum in the last formula by the supremum over Sα·n(Mn)-admissible familiesand relabeling the subsequence by (w′

i) we get, for n ∈ N and w ∈ [w′i]∞n ,

‖w‖ ≥ (1/4 D) sup{

θn∑

‖Eiw‖ : (Ei) is α · n-admissible}

.

It should be noted that in this last estimate, the admissibility condition is under-stood with respect to the above subsequence (w′

i) of (wi) which indeed correspondsto the subsequence M of N.

We relabel once more, denoting (w′i) simply by (wi). Set |||w|||n = sup

{θn∑‖Eiw‖ :

(Ei) is α · n-admissible}

, for w ∈ [wi]∞n and n ∈ N. These are equivalent normson the subspaces where they are defined. Therefore stabilizing all norms ||| · |||non a nested sequence of block subspaces, using the assumption of bounded dis-tortion, and passing to a diagonal subspace we get (zi) ≺ (wi) and An such that[zi]∞n ≺ [wi]∞n and An|||z|||n ≤ ‖z‖ ≤ 2 DAn|||z|||n for z ∈ [zi]∞n . Since for all(zi) ≺ (wi) we have δα·n([zi], ‖ · ‖) ≤ δα·n(zi) = θn < 2θn, then for all (zi) ≺ (wi)and all n ∈ N, there exists vn ∈ [zi]∞n such that ‖vn‖ ≤ 1 and |||vn|||n ≥ 1/2. HenceAn ≤ 2, thus ‖z‖ ≤ 4 D|||z|||n on [zi]∞n .

We have shown that for all k ∈ N, ‖z‖ ≥ (1/4 D) sup1≤n≤k |||z|||n on [wi]∞k �[zi]∞k ; and ‖z‖ ≤ 4 D inf1≤n≤k |||z|||n on [zi]∞k . �

We would like to directly relate the norm of an asymptotic `1 space of boundeddistortion with a norm in some Tsirelson space. While we were unable to obtaintwo-sided estimates we did obtain the following lower estimate.

Proposition 6.11. Let X be an asymptotic `1 space of D-bounded distor-tion, α ≺ ω1 and suppose that δα(Y ) = θ ∈ (0, 1) for all Y ≺ X. Let εn ↓ 0.There exist (wi) ≺ X so that for all n if w ∈ [wi]∞n then ‖w‖ ≥ (1 − εn)(D +εn)−1‖

∑‖Eiw‖epi

‖T (Sα,θ−εn), whenever E1 < E2 < · are adjacent intervals, Eiwdenotes the restriction of w w.r.t. (wi) and pi = minEi.

Note that the first paragraph of the proof of Theorem 6.10 shows how to choosea subspace X satisfying the above hypothesis in an asymptotic `1 space of boundeddistortion.

Proof. Choose (zi) ≺ X so that for all n there exists an equivalent bimonotonenorm | · |n on [zi]∞n with δα((zi)∞n , | · |n) > θ − εn. This can be done by Propo-sition 4.17 using that δα(Z) = θ for all Z ≺ X. Hence by a diagonal argument,

6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION 49

applying Corollary 5.8, we may assume also that if z ∈ 〈zi〉∞n and E′1 < · · · < E′

`

are adjacent intervals then

|z|n ≥ (1− εn)∥∥∥∑

1

|E′iz|neri

∥∥∥T (Sα,θ−εn)

,

where ri = minE′i and E′

iz is the restriction of z w.r.t. (zi). Using that X is ofD-bounded distortion and scaling | · |n we may obtain (wi) ≺ (zi) so that for all nand w ∈ 〈wi〉∞n , ‖w‖ ≥ |w|n ≥ 1

D+εn‖w‖. We thus obtain for w ∈ 〈wi〉∞n ,

‖w‖ ≥ (1− εn)∥∥∥∑ |E′

iw|nepi

∥∥∥T (Sα,θ−εn)

≥ 1− εn

D + εn

∥∥∥∑ ‖E′iw‖eri

∥∥∥T (Sα,θ−εn)

.

Now given adjacent intervals E1 < E2 < · · · , take intervals E′1 < E′

2 < · · · suchthat for all w ∈ 〈wi〉∞n , and all i, the restriction Eiw w.r.t. (wi) coincides with therestriction E′

iw with respect to (zi). Then we have ri = minE′i ≥ pi = minEi for all

i and since Sα is invariant under spreading we easily get that ‖∑

aieri‖T (Sα,θ′) ≥

‖∑

aiepi‖T (Sα,θ′) for all (ai) and all 0 < θ′ < 1. Thus the final lower estimate

follows. �

The following proposition generalizes the fact that for the Tsirelson space Tθ,D(Tθ) ≥ θ.

