Banach Spaces Determined by Their Uniform Structures

BANACH SPACES DETERMINED BY THEIR UNIFORM STRUCTURES

yz z z

by William B Johnson Joram Lindenstrauss and Gideon Schechtman

Dedicated to the memory of E Gorelik

Abstract

Following results of Bourgain and Gorelik we show that the spaces p as

well as some related spaces have the following uniqueness prop erty If X is a Banach

space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the

same space We also prove that if a C K space is uniformly homeomorphic to c then

it is isomorphic to c We show also that there are Banach spaces which are uniformly

homeomorphic to exactly isomorphically distinct spaces

Subject classication B Hxx Keywords Banachspaces Uniform homeomor

phism Lipschitz homeomorphism

Erna and Jacob Michael Visiting Professor The Weizmann Institute

Supp orted in part by NSF DMS

Supp orted in part by the USIsrael Binational Science Foundation

Participant Workshop in Linear Analysis and ProbabilityTexas AM University

Intro duction

The rst result in the sub ject we study is the MazurUlam theorem which says that

an isometry from one Banach space onto another whichtakes the origin to the origin must

be linear This result which is nontrivial only when the Banach spaces are not strictly

convex means that the structure of a Banach space as a metric space determines the linear

structure up to translation On the other hand the structure of an innite dimensional

Banach space as a top ological space gives no information ab out the linear structure of the

space Kad Tor In this pap er we are concerned with equivalence relations of Banach

spaces which lie b etween isometry and homeomorphism mainly Lipschitz equivalence and

uniform homeomorphism There exists a considerable literature on this topic see Ben

for a nice survey to ab out Nevertheless the sub ject is still in its infancy as many

fundamental basic questions remain unanswered What we nd fascinating is that the

sub ject combines top ological arguments and constructions with deep facts from the linear

structure theory of Banach spaces The present pap er is also largely concerned with this

interplay

Early work in this sub ject esp ecially that of Rib e Rib showed that if two Banach

spaces are uniformly homeomorphic then each is nitely crudely representable in the other

Recall that Z is nitely crudely representable in X provided there is a constant so that

each nite dimensional subspace E of Z is isomorphic to a subspace of X In a short but

imprecise way this means that the two spaces have the same nite dimensional subspaces

Since the spaces and L have the same nite dimensional subspaces a natural

p p

question was whether they are uniformly homeomorphic for p p This

question was answered in the negative rst for p by Eno Ben then for p

by Bourgain Bou and nally for p by Gorelik Gor The main new point

in Goreliks pro of is a nice top ological argument using the Schauder xed point theorem

In section we formulate what Goreliks approach yields as The Gorelik Principle

This principle is most conveniently applicable for getting information ab out spaces which

are uniformly homeomorphic to a space which has an unconditional basis with a certain

convexity or concavity prop erty see eg Corollary

In section we combine Bourgains result Bou and the Gorelik Principle with struc

tural results from the linear theory to conclude that the linear structure of p

is determined by its uniform structure that is the only Banach spaces which are uni

formly homeomorphic to are those which are isomorphic to The case p was done

p p

twentyveyears ago by Eno Enf The main part of section is devotedtoproving that

some other spaces are determined by their uniform structure and also to investigating the

p ossible number of linear structures on spaces which are close to in an appropriate

lo cal sense

In section we use the Gorelik Principle to study the uniqueness question for c

answering in the pro cess a question Aharoni asked in Aha

Very roughly the passage from a uniform homeomorphism U between Banach spaces

to a linear isomorphism involves two steps

Passage from a uniformly continuous U to a Lipschitz map F via the formula

F x lim n U nx

Passage from a Lipschitz map F to its derivative

Of course b oth steps lead to diculties In step one a ma jor problem is that the limit

do es not exist in general and thus one is forced to use ultralters and ultrapro o ducts In

step two the problem is again the the existence of derivatives Derivatives in the sense of

Gateaux of Lipschitz functions exist under rather mild conditions but they usually do not

suce on the other hand derivatives in the sense of Frechet which are much more useful

exist or at least are known to exist only in sp ecial cases

It is therefore natural that in section ultrapro ducts are used as well as a recent

result LP on dierentiation which ensures the existence of a derivative in sense whic h is

between those of Gateaux and Frechet

It is evident that for step one of the pro cedure ab ove it is imp ortantthatU b e dened

on the entire Banach space In fact it often happ ens that the unit balls of spaces X and

Y are uniformly homeomorphic while X and Y are not The simplest example of this

phenomenon go es back to Mazur Maz who noted that the map from L p

to L dened by f jf j sign f is a uniform homeomorphisms between the unit balls

of these spaces while in Lin and Enf it is proved that the spaces themselves are

not uniformly homeomorphic this of course also follows from Rib es result mentioned

ab ove The Mazur map was extended recently to more general situations by Odell and

Schlumprecht OS In section we obtain estimates on the mo dulus of continuity of these

generalized Mazur maps The pro ofs are based on complex interp olation

In the rather technical section wecombine the results of sections and with known

constructions to pro duce examples for each k of Banach spaces which admit

exactly linear structures The most easily describ ed examples are certain direct sums

of convexications of Tsirelsons space see Tsi CS

Although our interest is mainly in the separable setting we present in section some

results for nonseparable spaces

As p ointed out by Rib e Rib Enos result on Enf mentioned earlier follows

from the fact that is determined by its nite dimensional subspaces Section whichis

formally indep endent of the rest of the pap er is motivated by the problem of characterizing

those Banach spaces which are determined by their nite dimensional subspaces We

conjecture that only has this prop erty Weshowthatany space which is determined by

its nite dimensional subspaces must be close to and if we replace determined by its

nite dimensional subspaces by a natural somewhat stronger prop erty then b esides

there are other spaces which enjoy this prop erty

The pap er ends with a section which mentions a few of the many op en problems

connected to the results of sections

We use standard Banach space theory language and notation as may be found in

LT and TJ

The authors thank David Preiss for discussions which led to the formulation of Prop o

sition

As is clear from this intro duction our work on this pap er was motivated by Goreliks

Gor A few days after he submitted the nal version of Gor Gorelik was fatally pap er

injured by a car while he was jogging We all had the privilege of knowing Gorelik

unfortunately for only a very short time

The Gorelik Principle

A careful reading of Goreliks pap er Gor leads to the formulation of the following

principle

The Gorelik Principle A uniform homeomorphism b etween Banach spaces cannot take

a large ball of a nite co dimensional subspace into a small neighborhood of a subspace of

innite co dimension

The precise formulation of the Gorelik Principle in the form we use is

Theorem Let U b e a homeomorphism from a Banach space X onto a Banachspace

Y with uniformly continuous inverse V Supp ose that d b are such that there exist a nite

co dimensional subspace X of X and an innite co dimensional subspace Y of Y for which

U dBall X Y bBall Y

Then V b d where V t sup fkVy Vy k ky y k tg is the mo dulus

of uniform continuity of V

The pro of of the Gorelik Principle is based on two lemmas which are minor variations

of Lemmas and in Goreliks pap er Gor The rst lemma is obvious while the second

is a simple consequence of Brouwers xed point theorem

Lemma let Y be an innite co dimensional subspace of the Banach space Y and B

a compact subset of Y Then for every there is a y in Y with ky k so that

dB y Y

Lemma Let X b e a nite co dimensional subspace of the Banach space X For every

there is a compact subset A of Ball X so that whenever is a continuous map

A X from A into X for which kx xk for all x in A then

Pro of of Lemma By the BartleGraves theorem or Michaels selection theorem

Mic there is a continuous selection f Ball XX Ball X of the inverse of the

quotient mapping Q from X to XX Set A f Ball XX and apply Brouwers

theorem to the mapping x x Qf x from Ball XX to itself

Pro of of Theorem Get the set A from Lemma for the value d

Applying Lemma to the compact set U A we get y in Y with ky k b so that

dU A y Y b The mapping from A into X dened by a V Ua y moves each

p oint of A a distance of at most V ky k V b so if this is less than d we

have from Lemma that there must be a p oint a in A for which V Ua y is in X

and hence in dBall X But then Ua y would be in Y bBall Y which contradicts

the choice of y Therefore V b d as desired

Recall that an unconditional basis is said to have an upp er pestimate resp ectively

lower pestimate provided that there is a constant C so that for every nite

of vectors which are disjointly supp orted with resp ect to the uncondi sequence fx g

tional basis the quantity x is less than or equal to resp ectively greater than or

equal to C kx k Tocharacterize the uniform homeomorphs of and the Tsirelson

k p

spaces we need the following implementation of the Gorelik Principle

Theorem Supp ose that X has an unconditional basis which has an upp er pestimate

and X is uniformly homeomorphic to Y Then no quotientofY can have an unconditional

basis which has a lower r estimate with rp

For the pro of of Theorem we need to recall the concept of approximate metric

midp oint which plays a role also in the pro ofs of Eno Enf and Bourgain Bou Given

p oints x and y in a Banach space X and let

kx y k

Mid x y z X kx z kkz y k

We also need a quantitative way of expressing the wellknown fact that a uniformly

continuous mapping from a Banach space is Lipschitz for large distances If U is a

uniformly continuous mapping from a Banach space X into a BanachspaceY set for each

kUx Ux k

u sup x x X

t kx x k

The statement that U is Lipschitz for large distances is just that u is nite for each

t Obviously u is a decreasing function of t denote its limit by u

Lemma Let U be a uniformly continuous mapping from a Banach space X and let

d Supp ose that x y in X satisfy for certain

i kx y k

ii kUx Uyk kx y k

Then U Mid x y MidUxUy

Pro of Let z be in Midx y Then

kx y k kx y k

u d kUx UzkkUz Uyk u

d d

kUx Uyk

In a space which has an unconditional basis which has an upp er or lower pestimate

the set of approximate metric midp oints between two points has some obvious structure

