Lipschitz Quotients from Metric Trees and from Banach Spaces Containing
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Lipschitz quotients from metric trees and from Banach spaces containing William B Johnson Department of Mathematics Texas AM University Col lege Station TX USA Email johnsonmathtamuedu Joram Lindenstrauss Institute of Mathematics Hebrew University of Jerusalem Jerusalem Israel Email jorammathhujiacil David Preiss Department of Mathematics University Col lege London London Great Britain Email dpmathuclacukj and Gideon Schechtman Department of Mathematics Weizmann Institute of Science Rehovot Israel Email gideonwisdomweizmannacil INTRODUCTION ALipschitz map f between the metric spaces X and Y is called a Lips chitz quotient map if there is a C the smallest such C the coLipschitz constant is denoted coLipf while Lipf denotes the Lipschitz constant of f so that for every x X and r fB x r B f xrC X Y Thus Lipschitz quotient maps are surjective maps whichby denition have the prop erty ensured by the op en mapping theorem for surjective linear op erators b etween Banach spaces WB Johnson was supp orted in part by NSF DMS and the USIsrael Binational Science Foundation J Lindenstrauss and G Schechtman were supp orted in part by the USIsrael Binational Science Foundation and were participants in the NSF Workshop in Linear Analysis and ProbabilityatTexas AM University JOHNSON LINDENSTRAUSS PREISS AND SCHECHTMAN If there is a Lipschitz quotient mapping f from X onto Y with Lipf coLipf C wesaythat Y is a C Lipschitz quotientofX If Y is a C Lipschitz quotientof X for some C wesaythat Y is a Lipschitz quotient of X Lipschitz quotient maps were intro duced and studied in Among other results the following prop osition was proved in seealso The orem Proposition Let X be a superreexive Banach space and let the Banach space Y bea Lipschitz quotient of X Then Y is crudely nitely representable in X The conclusion of the Prop osition means that there is a constant so that for every nite dimensional linear subspace F of Y there is a nite dimensional subspace E of X so that dE F where d denotes here the Banach Mazur distance An equivalentway to express the conclusion of Prop osition is that Y is linearly isomorphic to a subspace of an ultrapro duct of X An immediate consequence of Prop osition is that any Banachspace Y which is a Lipschitz quotient of a Hilb ert space must b e linearly isomorphic to a Hilb ert space Prop osition was proved in by using a sp ecial result on approxi mating Lipschitz functions from a sup erreexive Banach space into nite dimensional Banach spaces the socalled UAAP see also Prop osi tion It was proved in that this approximation prop erty actually characterizes sup erreexive spaces and thus this pro of of Prop osition do es not work b eyond the class of sup erreexive spaces In itwas shown that if X is asymptotically uniformly smo oth AUS then every Lipschitz map from X to a nite dimensional space has for every points of Frechet dierentiability We do not intend to recall here the denitions of AUS or of Frechet dierentiability Wejust mention that the class of spaces ha ving an equivalentAUS norm is much richer than that of sup erreexive spaces and contains for example the non reexive space c On the other hand anyspaceX with an AUS norm is an Asplund space any separable subspace of X has a separable dual The existence of p oints of Frechet dierentiabilityforevery provides an approximation of Lipschitz functions by ane functions which also suces for proving the conclusion of Prop osition Thus Prop osition is valid also if X has an AUS norm In the result on existence of points of Frechet dierentiabilityandthus Prop osition is proved also for some spaces X which do not admit an equivalentAUS norm for example every table unless such X is X which is a C K space with K compact coun isomorphic to c it do es not have an equivalentAUS norm LIPSCHITZ QUOTIENTS FROM METRIC TREES It is p ossible that the Frechet dierentiability result and thus Prop o sition holds for every Asplund space X The Frechet dierentiability result is known to b e false for every nonAsplund space see eg Prop o sition or Chapter I I I of In this connection wemention also that in itisproved that a Banach space which is a Lipschitz quotientofan Asplund space is itself an Asplund space All the results mentioned so far leave op en the question whether the conclusion of Prop osition is valid for an arbitrary Banach space X Recall that in the formally dual setting of Lipschitz emb eddings the dual version of Prop osition is valid for an arbitrary Banachspace X ie if X Lipschitz emb eds in Y then X is crudely nitely representable in Y This fact is due to Rib e seealso Corollary We prove in this pap er that Prop