Proposition 6.12. Let X be an asymptotic `1 space. Then sup{D(Y ) : Y ≺X} ≥ sup{γ−1

1 : γ ∈ ∆(X)}.

Proof. Let γ ∈ ∆(X) and let (xi) ≺ X ∆-stabilize γ. Thus for some εn ↓ 0,all n and all (yi) ≺ (xi)∞n ,

γ1 ≥ δ1(yi) ≥ γ1 − εn .

For n ∈ N and (yi) ≺ (xi) define

δ1(n)(yi) = sup{

δ : ‖y‖ ≥ δn∑1

‖Eiy‖ : y ∈ [yi], Ey is a restriction w.r.t.(yi),

E1 < · · · < En0 are adjacent intervals with⋃

Ei = supp(y)}

.

Now observe that given ε > 0 there exists n0 ∈ N and (yi) ≺ (xi) so thatδ1(n0)(wi) < γ1 + ε for all (wi) ≺ (yi). Indeed, if not, we could, by a diagonalargument, produce (yi) ≺ (xi) with δ1(yi) ≥ γ1 + ε.

On [yi] define the norm

|y| = sup{ n0∑

1

‖Eiy‖ : E1y < · · · < En0y w.r.t. (yi) and

E1 < · · · < En0 are adjacent intervals with⋃

Ei = supp(y)}

.

Thus, by the choice of (yi), for all W ≺ Y = [yi]N, there exists w ∈ W ,‖w‖ = 1 and |w| > 1

γ1+ε . Also by considering long `k1-averages (see the proof of

Proposition 2.7) there exists x ∈ W , ‖x‖ = 1 and |x| < 1 + ε. Thus D(Y ) ≥d(X, | · |) ≥ (1 + ε)/(γ1 + ε). �

50 3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES18

More generally, we have

Proposition 6.13. Let X be asymptotic `1 and suppose that I∆(X) = α0.then

sup{D(Y ) : Y ≺ X} ≥ sup{γ−1α0

: γ ∈ ∆(X)} .

Proof. We may assume α0 > 1 by Proposition 6.12. Thus by Theorem 4.23,α0 is a limit ordinal. Let αn ↑ α0 be the ordinal sequence used in defining Sα0 . Letγ ∈ ∆(X), ε > 0. Then for some n0, γαn0

< γα0 +ε. Let (xi) ∆-stabilize γ. Choose(yi) ≺ (xi) and an equivalent norm |·| on [yi] with δαn0

((yi), |·|) > 1−ε. By passingto a block basis of (yi) and scaling | · | if necessary we may assume that for some Dwe have ‖ · ‖ ≤ | · | ≤ D‖ · ‖ on [yi], and for all W = [wi] ≺ Y there exists w ∈ W ,‖w‖ = 1 and |w| < 1 + ε. Since γαn0

< γα0 + ε there exists z ∈ W with ‖z‖ = 1and

∑`1 ‖zi‖ ≥ 1/(γα0 + ε), for some decomposition z =

∑`1 zi where (zi)`

1 is αn0-admissible w.r.t. (wi). Hence |z| ≥ (1−ε)

∑|zi| ≥ (1−ε)

∑‖zi‖ ≥ (1−ε)/(γα0+ε).

Comparing the norms |z| and ‖z‖ we get D(Y, | · |) > (1− ε)(1 + ε)/(γα0 + ε). �

We have a simple corollary.

Corollary 6.14. Let X be asymptotic `1 with I∆(X) = I∆(Y ) = α0 for allY ≺ X. If δα0(X) = 0 then no subspace of X is of bounded distortion.

CHAPTER 4

Asymptotic Versions of Operators and OperatorIdeals1

1. Preliminaries

The goal of this note is to introduce new classes of operator ideals, and, more-over, a new way of constructing such classes through employing the asymptoticstructure recently introduced in [MMiTo].

1.1. Konig’s Unendlichkeitslemma. An added technicality relative to [MMiTo] responsible for the use of the follow-ing lemma is that for our proofs we need to be able to find asymptotic versions ofspaces and operators, not only in the entire space, but also inside any set whichlarge enough to be asymptotic. Indeed, suppose we know we can extract asymp-totic subsets approximating some fixed asymptotic version arbitrarily well from anycollection of arbitrarily long asymptotic set. If we have that the set of sequenceswith a certain property is large enough to be asymptotic, we immediately knowthat we can find an asymptotic version of the space or of the operator with thesame property.

