Coarse embeddability into Banach spaces

COAR SE EMBEDD AB ILITY I NTO BANA CH SP A CES M.I. OSTRO VSKI I Abstra ct. The main p u rp oses of this pap er are (1) T o sur- vey the area of coarse em b eddability of metric spaces into Banac h spaces, and, in particular, coarse embeddab ility of different Banac h spaces in to each other; (2) T o present new re- sults on the problems: (a) W h ether coarse non-embeddability into ℓ 2 implies presence of expand er- like structu res? (b) T o what extent ℓ 2 is the most difficult space to embed into? 1. Introduction 1.1. Basic definitions. Let A and B b e metric spaces with metrics d A and d B , respectiv ely . Definition 1.1. A mapping f : A → B is called a c o arse emb e dding (or a uniform emb e dding ) if there exist functions ρ 1 , ρ 2 : [0 , ∞ ) → [0 , ∞ ) suc h that 1. ∀ x, y ∈ A ρ 1 ( d A ( x, y )) ≤ d B ( f ( x ) , f ( y )) ≤ ρ 2 ( d A ( x, y )). 2. lim r →∞ ρ 1 ( r ) = ∞ . R emark 1.2 . W e prefer to use the term c o arse emb e ddin g b e- cause in t he Nonlinear F unctional Ana lysis the term uniform emb e dding is used for uniformly contin uous injectiv e maps who- se inv erses are uniformly contin uous on their domains of defi- nition, see [6, p. 3]. In some of the pap ers cited b elow the term uniform emb e dding is used. 2000 M athematics Subje ct Classific ation. Primary 46B20, 54E40; Secondary 05C12. Key wor ds and phr ases. Coarse embedding, uniform em b edding, Banach space, expand er graph, lo cally fi nite metric space, b ound ed geometry . 1 2 M.I. OSTRO VSKII Definition 1.3. A mapping f : A → B is called Lips c hitz if there ex ists a constan t 0 ≤ L < ∞ su c h that (1.1) d B ( f ( x ) , f ( y )) ≤ L · d A ( x, y ) . The infim um o f all L > 0 for whic h the inequalit y in (1.1) is v alid is called the Lipschitz c onstant of f and is denoted b y Lip ( f ). A Lipschitz mapping is called a Lipschitz emb e dding if it is one- t o -one, and its in vers e, defined as a mapping from the image of f in to A , is a lso a Lipsc hitz mapping. Definition 1.4. A metric space A is said to ha v e b ounde d ge ometry if for each r > 0 there exist a p ositiv e in teger M ( r ) suc h that eac h ball B ( x, r ) = { y ∈ A : d A ( x, y ) ≤ r } o f radius r contains at most M ( r ) elemen ts. Definition 1.5. A metric space is called lo c al ly finite if all balls in it hav e finitely many elemen ts. Our terminology and notation of Banac h space theory fol- lo ws [6] and [27]. 1.2. Some history and motiv ation. M. Gromov [20] sug- gested to use coa rse embeddings of Ca yley graphs of infinite groups with finitely man y generators and finitely man y rela- tions (with their gra ph-theoretical metric) in to a Hilb ert space or into a uniformly con v ex Banac h space as a to ol for working on suc h w ell-kno wn conjectures as the Novik o v conjecture and the Baum–Connes conjecture (discussion of these conjectures is b ey ond the scope of this pap er). G. Y u [58] and G. Kas- paro v and G. Y u [32] ha v e sho wn t ha t this is indeed a v ery p ow erful to ol. G . Y u [58] used the condition of coa r se em b ed- dabilit y of metric spaces with b ounded geometry in to a Hilb ert space; G. Kasparov and G. Y u [32] used the condition of coarse em b eddability of metric spaces with bounded geometry in to a uniformly con v ex Banac h space. These results made the fol- lo wing problem p o sed b y M. Gromov in [20, Problem (4)] v ery imp ortant: “Do es every finitely gener ate d or finitely pr e sente d gr oup ad- mit a uniformly m e tric al ly pr op er Lipsc hitz emb e ddin g into a COARSE EMBEDDABILITY INTO BANA CH SP ACES 3 Hilb e rt sp ac e? Even such an emb e d d ing into a r eflexiv e uni- formly c onvex Banach sp ac e wo uld b e in ter esting. Thi s se ems har d.” Also, they attracted atten tion to the follo wing generalized v ersion of the problem: Whether e ach metric sp ac e with b ounde d ge ometry is c o arsely emb e ddable into a uniformly c onvex B anach sp ac e? The result of G. Kasparov and G . Y u [3 2] also made it in- teresting to compare classes of metric spaces em b eddable into differen t Ba na c h spaces ( with particular intere st to spaces with b ounded geometry). 