We state the next lemma only for p oints symmetric around by translation one obtains

a similar statement for general sets Midx y

Lemma Supp ose that x is in X and X has a basis fx g whose unconditional

constant is one

has an upp er pestimate with constant one then for each there is a i If fx g

nite co dimensional subspace X of X so that Midx x kxk kxk Ball X

ii If fx g hasalower r estimate with constant one then for each there is a nite

dimensional subspace X of X so that Midx x kxk X r kxk Ball X

Pro of We assume p and x since otherwise the conclusion is trivial To prove

kxk and y is disjoint from x relative to the basis i supp ose that y is in X ky k

fx g Then ky xk kxk kxk So when x is nitely supp orted

relative to fx g we can take for X the subspace of all vectors in X which vanish

on the supp ort of x The general case follows by approximating x by a vector with nite

supp ort the degree of approximation dep ending on

n n

The pro of of ii is similar If x is in span fx g X spanfx g then works

k k

indep endently of the general case follows by approximation

Pro of of Theorem If the conclusion is false we may assume after renorming

X and Y that the basis fx g for X has unconditional constant one and also the upp er

pestimate constant for fx g is one and that there is a quotient mapping Q from Y

onto a space Z having a basis fz g with unconditional constant one and with lower

r estimate constant one Let U be a uniform homeomorphism from X onto Y Since it is

easy to c heck that the Lipschitz for large distance constants s of S QU are b ounded

away from we can assume without loss of generality that s

Fix later we shall see howsmall need b e to yield a contradiction Since s

as t we can nd a pair x y of vectors in X with kx y k as large as we please for

kSxSyk

is as close to one as we please From one thing we want kx y k so that

kxy k

Lemma we then get as long as kx y k is suciently large that

S Midx y kx y k Mid Sx Sy kx y k

By making translations we only need to consider the case when y x and U x

Ux that is we can assume that there is x in X with kxk as large as we like and

S Mid x x kxk MidSx Sx kx y k

From Lemma we get a nite co dimensional subspace X of X and a nite dimen

sional subspace Z of Z so that

i h

kxk Ball X Z r kxk Ball Z S

Set Y Q Z Then Y has innite co dimension in Y and

h i

U kxk Ball Y kxk Ball X Y r

The Gorelik Principle tells us that

kxk

kxk V r

where V U

However keeping in mind that kxk we have

r r

kxk v r kxk V r

Putting and together gives

v r

which is a contradiction for small enough

In Section we need the next corollary

Corollary Supp ose that X has an unconditional basis and X is uniformly homeo

morphic to Y Assume that either the unconditional basis for X has a lower r estimate

for some r or that X is sup erreexive and the unconditional basis for X has an upp er

pestimate for some p Then is not isomorphic to a subspace of Y

Pro of Bourgain Bou proved that for r there is no homeomorphism U from into

for which b oth U and U are Lipschitz for large distances a nonessential mo dication

yields the same result for any space X which has an unconditional basis which hasalower

r estimate for some r Consequently is not isomorphic to a subspace of any space

which is uniformly homeomorphic to such an X So assume that X is sup erreexive and

the unconditional basis for X has an upp er pestimate for some p This implies that

X is of typ e in the sense of MaureyPisier see LT f and hence so is Y since

Y is nitely crudely representable in X by Rib es theorem Rib or see Ben Thus

by Maureys theorem Mau any isomorphic copy of in Y is complemented in Y so

Theorem implies that is not isomorphic to a subspace of Y

Uniform Homeomorphs of T and related spaces

The rst result in this section is an immediate consequence of Corollary and

previously known results

Theorem If X isaBanach space which is uniformly homeomorphic to p

then X is isomorphic to

Pro of By Rib es theorem Rib X is isomorphic to a complemented subspace of L

But X do es not contain an isomorphic copy of by Corollary if p so by the

results of JO X is isomorphic to

Remark The case p in Theorem was proved by Eno Enf A simple pro of using

ultrapro ducts was provided by Heinrich and Mankiewicz HM Ben They also used

ultrapro ducts to give a simple pro of of Rib es theorem Rib Since we need the typ e of

reasoning intro duced in HM to prove the remaining theorems we recall some moreorless

standard facts ab out ultrapro ducts of Banach spaces Much more information as well as

references to the original sources can be found in Hei

Given a sequence fX g of Banach spaces and a free ultralter U on the natural

numbers denote by X or just X if all the X s are the same the Banach space

n n

U U

ultrapro duct of fX g dened as the collection of all b ounded sequences fx g with

n n

x X under the norm lim kx k Here we identify two sequences fx g and fy g

n n n n n

n n

as b eing the same if lim kx y k The space X is nitely representable in X A

n n

less commonly used fact which is imp ortant for us is that nite dimensional complemented

subspaces of X pull down to X Precisely if E is a nite dimensional subspaces of X

U U

whichiscomplemented in X then for each E is isomorphic to a subspace

of X which is complemented in X This follows from the ultrap ower version of

lo cal duality Hei p which says that every nite dimensional subspace of X

embeds into X in such a way that the action on any xed nite subset of X

U U

is preserved The space X is always a subspace of X but these spaces coincide

U U

only when X is sup erreexive

The space X is naturally emb edded into X as the diagonal If X is reexive it is

norm one complemented in X map fx g in X to the U weak limit in X of fx g

n n

U U

the kernel X of this pro jection consists of those b ounded sequences in X which tend

weakly to zero along the ultralter U

An ultrap ower of a Banach lattice X is again a Banach lattice moreover if X is a L

space then the norm in X is padditive for disjointvectors and then so is the norm in the

ultrap ower Thus bythe generalized Kakutani representation theorem the ultrap ower is

also a L space if p

Heinrich and Mankiewicz HM see also Ben used ultrapro ducts to give simple

pro ofs of a number of previously known theorems concerning uniformly homeomorphic

Banach spaces and to answer a number of op en problems Here we just recall the basic

approach in a situation which is general enough to meet our needs

Supp ose that X and Y are separable uniformly homeomorphic Banach spaces Using

the fact that a uniformly continuous mapping from a Banach space is Lipschitz for large

hitz equivalent ultrap owers X and Y distances one sees easily that X and Y have Lipsc

U U

Supp ose now that X hence also X is sup erreexive Using a backandforth pro cedure

and a classical weak compactness argument Lin one gets a separable norm one com

plemented subspace X of X whichcontains X and is Lipschitz equivalent to a subspace

Y of Y which contains Y A dierentiation argument combined with a technique from

Lin now yields that Y isomorphically emb eds into X as a complemen ted subspace

This shows that Y is also sup erreexive and so X embeds into Y as a complemented

subspace Moreover now that we know that Y is reexive we can make the earlier

construction pro duce a Y which is norm one complemented in Y

One of several consequences of this construction which we use later is that uniformly

homeomorphic spaces X and Y have the same nite dimensional complemented subspaces

if X is separable and sup erreexive in fact without any restriction on X but this requires

more work

necessary background on ultrap owers analysis only slightly more involved Given the

than that of Theorem yields

Theorem Let p p p or p p p and set

n n

X If X is uniformly homeomorphic to a Banach space Y thenX is isomorphic

to Y

Pro of For simplicity of notation we treat the case n By one of the results of

HeinrichMankiewicz HM or see Ben X Y have Lipschitz equivalent ultrap owers

X Y resp ectively and X splits as the direct sum of an L space with an L space

p p

U U U

X L L We do not need that the measures for p and p are the same

p p

but they are Rib e proved that Y is nitely crudely representable in X Rib HM

Ben so Y is sup erreexive and Y splits as the direct sum of Y and some space Z

which is incidentallyemb eddable as a subspace of L L for some measure

p p

Since Y is separable there exists a separable subspace Z of Z so that the image of Y Z

under the Lipschitz isomorphism from Y on to X is of the form L L where

p p

U U

is the restriction of to a separable sigma subalgebra Hence by HM Y is isomorphic

to a complemented subspace of L L whence of L L

p p p p

Supp ose now that p and p are larger than two Let J denote the emb edding of

and let S S be the natural L Y onto a complemented subspace of L

p p

resp ectively We know and L onto L L pro jections from L

p p p p

from Corollary that do es not embed into Y soby the generalization in Joh of the

theorem from JO used earlier for k the op erator S J factors through Hence

k p

J factors through whence Y emb eds into as a complemented subspace

p p p p

However by a result of EdelsteinWo jtaszczyk Ede Woj EW or see LT p

every complemented subspace of is isomorphic to or Theorem

p p p p p p

eliminates the rst two p ossibilities

When p and p are smaller than two we pass to the duals Y is isomorphic to a

where q is the conjugate index to p complemented subspace of L L

q q

k k

Y also do es not contain an isomorph of since Y has typ e every copy of in Y

is complemented The reasoning in the last paragraph shows that Y is isomorphic to

q q

In sections and we prove that for p the pconvexied version T of

Tsirelsons space is uniformly homeomorphic to T Here weshow that for p

T is uniformly homeomorphic to at most two isomorphically distinct spaces Since it is

desirable to have general conditions which limit the isomorphism class of spaces uniformly

homeomorphic to a given space it seems worthwhile to prove some results in this direction

which might b e used elsewhere The reader who is interested only in the example can skip

to Prop osition and substitute T for X in the statement

We b egin with a denition

Denition A Banach space X is said to b e as L p provided there exists

so that for every n there is a nite co dimensional subspace Y so that every n dimensional

subspace of Y is contained in a subspace of X whichisisomorphic to L for some

A space is as L if and only if it is asymptotically Hilb ertian in the sense of Pisier