The proof of the said combinatorial lemma is an elementary exercise, and canbe found in [Ko]. A rooted tree is a connected tree with some vertex labeled as’root’.

Lemma 1.1. A rooted tree has an infinite branch emanating from the root if:(1) There are vertices arbitrarily far from the root, and(2) The set of vertices with any fixed distance to the root is finite.

1.2. Extracting an asymptotic version from asymptotic sets. We will now prove the extraction theorem for asymptotic versions of operators.

We use the following technical terminology.

Definition 1.2. A truncation (of length k) of a given collection, Σ ⊆ S(X)n<, is

the collection of sequences of the leading k blocks from sequences in Σ. A truncationof an asymptotic set is obviously an asymptotic set.

Theorem 1.3. Let X be a space with a shrinking basis. For every operatorT ∈ L(X) and every sequence, {Φn}∞n=1, of asymptotic sets with increasing lengthsthere exists T ∈ {T}∞ approximated arbitrarily well by asymptotic sets which aretruncations of asymptotic subsets of Φn’s.

Proof. The proof will split into three parts. First we will extract asymptoticsubsets of block sequences, whose normalized images under T are closely equiva-lent to asymptotic spaces of X (this is the only part where we use the shrinking

1Joint work with Professor V. Milman. To appear in ???

51

52 4. ASYMPTOTIC VERSIONS OF OPERATORS AND OPERATOR IDEALS2

property of the basis). Then we will use a compactness argument and Lemma 4.4(chapter 1) to extract asymptotic subsets of block sequences, where the norm oflinear combinations, the image under T and the action of T are stabilized. Finallywe will pipe such asymptotic sets of different lengths together by means of lemma1.1, to approximate an asymptotic version of T .

First step: For every δ > 0 and for every asymptotic set of length n there isan asymptotic subset of sequences, whose normalized images under T are (1 + δ)-equivalent to elements of {X}n.

Proof of first step: We want to show that the set Φ of sequences mapped (up toδ) to sequences in the collection Σn,ε(X) from Remark 6.3 (chapter 1) is asymptotic.To do that, we must show that V has a winning strategy in the game for Φ.

To produce a sequence from Φ, in the first step of the game the vector playermust pick a block whose image is essentially (up to a norm δ perturbation) sup-ported far enough to fit into a sequence from Σn,ε(X).

Suppose this cannot be achieved. Specifically, suppose that for a sequence,{xk

1}∞k=1, of normalized vectors supported arbitrarily far, and for some natural n1

and δ > 0, we have ‖Pn1(T (xk1))‖ ≥ δ. This means that one of the bounded

functionals P{ei}(T (x)), 1 ≤ i ≤ n1, does not go to zero when applied to somesequence of norm bounded vectors supported arbitrarily far. This contradicts theshrinking property.

So, the vector player can choose a vector x1 with ‖Pn1(T (x1))‖ < δ, for arbi-trary n1 and δ, and therefore insure that that it fits essentially as the first vectorfrom a sequence in Σn,ε(X).

The same reasoning allows the vector player to choose x2 with image which isessentially supported far enough to fit as the second vector of a Σn,ε(X)-admissiblesequence beginning with a slight perturbation of T (x1). Repeating this argument,we achieve a sequence of essentially-consecutive vectors, arbitrarily-well equivalentto a Σn,ε(X)-admissible sequence (with equivalence constant going to 1 as δ and εgo to zero), and we are through.

Second step: Consider two copies of the Minkowski compactum of order n,M and N . Consider finite coverings of M and N , {Vi}i and {Wj}j respectively.Consider a finite covering of the cube [0, ‖T‖]n, {Ik}k. For every asymptotic set oflength n, Φ, there is an asymptotic subset, Φ′, of block-sequences with the followingadditional properties:

(1) The sequences in Φ′ are contained in some fixed Vi0 .(2) The normalized images under T of sequences from Φ′ are contained in

some fixed Wj0 .(3) For all {xi}n

i=1 ∈ Φ′, the sequences {‖T (xi)‖}ni=1 are contained in some

fixed Ik0 .

Proof of second step: This is an easy application of Lemma 4.4 (chapter 1) andthe fact that the covering is finite. Split the sequences in the asymptotic subsetfrom step 1 into the collections:

Φi,j,k = {{xi}ni=1|[xi]ni=1 ∈ Vi,

[ T (xi)‖T (xi)‖

]ni=1

∈ Wj , {‖T (xi)‖}ni=1 ∈ Ik}.

By Lemma 4.4 (chapter 1) one of those collections must be an asymptotic set.