osition may fail to hold for many common Banach spaces Wedothisbyproving in Theorem that if X is any separable Banach space containing then there is a Lipschitz quotient map from X onto any separable BanachspaceY In particular there is a Lipschitz quotient map from C onto in fact known results in the linear theory reduce the general theorem to this sp ecial case This provides also the rst known examples of pairs of separable Banach spaces X and Y so that there is a Lipschitz quotient map from X to Y butnosuch linear quotient map Theorem can b e considered in a way as a dual result to a theorem of Aharoni see also or Theorem that every separable metric space is Lipschitz equivalent to a subset of c There is however no obvious connection between the pro of of Theorem and the known pro ofs of Aharonis theorem The pro of of Theorem dep ends strongly on the sp ecial structure of Theorem actually holds for a general complete separable metrically convex metric space Y Recall that a metric space Y is metrical ly convex provided that for eachpairfx y g of p oints in Y and there exists a p oint z in Y so that dx z dx y and dy z dx y In other words for each x and y in Y there is a geo desic arc from x to y ie an arc which is the range of an isometry from an interval It is not simply for the sake of generalization that we mention metrically conv ex spaces The Lipschitz quotient maps f constructed in the pro of of Theorem factor through a sp ecial metrically convex metric space which wecall an tree This factorization forces the mapping f to have the prop erty that whenever the Gateaux derivative f xoff exists at a p oint x X then the rank of f x is see Prop osition b elow Note that if X has an AUS norm and dim Y such a situation is imp ossible bythe results in onFrechet dierentiability mentioned ab ove JOHNSON LINDENSTRAUSS PREISS AND SCHECHTMAN An tree is a sp ecial kind of metric tree Metric trees are used in geometry metric top ology and group theory but as far as weknow this is the rst time they have b een used in the geometry of Banach spaces There is a weakening of the notion of Lipschitz quotient map whichhas b een studied byDavid and Semmes Say that a Lipschitz map f from the metric space X into the metric space Y is bal l non col lapsing provided there is a c so that for every r the image under f of any ball of radius r contains a ball of radius cr If f is a Lipschitz map from a metric n space X into R say that f is measure non col lapsing provided there is a csothat for ev ery r the image under f of anyball of radius n r has Leb esgue measure at least cr David and Semmes showed that n that if X is nite dimensional and f X R is measure non collapsing then f is ball non collapsing In this was extended to the case where X is any sup erreexive Banachspace In Theorem weshow that if X is anyBanach space which contains a subspace isomorphic to then there is a measure non collapsing map from X onto the plane whose range has emptyinterior and hence the mapping is not ball non collapsing In Section we prove non separable versions of Theorem Theo rem implies in particular that if Y is a Banach space or more generally a complete metrically convex space whose density character is at most that of the continuum then there is a Lipschitz quotient map from onto Y PROOFS OF THE SEPARABLE RESULTS We b egin by recalling the denition of a metric tree A metric space X is a metric tree provided it is complete metrically convex and there is a unique arc which then by metric convexitymust be a geo desic arc joining each pair of p oints in X For an intro duction to metric trees see However except for the pro of of Prop osition wedonotusethe theory of metric trees Instead we dene and use a concept which we denote by SMT which is sucient for our purp oses Later in Section we show that this notion actually coincides with the notion of separable metric tree Similarlywe dene a notion of MT sucient for our purp oses and in Section show that it coincides with the notion of a metric tree The concept of SMT is dened by describing how to construct one In order to explain the construction we need a sp ecial case of the concept of the union of two metric spaces Supp ose that X d and Y d are X Y two metric spaces whose intersection is a single p oint p We dene the union X Y of X and Y to b e X Y d where the metric d agrees with x y is d on X d agrees with d on Y and if x X y Y then d X Y dened to b e d x pd p y We can now describ e the construction of X Y an SMT Let I b e a closed interval or a closed ray and dene T I The LIPSCHITZ QUOTIENTS FROM METRIC TREES metric space T is the rst approximation to our SMT Having dened T n let I b e a closed interval or a closed ray whose intersection with T is n n an end p oint p ofI and dene