Pis We avoided the term asymptotically L because when p as L may not be the

p p

right denition for asymptotically L The denition we give is also not the denition

one gets by sp ecializing Pisiers Pis p as prop erty P to P L structure

however Pisiers denition seems right only for hereditary prop erties

In section we review some of the prop erties of the Tsirelson spaces Here we just

mention that T has an unconditional basis fe g for which there is a constant such

that for every n there exists an m so that every ntuple of disjointly supp orted unit

v is equivalent to the unit vector basis in It is easily seen that ectors in span fe g

k m

a space X with this prop ertyisas L and also that X is isomorphic to L for some

p p

measure This prop erty of such spaces can be generalized to the class of as L spaces

Prop osition a Supp ose that X is as L p Then for every ultralter U

on the natural numbers the space X is a L space

Pro of First use the James characterization of reexivitysup erreexivity to see that

X is sup erreexive see for example the argument in Pis pp for the similar

case of asymptotically Hilb ertian spaces Therefore X splits as the direct sum of X and

X Now if E is a nite say n dimensional subspace of X then it is not hard to

U U

of ndimensional subspaces of X which converges see that there exists a sequence fE g

weakly to zero along U in the sense that every b ounded sequence fx g x in E with

k k k

converges weakly to converges weakly to along U so that E E Since fE g

k k

zero along U and have b ounded dimension given any nite co dimensional subspace Y of

X a standard small p erturbation argument shows that there are space automorphisms T

on X with lim kI T k and T E Y for all k Consequentlysince X is as L with

k k k p

constant say there are sup erspaces F in X of E with F isomorphic to L

k k k k p k

and lim In particular the inclusion mappings from the E s into X uniformly

k k

factor through L spaces for a U large set of k s Passing to ultrapro ducts and using the

fact that an ultrapro duct of L spaces is again an L space we conclude that the inclusion

p p

mapping from E into X factors through an L space with the norms of the factoring maps

indep endent of k The same can be said ab out the inclusion mapping from E into X

since X is complemented in X Finally we conclude from LR that X is either a

U U U

L or isomorphic to a Hilb ert space But the latter is imp ossible for p Indeed is

p p

nitely crudely representable in X and hence also in every nite co dimensional subspace

of X and this implies that embeds into X

Prop osition b Let X be a reexive space for which there exists an ultralter U on

the natural numbers so that X is L p Then X is as L

p p

Pro of Assume that the L constant of X is smaller than but that X is not as

U U

L with constant Let n b e such that every nite co dimensional subspace of X contains

an ndimensional subspace no sup erspace of whichisisomorphic to an L space Using a

standard basic sequence construction we can construct a nite dimensional decomp osition

g of ndimensional subspaces of X so that no sup erspace of any E is isomorphic fE

k k

to an L space But since X is reexive the sequence fE g converges weakly to zero

p k

and hence the ultrapro duct E E is an ndimensional subspace of X Then there

U U

is a nite say m dimensional subspace of X which contains E and whose Banach

Mazur distance from is less than It follows that F can be represented as F

where for each k F is an mdimensional subspace of X which contains E Necessarily

k k

the set of k s for which F is isomorphic to is in U hence is nonempty

The next lemma is an as version of the result LR that a complemented subspace of

a L space is either a L space or a L space

p p

Lemma If X is as L p and Y is a complemented subspace of X then

Y is as L or as L Y is as L i is nitely crudely representable in Y i Y contains

p p p

uniformly complemented s for all n

Pro of The ultrapro duct of a pro jection from X onto Y denes a pro jection from X onto

Y which pro jects X onto Y Thus the rst conclusion follows from Prop osition

U U U

and the classical theory of L spaces LPe LR The last sentence follows from the fact

KP that a complemented subspace of an L space which is not isomorphic to a Hilb ert

space contains a complemented subspace isomorphic to and the fact mentioned earlier

that one can pull down nite dimensional wellcomplemented subspaces of an ultrap ower

to the base space

We also need the as version of Rib es Rib theorem that a uniform homeomorph of

a L space pis again a L space

p p

Lemma If X is separable as L p and Y is uniformly homeomorphic to

X then Y is as L

Pro of Y is nitely crudely representable in X by Rib or see HM so Y is sup erreex

ive Reasoning as in the pro of of Theorem we get Lipschitz equivalent ultrapro ducts

X X X and Y Y Y of X and Y resp ectively and by Prop osition

U U U U

X is L Continuing as in the earlier pro of we get separable subspaces X of X and

U U

Y of Y with X a L space so that X X is Lipschitz equivalenttoY Y By another

result from HM Y is isomorphic to a complemented subspace of X X which of course

is as L Hence by Lemma Y is as L or as L But X is nitely representable in

p p

Y so the latter is imp ossible

Remark The separability assumption is not needed in Lemma One way to see

this is to check that if X is as L p then X X X with X separable and

X L Casazza and Shura CS p proved this for p and the general case can be

done similarly

The pro of of Lemma yields that if the separable as L space X is uniformly

homeomorphic to Y then X and Y are Lipschitz equivalent and hence by HM

each is isomorphic to a complemented subspace of the other Prop osition givesaversion

of this for certain direct sums of as L spaces For its pro of we need a consequence of the

preceding Supp ose that X Y contains uniformly complemented copies of for all n X is

as L p and r p Then Y contains uniformly complemented copies of

for all n Indeed one observes that emb eds complementably into X Y X Y

U U U

and hence complementably into either X or Y But X is as L for example by

U U U

Prop osition a so if emb eds complementably into X then is as L or as L by

r r p

Lemma The latter is clearly imp ossible and so is the former by the last statementin

Lemma and wellknown prop erties of Hence embeds complementably into Y

r r

whence Y a fortiori Y itself contains uniformly complemented copies of for all n

Prop osition Let p p p or p p p and

n n

for k n assume that X is a separable as L space with an unconditional basis

k p

which has an lower r estimate for some r or an upp er pestimate for some p Set

X X If X is uniformly homeomorphic to a Banach space Y then X

k p

and Y are isomorphic to complemented subspaces of each other

Pro of As in the pro of of Theorem we assume for simplicity of notation that n

Reasoning as in the pro of of Lemma we get Lipschitz equivalent ultrapro ducts X and

Y of X and Y resp ectively X is X X X where X is L for k Continuing

k p

U U

t to as in the pro of of Lemma we conclude that X Z Z is Lipschitz equivalen

Y W for some separable L spaces Z and some W By enlarging the Z s and W if

p k k

X L necessarywe can assume that Z L So Y emb eds into X L

p k p p

as a complemented subspace However we also know from Corollary that is not

isomorphic to a subspace of Y so reasoning as in the pro of of Theorem w e conclude

The two X that Y is isomorphic to a complemented subspace of X

p p

spaces in parentheses are totally incomparable so by the theorem of EdelsteinWo jtaszczyk

EW or LT p Y Y Y with Y isomorphic to a complemented subspace of

X Lemma now tells us that Y is as L or as L But Y is uniformly

k p k p

k k

homeomorphic to X and X contains uniformly complemented s for all n hence so do es

Y by HM see the earlier discussion on ultrapro ducts But Y cannot contain uniformly

and similarly Y is as complemented s for all n and so Y must Thus Y is as L

L This shows that the situation with X and Y is symmetrical and we can conclude

that X is isomorphic to a complemented subspace of Y

p p

Remark In Prop osition even when n thehyp othesis on X cannotbeweakened

to X is L X has an unconditional basis and no subspace of X is isomorphic to

T is as L for p and thus is a counterexample by Prop osition Indeed T

The conclusion in Prop osition is not completely satisfactory since it is op en

p p

whether a complemented subspace of T which contains a complemented copyofT must

is p ossible to improve Prop osition for spaces of be isomorphic to T Fortunately it

Tsirelson typ e without solving this problem ab out T However this requires rather more

work so we treat rst the simpler case of T and related spaces which already provides

examples of spaces that are isomorphic to exactly two isomorphically distinct spaces Basic

to this argument is a recent result concerning dierentiation LP

Assume that X and Y are separable sup erreexive Banach spaces Let F b e a Lipschitz

map from X into Y and G a Lipschitz map from X into a nite dimensional normed space

Then for every there is a p oint x in X so that F and G are Gateaux dieren tiable

at x and moreover there exists so that for jjujj

jjGx u Gx Sujj jjujj

where S is the Gateaux derivative of G at x

In other words G is almost up to even Frechet dierentiable at x An easy

consequence of this result and the reasoning used in HM is the following

Prop osition Supp ose that F is a mapping from the separable sup erreexive space

X onto a Banach space Y the Lipschitz constantofF is one and the Lipschitz constantof

F is C Then for every nite dimensional subspace E of Y and each

T C TX is C complemented there is a norm one op erator T from X into Y so that

in Y and T C

Pro of Note that the space Y is also sup erreexive eg by Rib We shall see that for

T one may take the Gateaux derivativeofF at a suitable p oint x in X thepoint dep ends

on the subspace E of Y In view of the pro of of Prop osition of HM and the fact

T C and that T is the Gateaux derivativeof F at some p oint the conclusions that

TX is C complemented in Y are automatic To nd the point x apply the result from

LP to the mapping F and an auxiliary mapping G which we now dene Let G be the

map F followed by the evaluation quotient mapping Q from Y onto E so that for x in