2. ASYMPTOTIC VERSIONS OF OPERATOR IDEALS 53

Third step: For every operator T ∈ L(X) and every collection, {Φn}∞n=1, ofasymptotic sets of increasing lengths there exists a formally diagonal operator,T ∈ {T}∞, approximated arbitrarily well by truncated asymptotic subsets of {Φn}n

Proof of the third step: Fix a positive sequence converging to zero, {εn}n.Choose inductively open finite coverings by cells of diameter less than εn of thecompact product of the two copies of the Minkowski compactum of order n and thecube [0, ‖T‖]n (as in step 2 above), such that the projection of the covering of theorder n product space onto the order n − 1 product space refines the covering ofthe order n− 1 product space. The cells of this covering form a tree in an obviousway.

Take the collection {Φn}n of asymptotic sets. Applying steps 1 and 2, we getthat for every k ≤ n, on the k-th level of the tree some vertex (i.e. covering cell)contains an asymptotic subset of k-truncations of Φn (note that we take not onlyan asymptotic subset of each Φn, but also an all truncations of these subsets, toguarantee we indeed have a subtree).

The subtree spanned by such vertices has properties 1 and 2 from Lemma1.1. Therefore it has an infinite branch. Let {Φ′n}n be the truncated asymptoticsubsets contained in the vertices of the infinite branch. For every n, the truncatedasymptotic subsets {Φ′n}n of {Φn}n have the following properties:

(1) The sequences of blocks from {Φ′n}n are (1 + εn)-equivalent to the basisof [Y ]n for some fixed Y ∈ {X}∞.

(2) The normalized images under T of all sequences in {Φ′n}n are (1 + εn)-equivalent to the basis of [Z]n for some fixed Z ∈ {X}∞.

(3) The norm of the image of a block from a sequence in {Φ′n}n depends (upto εn) only on the place of this vector in the sequence.

Therefore, the result of this process is a sequence of asymptotic sets approxi-mating arbitrarily well a formally diagonal asymptotic version.

�

Remark 1.4. Note that since we may start with any collection of asymptoticsets with increasing lengths, we may choose to extract subsets approximating anasymptotic version of T arbitrarily well from asymptotic sets approximating a givenasymptotic version of X. We thus have for any operator T ∈ L(X) and for anyX ∈ {X}∞ an asymptotic version of T whose domain is X.

2. Asymptotic versions of operator ideals

2.1. Compact operators.

Proposition 2.1. If T is compact then {T}∞ = {0}. If T is non compact thennot all operators in {T}∞ are compact.

Proof. If T is compact, take asymptotic sets, {Φn}n, approximating an as-ymptotic version of the operator. Let V play his winning strategy for Φn, and letS play tail subspaces of [X]>n with n such that ‖T‖[X]>n

< ε.The block-sequence resulting from this game will still be an approximation of

the same asymptotic version. This shows that any asymptotic version of T can beapproximated arbitrarily well by operators with norm smaller than any positive ε.Therefore the only asymptotic version of T is zero.

54 4. ASYMPTOTIC VERSIONS OF OPERATORS AND OPERATOR IDEALS3

If T is non compact, there is an ε > 0 such that for every n one can choose anasymptotic set of length n, Φn, with every {x1, . . . , xn} ∈ Φn having ‖T (xi)‖ ≥ εfor all 1 ≤ i ≤ n.

Using theorem 1.3, extract asymptotic subsets approximating an asymptoticversion of T arbitrarily well. The norm of this asymptotic version will not besmaller than ε on any element of the basis of its domain, and will therefore benon-compact.

�

Asymptotic versions induce a seminorm on operators, through the formula:

|||T ||| = sup ‖T‖

where the supremum is taken over all asymptotic versions of T and the double-barnorm is the usual operator norm.

It is interesting to note that this gives a way of looking at the Calkin algebrasL(X)/K(X) (c.f. [CaPfY]).

Proposition 2.2. Suppose X is a Banach space with a shrinking basis suchthat the norm of all tail projections is exactly 1 (this can always be achieved byrenorming, see [LTz]). Then the norm of the image of an operator T in the Calkinalgebra is equal to |||T |||.

Proof. One direction is clear. If K is a compact operator on X, the norm ofT + K is at least the supremum of norms of asymptotic versions of T + K. Thelatter, by the proof of Proposition 2.1, are the same as asymptotic versions of T .

For the other direction we will show that for every T there exist compactoperators K such that the norm of T +K is almost achieved by asymptotic versionsof T + K, which, again, are the same as asymptotic versions of T .