X and y in E Gxy y Fx Given set and get x and S from

the LP result stated ab ove For notational simplicity assume that x and F

this is without loss of generality b ecause it amounts to making one translation in X

and another in Y Of course S QT where T is the Gateaux derivative of F at

Only the last conclusion that is the T C needs to b e checked Since Q

inclusion mapping from E into Y this is the same as checking that S C

So supp ose that y is a norm one vector in E and cho ose a vector y in Y of norm so

that y y is equal to an approximation would be OK but is not needed b ecause Y is

reexive Set x F y so that jjxjj Then

jS y x j jS y x y Fxj jy Sx Gxy j

jjy jj jjSx Gxjj jjxjj

C C

so that S C as desired Since jjxjj this gives jjS y jj

Prop osition Supp ose that X is separable as L do es not contain a complemented

subspace isomorphic to and is isomorphic to its hyp erplanes If X is uniformly home

omorphic to Y then Y is isomorphic either to X or to X

Pro of The main part of the pro of is devoted to proving the following

Claim X is isomorphic to Y

Assuming the Claim we complete the pro of of Prop osition as follows If Y contains

a complemented subspace isomorphic to then Y is isomorphic to Y and so we are

done so assume otherwise By a slight abuse of notation write X Y W with

W isomorphic to Since X do es not contain a complemented subspace isomorphic to

every op erator from X into is strictly singular that is not an isomorphism on any

innite dimensional subspace of Y Hence by the result of EdelsteinWo jtaszczyk EW

by applying a space automorphism to X we can assume without loss of generality

that Y Y Y W W W with Y W subspaces of X and Y W subspaces of

X X X X

But Y must b e nite dimensional since Y do es not contain a complemented subspace

isomorphic to Similarly W is nite dimensional since since X do es not contain a

complemented subspace isomorphic to Thus Y is isomorphic to X plus or minus a

nite dimensional space which is isomorphic to X

We turn to the pro of of the Claim

As mentioned earlier the pro of of Lemma shows that Y Y is Lipschitz

equivalent to X X Let f E g be an increasing sequence of nite dimensional

subspaces of X whose union is dense in X and using Prop osition get for each n a

T norm one isomorphism T from Y onto a C complemented subspace of X with

C T C The ultrapro duct T of the isomorphisms T is an isomorphism from

the ultrap ower Y onto a complemented subspace of the ultrap ower X T y y

U U

T y T y Since X is sup erreexive X X and the adjoint T is dened

U U

for fx g a b ounded sequence in X by T x x T x T x Identify X

with fx x x X g evidently T is an isomorphism By Prop osition a

is isomorphically just X direct summed with a nonseparable Hilb ert space Thus X

by coming down to appropriate separable subspaces of the ultrap owers we get that there

is a pro jection P from X onto a subspace isomorphic to Y for which P is an

isomorphism This implies that the pro jection from X onto is an isomorphism on

the subspace RP of X in particular RP is isomorphic to a Hilb ert space

But RP is isomorphic to the dual of the null space of P so the null space of P is also

isomorphic to a Hilb ert space

Remark The last two hyp otheses on X were not used in the pro of of the Claim

Remark The pro of of Prop osition yields that if X is a separable as L space

and every innite dimensional subspace of X contains a complemented subspace which is

isomorphic to then X is determined by its uniform structure The simplest examples

of such spaces which are dierent from are sums of with p and k

n n

suciently quickly

Prop osition Let p p p or p p p and

n n

for k n assume that X is a separable as L has an unconditional basis which

k p

has an upp er pestimate for some p or a lower r estimate for some r is isomorphic

to its hyp erplanes and X do es not contain a subspace isomorphic to for any s Set

X If X is uniformly homeomorphic to a Banach space Y then Y is isomorphic X

to X for some subset F of f ng

k F

Pro of As usual assume n From the pro of of Prop osition we know that

is Lipschitz equivalent to L and that X L Y Y Y with Y as L

p p k p

L Y L The main part of the pro of is devoted to proving the following

p p

Claim There exists a quotient W of L L so that Y W is isomorphic to

p p

X L L

p p

Assuming the Claim we complete the pro of as follows Every op erator from X

to L L where q is the conjugate index to p is strictly singular by

q q

k k

KP Ald KM so the EdelsteinWo jtaszczyk theorem EW LT p says that by

applying a space automorphism to X L L wemay assume without loss

p p

of generality that Y Y Y W W W with Y W subspaces of X and

X L X L X X

Y W subspaces of L L As in the pro of of Theorem Theorem

p p

L L

and old results from the linear theory imply that Y is isomorphic to

p p p p

or is nite dimensional in which case we can assume the dimension is zero b ecause X is

isomorphic to its hyp erplanes So the pro of is complete once we observethatW must b e

nite dimensional But W embeds into L L and hence by KP Ald

q q

KM would contain a subspace isomorphic to for some s if it were innite dimensional

embeds into X This would contradict the hyp otheses on X since W

We turn to the pro of of the claim which is only slightly more complicated than the

pro of of the Claim in Prop osition

Since X X L L is Lipschitz equivalent to Y Y L

p p p

L we use Lemma just as in the pro of of Prop osition to get go o d isomorphisms

T from Y onto wellcomplemented subspaces of X so that T is a go o d isomorphism

where fE g is an increasing sequence of nite dimensional on a subspace E of Y

n n

subspaces of Y whose union is dense in Y Taking ultrap owers we get an isomorphism

T from an ultrap ower Y of Y onto a complemented subspace of X so that T is an

U U

isomorphism on X But we know that Y resp ectively X is just Y resp ectively X

U U

direct summed with the direct sum of a L space with a L space Therefore by coming

p p

down to appropriate separable subspaces of the ultrap owers and if necessary enlarging

the L spaces we conclude that there is a pro jection P from X L L onto

p p p

a subspace isomorphic to Y L L for which P is an isomorphism This

p p

is implies that the natural pro jection from X L L onto L L

q q q q

an isomorphism on the subspace RP in particular RP is isomorphic to a subspace

of L L But RP is isomorphic to the dual of the null space of P so

q q

the null space of P is isomorphic to a quotient of L L

p p

Uniform Homeomorphs of c

In this section we prove

Theorem Let X be a Banach space which has C as a quotient space Then X

is not uniformly homeomorphic to c

In particular c is not uniformly homeomorphic to C or any other C K space

except those which are isomorphic to c so Theorem solves the problem mentioned in

Aha whether c and C are Lipschitz equivalent In fact an immediate application

of Theorem and results of Alspach Zippin and Benyamini is

Corollary Let X be a complemented subspace of a C K space If X is uniformly

homeomorphic to c then X is isomorphic to c

Pro of By Als or see AB since X do es not have a quotient isomorphic to C

the Szlenk index of X is nite for each But in Ben pro of of Theorem it was

shown using among other things a small variation on a lemma in Zip that then X

must be isomorphic to a quotient of c hence X is isomorphic to c by JZ

Although it is known HM that a uniform homeomorph of c is a L space it is op en

whether it must be isomorphic to a complemented subspace of a C K space

Pro of of Theorem Denote by K the space and let K K be the

derived sets of K Let R be the restriction mapping from C K onto C K

Assume that there is a uniform homeomorphism U from c onto X with inverse V

and that there is a quotient mapping Q from X onto C K Without loss of generalitywe

assume that Q maps the op en unit ball of X onto the op en unit ball of C K

For each n let s be the limiting Lipschitz constant of the map S R QU So

n n n

s is the smallest constant so that for each there is a d d n so that if y z are

in c with ky z kd then k S y S z k s ky z k The sequence fs g is

n n n n

decreasing and tends to a limit s which is easily seen to b e p ositive so we can assume

without loss of generality that s

Fix small and take n so that s We can nd a pair x y of vectors

in c with kx y k as large as we please so that kx y k kS x S y k

n n

and since kx y k is large of necessity kS x S y k kx y k By making

n n

translations we can assume that y x and U x Ux so we have

kxk kS xkkS xk kxk

n n

Moreover since kxk can b e taken arbitrarily large we can assume in view of Lemma

that

S Midx x MidS x S x

n n n

Cho ose N so that the magnitude of the k th co ordinate of x is less than for kN

and let Y b e the nite co dimensional subspace of c consisting of those vectors whose rst

N co ordinates are zero so that

kxkBallY Mid x x

Now take t in K with jS xt j kS xk kxk The p oint t is

n n

a limit p ointofK so jS xtj kxk for innitely many p oints t in K say

for t B

Let W be the subspace of C K of functions which vanish on B Recalling that

kS xk kxk we observe that

Mid S x S x W kxk Ball C K

n n

Setting X R Q W we have from and that

U kxk Ball Y X kxk Ball X

which contradicts the Gorelik Principle since kxk is arbitrarily large and is arbitrarily

small

Uniform homeomorphisms between balls in close spaces

S Mazur Maz proved that the unit balls of any two p spaces are

uniformly homeomorphic This fact was extended by E Odell and T Schlumprecht OS

as a to ol in solving the distortion problem to anytwo spaces with unconditional bases and

nontrivial cotyp e It follows that anytwo balls in such spaces are uniformly homeomorphic

However the mo dulus of continuity of the uniform homeomorphism or its inverse generally

gets worse as the radius of the balls increases In Corollary we show that if the two

spaces are close to a common space then large balls are uniformly homeomorphic with

good mo dulus where large dep ends on p and how close the two spaces are to

We b egin by recalling the denition of the generalized Mazur map of Odell and

n n

Schlumprecht Let b fb g IR with kbk b and let X be a Banach

i i

space with a unconditional basis fe g X with x let For x x e

i i i i

G b x x G b sup G b x and F b F b the unique x

X X X X X

kxk

with the same supp ort as b for which the sup in the denition of G b is attained

P P

p p

Recall that for x j x e and p we denote jx jx j e and

i i i i

i i

that the unconditional basis fe g is said to be pconvex resp ectively q concave with