We will perturb T by a compact operator K, such that the set of normalizedblocks mapped by T +K to vectors of norm greater than ‖T +K‖−ε is asymptoticas a set of sequences of length 1. If we manage to do that, then an asymptoticversion of T + K, which can be approximated arbitrarily well by subsets extractedfrom asymptotic sets composed of sequences of the above blocks, will almost achievethe norm of T + K, as required.

Take λ to be (up to ε) the largest such that

{x ∈ S(X)1<; ‖T (x)‖ ≥ λ}

is asymptotic. By this we mean that the set

{x ∈ S(X)1<; ‖T (x)‖ ≥ λ + ε}

does not have elements in some tail subspace, [X]>m. Consider the compact per-turbation of T , T ′ = T − T ◦ Pm. By our assumption on the basis ‖T ′‖ ≤ λ + ε,and the set

{x ∈ S(X)1<; ‖T ′(x)‖ ≥ λ}

is still asymptotic. The proof is now complete.�

2. ASYMPTOTIC VERSIONS OF OPERATOR IDEALS 55

2.2. Finitely singular and asymptotically finitely singular operators.

Definition 2.3. An operator T on a sequence space X is asymptoticallyfinitely singular if for every ε there exists n(ε, T ) and an admission set Σn of lengthn, such that T takes some normalized block from the span of each sequence in Σ toa vector with norm less than ε.

T is called finitely singular if it satisfies the above definition with Σn = S(X)n<.

In other words, an operator T is asymptotically finitely singular, if, when re-stricted to the span of a sequence from Σ, T−1 is either not defined or has normlarger than 1

ε An operator ideal close to the ideal of finitely singular operators wasdefined in [Mi], and called σ0. The difference is that the original definition referredto all n dimensional subspaces, rather than just block subspaces, as we read here.

Proposition 2.4. Operators on a Banach-space X, which are asymptoticallyfinitely singular with respect to a given basis, form a Banach space with the usualoperator norm and a two sided ideal of L(X).

Proof. Let T be asymptotically finitely singular, and let S be bounded. STis obviously asymptotically finitely singular. Indeed, n(ε, ST ) ≤ n( ε

‖S‖ , T ), and theadmission sets for ST are the same as those for T .

To see that TS is asymptotically finitely singular we use the proof of step 1in Theorem 1.3, and find admission sets Σ′, whose normalized image under S isessentially contained in the admission sets Σ used to define T as an asymptoticallyfinitely singular operator. TS will take some normalized block from the span ofany sequence from Σ′ to a vector with arbitrarily small norm. Indeed, S takes(essentially) Σ′ to Σ, where T has an ’almost kernel’.

To show that the sum of two asymptotically finitely singular operators is alsoasymptotically finitely singular, we need to use the following:

Claim: If T is asymptotically finitely singular then for every ε and every k thereexists an N(k, ε, T ), and an admission set Σ of length N , such that every sequencefrom Σ has a k-dimensional block subspace on which T has norm less than ε.

This claim is standardly proved by taking concatenations of asymptotic setsfrom the definition of T as asymptotically finitely singular (it is easier to think hereof the finitely singular case: if T has an ’almost kernel’ on every n dimensional blocksubspace, then it has a k dimensional ’almost kernel’ in every N(k) dimensionalblock subspace).

To complete the proof of the proposition, fix ε > 0 and consider asymptoticallyfinitely singular operators, T and S. By definition of asymptotically finitely singularproduce an admission set, Σ, of length n( ε

2 , S), such that S takes some normalizedblock in the span of any Σ-admissible sequence to a vector with norm less than ε

2 .Take the admission set Ψ of length N(n, ε

2 , T ) from the above claim. It is possibleto extract an admission subset Ψ′ ⊆ Ψ, such that any n consecutive blocks of asequence in Ψ′ are also in Σ (similarly to Remark 4.2).

We therefore have that in any block sequence in Ψ′ there is an n-dimensionalblock subspace where T has norm less than ε

2 , and inside this subspace a normalizedvector, whose image under S has norm less than ε

2 . This means that T + S isasymptotically finitely singular.

The fact that asymptotically finitely singular operators form a closed subspaceof L(X) is straightforward.

�

56 4. ASYMPTOTIC VERSIONS OF OPERATORS AND OPERATOR IDEALS4

Remark 2.5.

(1) From Gowers’ combinatorial lemma (in [G5], see also chapter 2) it followsthat for every asymptotically finitely singular operator T , there is a blocksubspace Y where:

for every ε there exists an n, such that in the span of every sequencein S(Y )n

< supported after the n-th basic element, there is a normalizedvector whose image under T has norm less than ε.

It is easy to see that on this block subspace T is finitely singular; in-deed, every sequence of n blocks contains a sequence of [n

2 ] blocks supportedafter the [n

2 ]-th basic element.(2) It is not true, however, that the ideals of asymptotically finitely singular

and finitely singular operators coincide.Consider the following example: Let X be the `2 sum of increasingly

long `n2 ’s, and let Y be their `3 sum. The formal identity from X to Y is

not finitely singular, but is asymptotically finitely singular.(3) It is easy to extend Proposition 2.4 and show that asymptotically finitely

singular operators form a two sided ideal in the operator norm when re-stricting our attention to the category of Banach spaces with shrinkingbases. Finitely singular operators will only form a left-sided ideal in the(shrinking) basis context. This is because a bounded operator multipliedto the right side of a finitely singular operator does not have to preserveblocks.

(4) It is worth noting that in the Gowers-Maurey space (from [GM1]), alloperators are finitely singular perturbations of a scalar operator (this fol-lows from Lemma 22 and Lemma 3 in [GM1]). This point is even moreinteresting in light of Corollary 2.8 below.

The following proposition claims a strong dichotomy in the asymptotic struc-ture of operators: either it contains an isomorphism, or it is composed only offinitely singular operators.

Proposition 2.6. If T is an asymptotically finitely singular operator then alloperators in {T}∞ are finitely singular. If T is not asymptotically finitely singular,{T}∞ contains an isomorphism.

Proof. Let T be asymptotically finitely singular. Take asymptotic sets ap-proximating an asymptotic version of T , Φn. Let player V play the winning strategyfor {Φn}n while S plays the winning strategy for the admission sets Σ in the defi-nition of asymptotically finitely singular operators. The resulting vector sequencesmust approximate the same asymptotic version, but must also contain an ’almostkernel’ for T . Therefore any asymptotic version of T is finitely singular.

Suppose T is not asymptotically finitely singular. Then for some ε > 0 thesets Φn of all sequences in S(X)n

<, on which T is an isomorphism with ‖T−1‖ ≤ 1ε ,

are asymptotic. Indeed, if they weren’t, by Lemma 4.3 (chapter 1), for some ε andfor every n, Φc

n would be admission sets, and therefore T would be asymptoticallyfinitely singular, in contradiction. Using Theorem 1.3, extract from Φn asymptoticsubsets Φ′n, which approximate arbitrarily well an asymptotic version of T . Thisasymptotic version is an isomorphism. �

2. ASYMPTOTIC VERSIONS OF OPERATOR IDEALS 57

Remark 2.7. Note that the proof shows that if T is asymptotically finitelysingular, all its asymptotic versions will be finitely singular operators with the samen(ε) as T .

We offer here an application to the theory developed above. Recall that anasymptotic `p space is a space, all whose asymptotic versions are isomorphic to `p.

Corollary 2.8. Every asymptotically finitely singular operator from an as-ymptotic `p space X to itself is compact.

Proof. Consider the set of asymptotically finitely singular operatorsAFS(X) ⊂L(X). The asymptotic versions of these operators, by proposition 2.6 above, areall finitely singular operators in L(`p). In particular they are all asymptoticallyfinitely singular, and therefore belong to some proper closed two-sided ideal. It iswell known (cf. [GoMaFel]), that the only proper closed two sided operator idealin L(`p) is the ideal of compact operators, but let us sketch a very simple proof:

All asymptotic versions of T are diagonal finitely singular operators in L(`p).It is clear, then, that given any ε > 0, only finitely many entries on the diagonal arelarger than ε; otherwise, restricting to the span of the basic elements correspondingto the entries larger than ε we get an isomorphism. Therefore the entries on thediagonal go to zero, and the operator is compact.

We can now complete the proof of the corollary, using proposition 2.1 oncemore.

AFS(X) = {T ∈ L(X)|{T}∞ ⊆ FS(`p)} = {T ∈ L(X)|{T}∞ ⊆ K(`p)} = K(X).

where K(Z) is the set of all compact operators on Z, and FS(Z) is the set of allfinitely singular operators on Z. �

Note that, by this corollary, if an asymptotic-`p space with a shrinking basishas the the property of the Gowers-Maurey space from the last point of Remark2.5, than all bounded linear operators on this space will be compact perturbationsof scalar operators. We do not know whether such space exists.