P P P

N N N

n q q n p p n p p

jx j k kx k resp ectively k jx j k C constant C if k

n n n

n q q n N

C of vectors in X See LT sec kx k for all nite sequences fx g

tion d for a discussion of pconvexity and q concavity

Odell and Schlumprecht proved that if X is q concave with constant one for some

q then F extends naturally ie homogeneously to the p ositive quadrant in

then extending to the other quadrants so that F will commute with changes of signs and

nally bycontin uityfrom to to a uniform homeomorphism of the unit balls of and

X Moreover the mo dulus of continuityofF and F dep end only on q If X and Y both

X Y

have unconditional basis and nontrivial cotyp e we shall denote F F F

XY Y

The following lemma follows from a pro of of Lozanovskiis theorem Loz see for example

TJ Lemma but note that misleading notation should be corrected

Lemma Let X be a Banach space with a unconditional basis fe g let b be a

nitely supp orted vector in the unit sphere of with nonnegative co ordinates and set

e is a norming functional of x Here fe g denotes the x F b Then f

i i i

biorthogonal basis to fe g and is interpreted as

If Y and Z are two spaces with unconditional bases or more generally Banach

lattices of functions and we denote by X Y Z the space of all sequences x

for which jxj can b e written as jxj y z with y Y z Z the op erations are dened

p ointwise with the norm

kxk inf fky k kz k jxj y z g

By a theorem of Calderon Cal Y Z is the interp olation space in the complex metho d

between X and Y with parameter commonly denoted byY Z see for example

BL The following lemma relates the functions F F and F

X Y Z

Lemma Let b have nite supp ort with kbk Then for all spaces Y Z with

unconditional bases

bF b F b F

Y Z

Pro of

Y Y

G b sup y z

Y Z

i i

ky k kz k

Y Y

b b

i i

sup y sup z

i i

y Y

kz k ky k

sup ky k y G b

Y Z

y Y

G b G b

Let y F b z F b and x y z Then kxk and

Y Z

Y Y

G bG b G b z G b x y

Y Z Y Z

i i

Consequently kxk and F b x

Y Z

Recall that the pconvexication of a space X with a unconditional basis or a

general Banach lattice of functions is the space

p p p

kjxj k g X fx x e kxk

p i i

is a norm whenever p If p X is sometimes called the The expression kk

pconcavication of X It is still a Banach space if the original X is pconvex with

constantone See LT section d for more information on these op erations Note that

X X

b for all spaces X with Corollary If b is as in Lemma then F b F

unconditional basis

Pro of Since b has nite supp ort it is enough to prove the corollary when X is nite

dimensional This case follows from Lemma and the the observation that for b a strictly

p ositive vector in F b

For p p denotes the conjugate index to p

Lemma If X is pconvex and r p concave both with constant one then X

Y with Y pconvex and pconcave b oth with constant one and

Pro of Even if we just assume that X is a space of functions on the natural num

bers the convexity and concavity assumptions guarantee that the unit vector basis is a

unconditional basis for X This implies that the general case follows from the case where

X is nite dimensional which we assume in the sequel to avoid irrelevant top ological and

duality problems that arise in the innite dimensional setting

Assume rst that p Since for any space W with unconditional basis W

r r r r

so X X and X is concave with X X W

constant one

p p

p p p

For a general p X X Z with and Z

concave with constant one This can be rewritten as

p p p

X Z Z

We are now ready to investigate the mo dulus of continuityofF for spaces X close

Prop osition Let X be pconvex and r concave both with constant one and with

r p Then the mo dulus of uniform continuity of F on the unit ball of is

X p

b ounded by for some function dep ending only on p

Pro of Let u v be two nonnegativevectors of norm one in It follows from Lemma

that with the notation of Lemma

kF uF v k ku F u v F v k

X X

p p

Y Y

X X

p p

v u F ju v jF v u F F w

Y Y Y Y

p p p p

X X

where w fw g is dened by w u whenever the ith comp onent of F uis smaller

i i i Y

otherwise than that of F v and w v

i Y i

Now the second term in is dominated by

v k ku v k kF u F v k ku v k u F kju v j F

Y Y

p p p p

Y Y

p p

while the rst term is dominated by

F u F v

Y Y

p p

F u F v

Y Y

p p

F u F v F u F v

Y Y Y Y

p p p p

F u F v

Y Y

p p

F u F v F u F v

Y Y Y Y

p p p p

F u F v

Y Y

p p

By Corollary F uF u F u Thus by Lemma

Y Y

F u

and are norm one functionals on Y and the rst term in is dominated

F v

u v

F u F v F u F v

Y Y Y Y

p p p p

F u F v

Y Y

p p

F v F u F v F u

Y Y Y Y

p p p p

Y Y

F u F v

Y Y

p p

Combining this with and and using the Odell and Schlumprecht result we get

that

ku v k kF u F v k ku v k

p X X

p p p

ku v k ku v k

for some function dep ending only on p

Next weinvestigate the mo dulus of uniform continuityofF for spaces close to

X p

Prop osition Let X be pconvex and r concave both with constant one and with

F on the unit ball of r p Then the mo dulus of uniform continuity of F

X is bounded by for some function dep ending only on p

Pro of Write X Y with Y concave and Note that for x X

x F x x f where f is the norming with kxk and x F

X x x

functional of x Let x x be two norm one vectors in the p ositive cone of X Writing

x y f f we get from Lemma that y has norm one in Y and f norms

p y

y y

it Similarly let y be the norm one vector in Y corresp onding to x Let z z be dened

xi if f i f i

y y

z i

x i otherwise

and

x i if z ixi

z i

xi otherwise

Then

f f

x x

p p p p p

kx f x f k kjx x j k

x x p

f f f

x z x

p p

p p p p p p p

k kz j j k kjx x j

p p

z x

p p

I II

Now

f f

x x

p p

I kjx x j k gk k maxf kx x k

p p

p X X

by Lemma and

f f

x x

II kz j jk

kz jf f jk

p p

kz f jf f jk

y y

k maxfy y gjf f jk

y y

p p

kyf y f k kjy y j maxff f gk

y y y y

p p

ky y k ky y k

where is the uniformity function of F on the unit ball of Y Finally ky y k

kx x k where is the uniformity function of F Note that and can be

on p b ounded by functions dep ending only

Corollary Let p Then there is a function IR IR such that for

all R there exists an such that if X is pconvex and p concave both

with constant one then there is a map from the ball of radius R in onto the ball of

radius R in X with the mo dulus of continuity of and bounded by

Pro of Let be the natural homogeneous extension of F and use Prop ositions

and

Uniform homeomorphisms between X and X

In Corollary we give a useful sucient condition for a space X to be uniformly

homeomorphic to X This is used in Corollary to prove that T is uniformly

homeomorphic to T

Theorem Let p Let X X be a unconditional sum of spaces

with unconditional bases such that there are norm preserving homogeneous homeomor

onto

X with the prop ert phisms f y that for some sequence R the sequence

n n p n

f f n is equiuniformly continuous B R or just B R if

n Y

jB R

n jB R

there is no ambiguity denotes the closed ball of radius R in Y Then X is uniformly

homeomorphic to X

For X and p p this theorem was proved by M Rib e Rib for p

p q n

and extended by I Aharoni and J Lindenstrauss AL for p The pro of we sketch

closely follows a simplication of the pro of in AL given by Y Benyamini in his nice

exp osition Ben

In what follows we shall frequently use the spaces X X X and other

p n p n n

spaces of similar nature Since the isometric nature of the spaces we deal with is imp ortant

in the pro of we emphasize that by eg X X we mean the space of all triples

p n n

u x x u x X i n n with norm

n n p i i

p p

ku x x k kuk kx x k

n n n n

swemay assume that for each n X Y Z and there are norm By coupling the X

n n n n

onto onto

preserving homogeneous homeomorphisms g Y and h X Z such that the

n p n n n n

n is equiuniformly continuous h h sequence g g

n n

jB R jB R

n n

onto

Dene now I X by X

n n p n

ku xk

I u x I u y z g uh x

n n n n

kg uh xk

n n

I n s are norm preserving homogeneous and I Clearly the I

n n

jB S

n jB S

form an equiuniformly continuous family for some sequence S Since in order

we may assume that s to prove the theorem we may pass to a subsequence of the X

The following prop osition imitates Prop osition in Ben

Prop osition There is a family of homeomorphisms fF tg so that

n n

i For n and t t for n F maps X X

t p n n

onto X X

n n

ii F is homogeneous and norm preserving t

iii F x y z I x y z n

iv F x y z y I x z n

n n

Then the families fG ag v Denote for t G F

t t t

t jB

X X

p n

ag are equiuniformly continuous in b oth t and a fG

Before we sketch the pro of of the prop osition let us apply it to give the

onto

Pro of of Theorem Dene an homeomorphism f X X in the following

n n

way For a u x x X with kak kak for n let

f a x x x F u x x x

n n n n

kak

and

g a f akf ak

n n t t t

For x x X X and t denote u y y F x x

n n n n n n

n n t

Then it is easily checked that for x x x

kxk kxk

kxk

x x y y x u x

n n

g x

kxk kxk

kxk

x k y ku x x x y

n n

The unconditionality of the decomp osition of X to X and Prop osition now

show that g and g are uniformly continuous

Pro of of Prop osition Consider the maps

X X

n p n n p p p p p

and

X X

n n n p p p

given by

ku x y k

u x y u f xf y

n n n

ku f xf y k

n n

and

kx y k

x y f xf y

n n n

kf xf y k

n n

Note that these are norm preserving homogeneous maps and the sequence

is equiuniformly continuous Assume we can

n n

jB jB

P P

prove the prop osition in the sp ecial case where X for all n and X X

n p n p n

n n

and call the resulting maps H instead of F Then for n and t

t t

dene F H Clearly F satisfy the conclusions of the prop osition

t t n t

The construction of H is given in steps I IIV of the pro of of Prop osition of Ben