2.3. A general theorem. We conclude with a theorem which explains that the above phenomena are apart of a more general situation. When referring to operator ideals we invoke thecategorical algebraic definition from [P]. An injective operator ideal, J , has theproperty that if T : X → Y is in J , then the same operator with a revised range,T : X → Im(T ) is also in J . The following theorem states that the ’asymptoticpreimage’ in L(X) of an injective operator ideal is an ideal in the algebra L(X).

Theorem 2.9. Let J be an injective operator ideal, and let X be a space with ashrinking basis. The set of operators: J ′ = {T ∈ L(X)|{T}∞ ⊆ J} is an operatorideal in L(X).

Proof. If we multiply an operator S ∈ J ′ with an operator T ∈ L(X), anasymptotic version of the product will always be a product of asymptotic versions.

Indeed, take the asymptotic sets approximating an asymptotic version, R, ofR = ST (or R = TS), and extract (by the proof of Theorem 1.3) asymptoticsubsets, which approximate an asymptotic version T of T , and whose normalizedimages under T approximate an asymptotic version S of S. The product of these

58 4. ASYMPTOTIC VERSIONS OF OPERATORS AND OPERATOR IDEALS5

asymptotic versions, ST ∈ J , is also approximated by the same asymptotic subsets.But these asymptotic subsets must still approximate R.

Therefore R ∈ J , and R must be in J ′.Let T and S be in J ′. Let R = S+T , and find asymptotic sets approximating R,

an asymptotic version of R. Extract asymptotic subsets approximating asymptoticversions of S and T , S and T respectively. Note that we cannot say that R = S+ T ,since S and T may have different ranges. However, we do have:

(10) ‖R(x)‖ ≤ ‖S(x)‖+ ‖T (x)‖,This is enough in order to prove R ∈ J .Indeed, we can write:

R = P ◦ (i1 ◦ S + i2 ◦ T ),

where

R : X → W , S : X → Y , T : X → Z,

i1 : Y → Y ⊕ Z, i2 : Z → Y ⊕ Z,

i1(y) = (y, 0), i2(z) = (0, z),and P : Im(i1 ◦ S + i2 ◦ T ) → W is defined by: P

((i1 ◦ S + i2 ◦ T )(x)

)= R(x)

and by continuity. Inequality (10) assures that P is well defined and continuous.Now, T and S are in J , so i1 ◦ S + i2 ◦ T is also in J . By injectivity, we are

allowed to modify the range as we compose with P , and still get that the result ,R,is in J .

Therefore R ∈ J ′, and we conclude that J ′ is an ideal in L(X).�

Bibliography

[AAr] D. Alspach & S. Argyros, Complexity of weakly null sequences, Diss. Math. 321 (1992).

[AnO] G. Androulakis & E. Odell, Distorting mixed Tsirelson spaces, preprint.

[ArD] S. Argyros & I. Deliyanni, Examples of asymptotically `1 Banach spaces, preprint.[ArMeTs] S. Argyros, S. Merkourakis & A. Tsarpalias, Convex unconditionality and sumability

of weakly null sequences, preprint.[B] S. Bellenot, The Banach space T and the fast growing hierarchy from logic, Israel J. of Math.

47 (1984), 305–313.

[Bo] F. Bohnenblust, An axiomatic characterization of Lp-spaces, Duke Math. J. 6 (1940), 627–640.

[Bou] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Bel. 32

(1980), 235–249.[CaPfY] S.R. Caradus, W.E. Pfaffenberger & B. Yood, Calkin Algebras and Algebras of

Operators on Banach Spaces, Lecture Notes in Pure and Applied Mathematics 9, Dekker,

New-York, 1974.[C] P.G. Casazza, Some questions arising from the homogeneous Banach space problem, Con-

temporary Math. 144 (Bor-Luh Lin & W.B. Johnson, eds.), Amer. Math. Soc., 1993, 35–52.

[CJoTz] P. G. Casazza, W. B. Johnson & L. Tzafriri, On Tsirelson’s space, Israel J. Math. 47(1984), 81–98.

[CSh] P. G. Casazza & T. J. Shura, Tsirelson’s Space, Lecture Notes in Math. 1363, Springer-Verlag, 1989.

[DKut] I. Deliyanni & D. Kutzarova, On some asymptotic `1 Banach spaces, preprint.

[Dv] A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Symp. LinearSpaces, Jerusalem, 1961, 123–160.

[Fe] V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, preprint.

[FJo] T. Figiel & W. B. Johnson, A uniformly convex Banach space which contains no `p, Comp.Math. 29 (1974), 179–190.