The only additional thing to notice is that all the maps there are homogeneous

Corollary Let p and let X X b e an unconditional sum of spaces X

n n

with unconditional bases which are pconvex with constant C and p concave with

constant C for some p ositive sequence and some C Then X is

n p

uniformly homeomorphic to X

Pro of We may assume that the unconditionality constant of the sum X is By

a theorem from FJ each X is isomorphic with constant indep endent of n to a space

with a unconditional basis which is pconvex and p concave with constants one

Applying Corollary stated after Prop osition and the ab ove isomorphisms we get

the maps f X as in the statement of Theorem Now apply Theorem

n p n

We shall denote by T or just T when p the closed span of a certain subsequence

of the unit vector basis for the pconvexication of the mo died Tsirelson space In fact

it is known see CO CS that this space is up to an equivalent renorming the space

which is usually denoted by T that is the pconvexication of the dual to Tsirelsons

original space but we avoid using this for the convenience of the reader

The construction of T aswell as an elementary pro of that T is reexive is contained

space T is the completion of c under the norm dened implicitly by the in Joh The

formula

jjxjj max sup f jjE xjjg jjxjj

j c

where the sup is over all sequences of disjoint nite sets E of p ositiveintegers with j E

j j

for each j n Such sequences are called allowable The choice of the growth

n n is simply one of convenience the results of CJT CO imply that the space

is the same if e g n instead of n is used

In order to show that for all p T the pconvexication of T is uniformly

homeomorphic to T we need that for any q the span of a suciently thin

subsequence of the unit vector basis fe g for T has q concavity constant indep endent

of q In fact the subsequence can be just a tail of the basis The pro of uses an

elementary lemma

Lemma Supp ose q andm m with m m

k k

Then for all sequences a a

X X

sup a a

j m j

Pro of Assume without loss of generality that the left side is one Then

m m

k k

X X X X X

q q

a a a a

j j

j j m j m

k k

m m

k k

X X X X

q q q

A A

sup a sup a a a m m

j j k k

m m

k k

j m j m

k k

so the desired conclusion follows

the span of Corollary There is a constant M so that for any q

fe g in T has q concavity constant less than M where N q

nN q

Pro of A general result of Maureys Mau see LT pro of of Theorem f shows

that the q concavity constant of a Banach lattice is at most a constant multiple of the

lower s constant of the lattice where q s ie s Trace the constants in

the pro of in LT for the dual statement in the sp ecial case r p

Thus it is enough to prove the corollary with q concavity constant replaced with

lower q constant and N q replaced with M q N s where as ab ove s

so that M q

So let q be xed and supp ose that x x are disjoint vectors in T c

which are supp orted on M q We can assume that jjx jj jjx jj jjx jj If for

some m F is a set of at most m indices so that for each j in F x is supp orted in

X X X

m then by the denition of T jj x jj jj x jj jjx jj

j j j

j F j F

Dene m and m m M q for k Since for k m m

k k k k

m and since at most m of the x s i m fail to be supp orted on

k k i k

m we get

X X

sup jjx jj jj x jj

j j

j m j

satises the condition of the lemma with andthus the lemma The sequence fm g

has a lower q estimate with constant yields that that fe g

nM q

If X has a unconditional basis and is q concave with constant M then its p

convexication is pq concave with constant M Thus the next prop osition is an immediate

consequence of Corollary

Prop osition There is a constant M so that for any p r the span

r p

p p

in T has r concavity constant less than M where N p r of fe g

nN pr

As an immediate consequence of Prop osition and Corollary we get

p p

Prop osition For p T is uniformly homeomorphic to T

Finallyby combining Prop osition and Prop osition we get

Theorem Let p p p or p p p Set

n n

and only if Y X T If X is uniformly homeomorphic to a Banach space Y if

is isomorphic to X for some subset F of f ng Consequently X is

k F

uniformly homeomorphic to exactly mutually nonisomorphic Banach spaces

Pro of X is uniformly homeomorphic to spaces Y of the desired form by Prop osition

As mentioned in section the obvious fact that every ntuple even n tuple of

disjoint unit vectors in T whose rst n co ordinates are zero is equivalent to the unit

p p

vector basis of implies that T is as L Of course do es not embed into T for

p p r

any r the general case is immediate from the sp ecial case p proved in Joh So

it remains to observe that T is isomorphic to its hyp erplanes In fact the right shift is

an isomorphism on T hence on T for all p This is included in the rst result

in CJT for the classical Tsirelson space To avoid using the fact that this space is the

same space up to an equivalent renorming that we denote by T the reader will have to

check directly that the right shift is an isomorphism on T So Prop osition applies to

show that any uniform homeomorph of X must be of the form for Y in the statement of

Theorem Finally emb eds into X i and only if j is in F so all the

p p

k F

j k

Y s of the mentioned form are isomorphically distinct

The nonseparable case

Some of the uniqueness theorems proved in earlier sections have nonseparable ana

logues In particular there is a nonseparable version of Theorem

Theorem If X is a Banach space which is uniformly homeomorphic to

pthen X is isomorphic to

Pro of A backandforth argument yields that each separable subspace of X is contained

in another subspace of X which is uniformly homeomorphic to hence isomorphic to

is a so that each separable subspace of X is It is then easy to see that there

contained in another subspace of X which is isomorphic to So X is L and hence is

p p

isomorphic to a complemented subspace of L for some measure LPe and of course

do es not embed into X The conclusion then follows from the next lemma which is

implicit in JO although not stated there

Lemma Let X b e an innite dimensional complemented subspace of L for some

measure p and supp ose that no subspace of X is isomorphic to Then X

is isomorphic to where is the density character of X

Pro of Since every Hilb ertian subspace of L is complemented for p KP by

dualit ywe can assume that p By Theorem of JO X emb eds into and

as noted in the pro of of that theorem X is isomorphic to a space of the form X

with each space X separable and without loss of generality innite dimensional But

then the X s are uniformly isomorphic to uniformly complemented subspaces of

hence by JO and JZ or see Joh are uniformly isomorphic to

The isomorphic classication of C K for compact metric spaces K BP implies that

c is isomorphic to C K if and only if the th derived set K of K is empty In DGZ

it is proved that for any compact Hausdor space K if K is empty then C K is

uniformly homeomorphic even Lipschitz equivalent to c with the densitycharacter

of C K Part of the interest of this result is that in contrast to the separable case

in the nonseparable setting this can happ en with C K not isomorphic to c AL

Gilles Go defroy p ointed out to us that Corollary and previously known results yield a

converse to the mentioned theorem from DGZ and kindly suggested that we include the

result in this pap er

Theorem G Go defroy Supp ose that K is a compact Hausdor space and C K

is uniformly homeomorphic to c Then K is empty

Pro of The usual backandforth argumentshows that every separable subspace of C K

is contained in a subalgebra which is uniformly homeomorphic to c hence isomorphic

to c by Corollary This implies that K is scattered by PS or see Sem Theorem

that is whenever K is nonempty K is a prop er subset of K But if

K is scattered and K is nonempty then is a continuous image of K by a result of

Bakers Bak But then C is isomorphic to a subspace of C K and hence by the rst

sentence of this pro of C is isomorphic to a subspace of c which of course is false

Remark Theorem and the mentioned result in DGZ yield that if c is uniformly

homeomorphic to a C K space X then c is Lipschitz equivalent to X We do not

know whether this holds for a general space X even in the separable case

Spaces determined by their nite dimensional subspaces

We b egin with some notions and denitions

A paving of a separable Banach space X is a sequence E E of nite

say that the two Banach dimensional subspaces of X whose union is dense in X We

spaces X and Y have a common paving if there exist pavings fE g of X and fF g

n n

of Y so that the isomorphism constants between E and F are uniformly b ounded

n n

Example warning c and c have a common paving since both are paved by

Denition A separable Banach space X is said to be determined by its pavings

provided that X is isomorphic to every separable Banach space which has a common

paving with X

Denition X and Y are said to have the same nite dimensional subspaces if each

is nitely crudely representable in the other A separable Banach space X is said to be

determined by its nite dimensional subspaces provided that X is isomorphic to every

separable Banach space which has the same nite dimensional subspaces as X

Obviouslyevery Banach space which is determined by its nite dimensional subspaces

is determined by its pavings It is easy to see and well known that is determined by its

pavings In this section we shall show among other things that anyspace determined by

its pavings must b e close to but that there are spaces not isomorphic to which are

determined by their pavings On the other hand we makethe following conjecture

Conjecture Every separable innite dimensional Banach space which is deter

mined by its nite dimensional subspaces is isomorphic to

We start the discussion of the concepts intro duced ab ove by proving

Prop osition If X is determined by its pavings then for every X has typ e

and cotyp e

Pro of Otherwise is nitely representable in X for some p by the Krivine

MaureyPisier theorem Kri MP If fF g is a paving of any subspace of L then

n p

using the fact that is nitely representable in every nite co dimensional subspace of X

it is easy to see that X has a paving of the form fE F g with fE g itself a

n p n n

n n

paving of X This means that X Y has a common paving with X for every subspace Y

of L which implies that every subspace of L is isomorphic to a complemented subspace

p p

of X But in JS it was proved that for p no separable Banach space is

complementably universal for subspaces of Probably the same result is true when

p although wecan get by with the weaker fact that there is no separable Banach

complementably universal for subspaces of L for p in this range This space which is

weaker fact follows from the pro of of what is called in JS the Basic Result and two facts

which can be found in LT emb eds into L when p r and for

r p

each r has a subspace which fails the compact approximation prop erty

Following Joh wesay that X has prop erty H m n K provided that there exists an

mco dimensional subspace of X all of whose ndimensional subspaces are K isomorphic

to So a Banach space X is asymptotically Hilb ertian in Pisiers terminology Pis or as

L in our terminology if and only if there exists a K so that for every n X has H m n K

for some m mn

Lemma Given m k K and a separable Banach space X the following are equiva

lent

i X has prop erty H m k K

ii There isapaving fE g for X such that for every n E has prop erty H m k K

n n

iii Every nite dimensional subspace of X has prop erty H m k K

iv Every subspace of X has prop erty H m k K

v Every space which is nitely representable in X has prop erty H m k K

Pro of The implications viiviiiii are all obvious To prove iiv it

is enough to observe that ultrapro ducts of fE g have prop erty H m k K since any

separable space nitely representable in X embeds isometrically into an ultrapro duct of

any paving of X

Notice that the preceding lemma implies that asymptotically Hilb ertian is a lo cal

prop erty

The next prop osition is suggested by a result and pro of due to P Casazza CS The

orem Ae

Prop osition Supp ose that X is asymptotically Hilb ertian fE g is a paving for

X and E is an ultrapro duct of fE g Then E is isomorphic to X H for some

nonseparable Hilb ert space H

Pro of The subspace of E consisting of all Cauchy sequences fx g in X with x in

n n

E is easily seen to b e isometrically isomorphic to X E is the direct sum of this subspace

and the subspace of all fx g in E for which fx g weakly converges along U to

n n n

n n

zero this latter subspace is isomorphic to a Hilb ert space

Corollary If X is asymptotically Hilb ertian and Y has a common paving with X

then X is isomorphic to Y Consequently if also every innite dimensional

subspace of X contains a complemented subspace which is isomorphic to in particular

if X is an sum of nite dimensional spaces then X is determined by its pavings

Pro of If fE g is a paving of X which is uniformly equivalenttoapaving of Y and E

is an ultrapro duct of fE g thenwe get from Prop osition that X H is isomorphic

to Y H for some nonseparable Hilb ert space H from which the rst conclusion follows

easily For the consequently statement we see that X is isomorphic to X since X

contains a complemented copy of In fact every subspace of X in particular Y

contains a complemented copy of so also Y is isomorphic to Y

This corollary implies that there are spaces not isomorphic to which are determined

by their pavings In order to continue our investigation we intro duce a further notion

We say that X is nitely homogeneous provided there is a K so that X is nitely

with constant K in every nite co dimensional subspace of itself crudely representable

First we observe that nite homogeneity is determined by nite dimensional subspaces

Prop osition X is nitely homogeneous if and only if there is a K so that for every

nite dimensional subspace E of X there is a space Z Z E which has a monotonely

subsymmetric and unconditional Schauder decomp osition into spaces isometric to E and

which is nitely crudely representable in X with constant K

Pro of The forward direction involves just a combination of mo dern renements of the

Mazur technique for constructing basic sequences and the BrunelSucheston construction

see eg Pis Chpt The reverse direction is even easier if F is a nite subset of

X and E E E is a go o d Schauder decomp osition of some subspace of X into

N copies of E with N large relative to the cardinality of F then the action of F on two

dierent copies of E is essentially the same Now pass to dierences

Prop osition If the separable Banach space X is nitely homogeneous and X has

the same nite dimensional subspaces as the separable Banach space Y then X and Y

have a common paving

Pro of The space Y is also nitely homogeneous by Prop osition Now it is a simple

exercise to show that if fE g is a paving for X and fF g is a paving for Y then

n n

there is a subsequence of fE F g which is uniformly equivalenttopavings for b oth

n n

X and Y

The next prop osition crystalizes some of what has already b een used

Prop osition If there is a K so that the separable Banachspace Y is nitely crudely

representable with constant K in every nite co dimensional subspace of the separable

Banach space X then X Y and X have a common paving

Corollary If X is nitely homogeneous and determined by its pavings then every

separable Banach space which is nitely representable in X is isomorphic to a comple

mented subspace of X Moreover X has an unconditional Schauder decomp osition such

that the summands form a sequence which is dense in the set of all nite dimensional

subspaces of X and hence every separable Banach space which is nitely representable in

X has the b ounded approximation prop erty Finallyif X has GLlust resp ectively GL

then so do es every subspace

See TJ x for the denitions of GLlust and GL prop erties

Using James characterization of reexivitysup erreexivity Jam and the limiting

technique of BrunelSucheston again as in Pis Chpt we get from Prop osition

Prop osition If X is reexive and is determined by its pavings then X is sup er

reexive

We do not know whether there exist nonreexive spaces which are determined by their

pa vings

Prop osition implies that if X is nitely homogeneous then X X is nitely crudely

representable in X The prop erties are not equivalent however since every Tsirelson typ e

space is isomorphic to its square The weaker condition gives similar conclusions to those

in Corollary for spaces which are determined by their nite dimensional subspaces

Prop osition If X is determined by its nite dimensional subspaces and X X is

nitely crudely representable in X then X is isomorphic to X Y for every space Y which

is nitely representable in X Hence if X has GLlust then so do es every subspace

yield Corollary Prop osition and KT

Prop osition Let k appropriately set E spanfe g in T and let

n n j

j k

X be the sum of fE g Then X is determined by its pavings but X is not determined

by its nite dimensional subspaces

Pro of Corollary says that X is determined byitspavings no matter how fk g is

k k

n n

dened Next note that E E is uniformly isomorphic to E again no matter

n n n

is dened This follows from the equivalence of fe g and fe g how fk g

n n n

with fe g and the fact that the last half of a nite string of e s is equivalent to an

n j

orthonormal sequence This yields that X is isomorphic to its square Thus by Prop osition

if X were determined by its nite dimensional subspaces then every subspace of X

would have GLlust which easily implies that there is a K so that every nite dimensional

subspace of X has GLlust constant less than K But by KT T has a subspace

has nite dimensional subspaces with which fails GLlust so for each N span fe g

j N

arbitrarily large GLlust constant Consequently having dened k X will have a

n n

subspace with GLlust constant larger than n if k is suciently large

Prop osition Supp ose that p k and let X be the sum of If X

n n

is determined by its nite dimensional subspaces then X is isomorphic to

Pro of Assume without loss of generality that all the p s are smaller than Now if

k k k

BDGJN JSc which yields p then is uniformly isomorphic to

p p

k k

Now for p has that X is isomorphic to X W where W is the sum of

p p

k k

a subspace whose GLconstant is of the same order as the distance of to Thus if X

is not isomorphic to a Hilb ert space then W has a subspace V which do es not have the

GLprop erty But X W is nitely crudely representable in X Thus X is isomorphic to

a Hilb ert space if X is determined by its nite dimensional subspaces

Problems

Say that a Banach space is determined by its uniform structure if it is isomorphic to

every Banach space to which it is uniformly equivalent It seems that most natural or

classical separable spaces are determined by their uniform structure but at present the

to ols to establish this fact are limited Here are some sp ecic problems the rst three

b eing the most interesting

a Is L p p determined by its uniform structure

by their uniform structure For c partial results are given b Are c and determined

in section For wehave practically no information b eyond Rib es theorem Rib

c Are separable C K spaces and L determined by their uniform structure In

particular if K and K are compact metric spaces for which C K is uniformly

homeomorphic to C K must C K be isomorphic to C K

More generally one can ask

d Is every separable L space p determined by its uniform structure For the

p it is even op en whether a uniform homeomorph of a L space must itself case

be a L space For p p the simplest unknown case is the space

In connection with the results of section it is natural to ask

e Is for p r determined by its uniform structure

p r

A general question related to uniform homeomorphisms is whether the uniform struc

ture of a Banach space is determined by the structure of its discrete subsets Supp ose that

A is an aseparated bnet for a Banach space X with a b that is every two

p oints in A are of distance apart larger than a and every p ointinX is of distance less than

b from some p ointof A If T is a uniform homeomorphism from X onto Y then T Aisa

cseparated dnet for Y for some cd and since T and T are Lipschitz for

large distances the restriction of T to A is biLipschitz We ask whether the converse is

true

If there exist bnets A and B in X and Y resp ectively for some b so that

there is a biLipschitz mapping from A onto B then must X and Y be uniformly

homeomorphic

An armative answer to this problem would of course imply that ultrap owers of X

and Y are Lipschitz equivalent this much at least is true

Prop osition Supp ose that A and B are bnets in X and Y resp ectively and there

is a biLipschitz mapping T from X onto Y Let U be a free ultralter on the natural

numbers Then X is Lipschitz equivalent to Y

U U

Pro of Dene a mapping T A B by T x T nx Then for each n T

n n n

and T have the same Lipschitz constants as T and T resp ectively Given a b ounded

sequence x fx g in X we can select a sequence y fy g with y in A so that

n n n

n n

kx y k So in the ultrap ower X x y Since the T s are uniformly Lipschitz

n n n

if for x and y as ab ove we set T x fT y g then T is a well dened Lipschitz

n n

mapping from X to Y Doing the the analogous thing with the T s we see that T is

U U

in fact a biLipschitz mapping from X onto Y

U U

We recall here a well known problem Are any two separable Lipschitz equivalent

Banach spaces isomorphic

Besides uniformly continuous and Lipschitz maps there are several other interest

ing kinds of nonlinear maps between Banach spaces which app ear in the literature A

notion which arises naturally from the theory of quasiconformal maps and which was in

tro duced and studied in the context of general metric spaces by Tukia and Vaisala TV

that of quasisymmetric maps There are several variants of this notion but for our is

purp ose in the context of Banach spaces we can use the following denition A map

f from a subset G of a Banach space X into a Banach space Y is quasisymmetric if

there is a constant M so that whenever u v w are in G with ku v k ku w k then

M kf u f w k Vaisala asked the following For p q is kf u f v k

there a homeomorphism between and which is biquasisymmetric More generally

p q

one can ask if is determined by its quasisymmetric structure In this connection we

mention an unpublished observation of Vaisalaand the second author Supp ose that G is

an op en subset of a separable Banach space X and f is a quasisymmetric Lipschitz map

from G into a Banach space Y which has the RadonNikodym prop erty in particular Y

can be any separable conjugate space Then unless X is isomorphic to a subspace of Y

f must be a constant Indeed if T is the Gateaux derivative of f at some p oint then T

is either or an into isomorphism Hence if X is not isomorphic to a subspace of Y then

the Gateaux derivativeof f must b e whenever it exists The assumptions on X Y and

f guarantee that the Gateaux derivative exists almost everywhere eg in the sense that

the complementhas measure with resp ect to every nondegenerate Gaussian measure on