[GoMaFel] I.C. Gohberg, A.S. Markus & I.A. Feldmann, Normally solvable operators and ideals

associated with them (Russian), Bul. Akad. Stiince RSS Moldoven. 10 (76) (1960), 51–70.[G1] W.T. Gowers, Lipschitz functions on classical spaces, European J. Comb. 13 (1992), 141–

151.[G2] W.T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26

(1994), 523–530.[G3] W.T. Gowers, A hereditarily indecomposable space with an asymptotically unconditional

basis, Geometrical Aspects of Functional Analysis, Operator Theory: Advances and

Applications 77 (J. Lindenstrauss & V. Milman, eds.), Birkhauser, 111–120, 1995.[G4] W.T. Gowers, A Banach space not containing `1, c0 or a reflexive subspace, preprint.

[G5] W. T. Gowers, A new dichotomy for Banach-spaces, preprint.[G6] W.T. Gowers, Analytic sets and games in Banach spaces, preprint.[GM1] W.T. Gowers & B. Maurey, The unconditional basic sequence problem, Jour. A.M.S. 6

(1993), 851–874.

[GM2] W.T. Gowers & B. Maurey, Banach spaces with small spaces of operators, preprint.[H] P. Habala, A Banach space whose subspaces do not have the Gordon-Lewis property, preprint.

[J] R. C. James, Uniformly non-square Banach spaces, Ann. Math. 80 (1964), 542–550.[JuO] R. Judd & E. Odell, Concerning the Bourgain `1 index of a Banach space, preprint.[K] Kolmogorov’s Zippin thm.?????

[Ko] D. Konig, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsge-sellschaft, Leipzig, 1936; reprinted by Chelsea, New-York, 1950, page 81.

59

60 BIBLIOGRAPHY

[Kr] J.L. Krivine, Sous-espaces de dimension finis des espaces de Banach reticules, Ann. Math.104 (1976), 1–29.

[Ku] K. Kuratowski, Topology, vol. I, academic Press, New York 1966.

[LMi] J. Lindenstrauss & V. Milman, The local theory of normed spaces and its applications toconvexity, Handbook of Convex Geometry (P.M. Gruber & J.M.Wills, eds.), Elsevier

Science publishers B.V., 1993, 1149–1220.

[LTz] J. Lindenstrauss & L. Tzafriri, Classical Banach Spaces, vol. I, Ergebnisse der Mathe-matik und ihrer Grenzgebiete 92, Springer Verlag, 1977.

[M] B. Maurey, A remark about distortion, Geometrical Aspects of Functional Analysis,Operator Theory: Advances and Applications 77 (J. Lindenstrauss & V. Milman, eds.),

Birkhaser 131–142, 1995.

[MMiTo] B. Maurey, V. Milman & N. Tomczak-Jaegermann, Asymptotic infinite-dimensional the-ory of Banach spaces, Geometrical Aspects of Functional Analysis, Operator Theory:

Advances and Applications 77 (J. Lindenstrauss & V. Milman, eds.), Birkhauser, 149–175,

1995.[Mi] V. Milman, Spectrum of continuous bounded functions on the unit sphere of a Banach space,

Funct. Anal. Appl. 3 (1969), 67–79 (translated from Russian).

[MiTo] V. Milman & N. Tomczak-Jaegermann, Asymptotic `p spaces and bounded distortion,Contemporary Math. 144 (Bor-Luh Lin & W.B. Johnson, eds.), Amer. Math. Soc., 1993,

173–195.

[Mo] J. D. Monk, Introduction to Set Theory, McGraw-Hill, 1969.[O] E. Odell, private communication.

[OS1] E. Odell & T. Schlumprecht, The distortion problem, GAFA 3 (1993), 201–207.

[OS2] E. Odell & T. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281.[OS3] E. Odell & T. Schlumprecht, A Banach space block finitely universal for monotone bases,

preprint.[P] A. Pietsch, Operator Ideals, North-Holland Mathematical Library 20, North-Holland pub-

lishing company, 1980.

[R] H. Rosenthal, A characterization of Banach spaces containing `1, Proc. Nat. Acad. Sci.U.S.A. 71 (1974), 2411–2413.

[S] T. Schlumprecht, An arbitrarily distortable Banach-space, Israel. J. Math. 76 (1991), 81–95.

[To1] N. Tomczak-Jaegermann, Notes privately distributed, December 1994.[To2] N. Tomczak-Jaegermann, Banach spaces of type p > 1 have arbitrarily distortable subspaces,

preprint.

[T] B. S. Tsirelson, Not every Banach space contains `p or c0, Funct. Anal. Appl. 8 (1974),138–141.

[Z] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265–

272.