X see Ben By Fubinis theorem it follows that the restriction of f to a line x L has

derivative almost everywhere for almost all translates x L of L This implies that f is

constant

The assumption that Y has the RadonNikodym prop erty is essential since Aharoni

Aha proved that there is a Lipschitz emb edding of C into c Of course every

Lipschitz emb edding is quasisymmetric

Theorem gives examples of spaces whose uniform structure determine exactly

isomorphism classes for k From the construction of AL one easily gets

examples of spaces whose uniform structure determine isomorphism classes If is

y cardinal less than the continuum which is not a p ower of two in particular if or an

we do not knowhow to construct a space which determines exactly isomorphism

classes

If X is determined by its pavings as dened in section must X be determined by

its uniform structure More generally if two spaces are uniformly homeomorphic must

they have a common paving In the terminology of section they do have the same

nite dimensional subspaces by Rib es theorem Rib Notice that Corollary and the

second remark after Prop osition give examples of spaces which are determined both

by their pavings and by their uniform structure

References

Aha I Aharoni Every separable metric space is Lipschitz equivalent to a subset of c

Israel J Math

AL I Aharoni and J Lindenstrauss Uniform equivalence between Banach spaces Bull

Amer Math So c

AL I Aharoni and J Lindenstrauss An extension of a result of Rib e Israel J Math

nos

Ald D Aldous Subspaces of L via random measure Trans Amer Math So c

Als D E Alspach Quotients of C with separable dual Israel J Math no

AB D E Alspach and Y Benyamini C K quotients of separable L spaces Israel J

Math nos

Bak J W Baker Disp ersed images of top ological spaces and uncomplemented subspaces

of C X Pro c Amer Math So c

BDGJN G Bennett L E Dor V Go o dman W B Johnson and C M Newman On uncom

plemented subspaces of L p Israel J Math

Ben Y Benyamini An extension theorem for separable Banach spaces Israel J Math

Ben Y Benyamini The uniform classication of Banach spaces Longhorn Notes Texas

Functional Analysis Seminar Univ of Texas An electronic

version can be obtained via FTP from ftpmathokstateedu pubbanach

BL J Bergh and J Lofstrom Interp olation spaces An intro duction SpringerVerlag

Grundlehren Math Wissenschaften

Bou J Bourgain Remarks on the extensions of Lipschitz maps dened on discrete sets and

uniform homeomorphisms Lecture Notes in Math

BP C Bessaga and A Pelczynski Spaces of continuous functions IV Studia Math

Cal A Calderon Intermediate spaces and interp olation the complex metho d Studia

Math

CJT P G Casazza W B Johnson and L Tzafriri On Tsirelsons space Israel J Math

CO P G Casazza and E Odell Tsirelsons space and minimal subspaces Longhorn

Notes Texas Functional Analysis Seminar Univ of Texas

CS P G Casazza and T J Shura Tsirelsons space Lecture Notes in Math

SpringerVerlag

DGZ R Deville G Go defroy and V E Zizler The three space problem for smo oth parti

tions of unity and C K spaces Math Ann

Ede I S Edelstein On complemented subspaces and unconditional bases in l l Teor

Funkcii Functional Anal i Prilozen Russian

EW I S Edelstein and P Wo jtaszczyk On pro jections and unconditional bases in direct

sums of Banach spaces Studia Math

Enf P Eno On the nonexistence of uniform homeomorphisms b etween L spaces Ark

Mat

FJ T Figiel and W B Johnson A uniformly convex Banach space whichcontains no

Comp ositio Math

Gor E Gorelik The uniform nonequivalence of L and Israel J Math

p p

HM S Heinrich and P Mankiewicz Applications of ultrap owers to the uniform and Lips

chitz classication of Banach spaces Studia Math

Hei S Heinrich Ultrapro ducts in Banach space theory J fur Die Reine und Ange

wandte Math

Jam R C James Some selfdual prop erties of normed linear spaces Annals Math Stud

ies

Joh W B Johnson Op erators into L which factor through J London Math So c

p p

Joh W B Johnson A reexive Banach space which is not suciently Euclidean Studia

Math

Joh W B JohnsonBanach spaces all of whose subspaces have the approximation prop erty

Sp ecial Topics of Applied Mathematics NorthHolland

JO W B Johnson and E Odell Subspaces of L which embed into Comp ositio

p p

Math

n k

JSc W B Johnson and G Schechtman On the distance of subspaces of l to l Trans

p p

Amer Math So c

JS W B Johnson and A Szankowski Complementably universal Banachspaces Studia

Math

JZ W B Johnson and M Zippin On subspaces of quotients of G and G

Israel J Math nos and

Kad M I Kadec A pro of of the top ological equivalence of separable Banach spaces

Russian Funkcional Anal i Prilozen

KP M I Kadec and A Pelczynski Bases lacunary sequences and complemented sub

spaces in the spaces L Studia Math

KT R Komorowski and N TomczakJaegermann Banach spaces without lo cal uncondi

tional structure Israel J Math

Kri J L Krivine Sous espaces de dimension nite des espaces de Banachreticules Annals

Math

KM J L Krivine and B Maurey Espaces de Banach stables Israel J Math

Lin J Lindenstrauss On nonlinear pro jections in Banach spaces Mich J Math

Lin J Lindenstrauss On nonseparable reexive Banach spaces Bull Amer Math

So c

LPe J Lindenstrauss and A Pelczynski Absolutely summing op erators in L spaces and

their applications Studia Math

LP J Lindenstrauss and D Preiss On almost Frechet dierentiability of Lipschitz func

tions

LR J Lindenstrauss and H P Rosenthal The L spaces Israel J Math

LT J Lindenstrauss and L Tzafriri Classical Banach spaces I Sequence spaces Springer

Verlag

LT J Lindenstrauss and L Tzafriri Classical Banach spaces I I Function spaces Spring

erVerlag

Loz G Ya Lozanovskii On some Banach lattices Sib erian Math J

English translation

Mau B Maurey Theoremes de factorisation p our les operateurs lineaires avaleurs dans les

espaces L Asterisque

Mau B Maurey Typ e et cotyp e dans les espaces munis de structures lo cales incondition

nelles Seminaire MaureySchwartz Exp ose Ecole Polytechnique

Paris

Mau B Maurey Un theoreme de prolongement C R Acad Paris

MP B Maurey and G Pisier Series de variables aleatoires vectorielles indep endantes et

proprietes geometriques des espaces de Banach Studia Math

Maz S Mazur Une remarque sur lhomeomorphismie des champs fonctionnels Studia

Math

Mic E Michael Continuous selections I Annals Math no

OS E W Odell and T Schlumprecht The distortion problem Acta Math to app ear

PS A Pelczynski and Z Semadeni Spaces of continuous functions III Studia Math

Pis G Pisier The volume of convex bodies and Banach space geometry Cambridge

University Press

Rib M Rib e On uniformly homeomorphic normed spaces Ark Math

Rib M Rib e On uniformly homeomorphic normed spaces II Ark Math

Rib M Rib e Existence of separable uniformly homeomorphic non isomorphic Banach

spaces Israel J Math

Sem Z Semadeni Banach spaces of continuous functions Polish Scientic Publishers War

saw

TJ N TomczakJaegermann BanachMazur distances and nitedimensional op erator

ideals Pitman Monographs and Surveys in Pure and Applied Mathematics

Longman

Tor H Torunczyk Characterizing Hilb ert space top ology Fund Math

Tsi B S Tsirelson Not every Banach space contains an emb edding of or c Functional

Anal Appl translated from the Russian

TV P Tukia and J Vaisala Quasisymmetric emb eddings of metric spaces Ann Acad

Sci Fenn Ser A I Math

Woj PWo jtaszczyk On complemented subspaces and unconditional bases in l l Stu

p q

dia Math

Zip M Zippin The separable extension problem Israel J Math nos

W B Johnson J Lindenstrauss

Department of Mathematics Institute of Mathematics

Texas AM University The Hebrew University of Jerusalem

College Station TX USA Jerusalem Israel

email johnsonmathtamuedu email jorammathhujiacil

G Schechtman

Department of Theoretical Mathematics

The Weizmann Institute of Science

Rehovot Israel

email mtschechweizmannwei zma nn ac il