2. Obstructions to e mbeddability of sp aces with bounded ge ometr y M. Gromov [19, R emark (b), p. 218] wrote: “There is no kno wn geometric obstruction for uniform embeddings in to in- finite dimensional Banac h spaces.” W riting this M. Gromov w as una ware o f P . Enflo’s w ork [16] in whic h it was show n that there is no uniformly con tinuous em b edding with uniformly con tinuous in verse of the Banach space c 0 in to a Hilb ert space. A.N. D ranishnik ov, G. Gong , V. Lafforgue, and G . Y u [13 , Section 6] adjusted the construction of P . Enflo [1 6] in order to pro v e that there exist lo cally finite me tric spaces whic h are not coarsely em b eddable in to Hilb ert spaces. After [13] w as written, M. G romo v (see [2 1 , p. 158]) observ ed t ha t expanders pro vide examples of spaces with bounded geometry whic h are not coarsely em b eddable in to a Hilb ert space and into ℓ p for 1 ≤ p < ∞ . Recall the definition (see [11] for an access ible in tro duction to the theory of expanders). Definition 2.1. F o r a finite gra ph G with v ertex set V and a subset F ⊂ V b y ∂ F w e denote the set of edges connecting F and V \ F . The exp anding c onstant (also kno wn as Che e ger 4 M.I. OSTRO VSKII c o nstant ) o f G is h ( G ) = inf  | ∂ F | | F | : F ⊂ V , 0 < | F | ≤ | V | / 2  . A sequence { G n } of graphs is called a fam ily of exp and e rs if all of G n are finite, connecte d, k -regular for some k ∈ N (that is, eac h v ertex is adjacen t to exactly k other v ertices), their expanding constan ts h ( G n ) a re b ounded a w ay from 0 (that is, there exists ε > 0 suc h that h ( G n ) ≥ ε for all n ), and their orders (n um b ers of v ertices) tend to ∞ as n → ∞ . W e consider ( vertex sets of ) connected graphs as metric spaces, with their standard graph- theoretic distance: the dis- tance b et wee n tw o vertice s is the n umber of edges in the short- est path joining them. Let A b e a metric space con taining isometric copies of all graphs fro m some family of expanders. The Gromov’s o bser- v atio n is: A do es not em b ed coarsely into ℓ p for 1 ≤ p < ∞ (see [52, pp. 160 – 161] for a detailed pro of, it is w or th men tion- ing that the result can b e pro v ed using the argumen t whic h is w ell- kno wn in the theory o f Lipsc hitz em b eddings of finite metric space s, see [37, pp. 192–1 93]). M. Gromov [21] suggested to use random gro ups in order to pro v e that there exist Ca yley graphs of finitely presen t ed groups whic h a re no t coarsely embeddable in to a Hilb ert space. Man y details on this appro a c h w ere giv en in t he pap er M. Gro- mo v [22] (some details w ere explained in [18], [44], and [55 ]). Ho wev er, to the b est of m y kno wledge, the work on clarifica- tion of all of the details of the M. Gromov’s construction has not been completed (as of now). The p osed a b ov e problem ab o ut the existence of coarse em- b eddings of spaces with b ounded geometry in to uniformly con- v ex Bana c h spaces w as recen tly solv ed in the negativ e b y V. L a- fforgue [34], his construction is also expander-based. COARSE EMBEDDABILITY INTO BANA CH SP ACES 5 N. O za wa [47, Theorem A.1] pro v ed that a metric space A con taining isometric copies of a ll graphs from some family of expanders do es not em b ed coarsely in to an y Banach space X suc h that B X (the unit ball of X ) is unifo rmly homeomorphic to a subset of a Hilb ert space. See [6, Chapter 9, Section 2] for results on sp aces X suc h that B X is uniformly homeomorphic to B ℓ 2 . It w ould b e v ery in teresting to find out whether each metric space with b ounded geometry whic h is not coarsely em b ed- dable into a Hilb ert space con tains a substructure similar to a family of expanders. A v ersion of this problem was p osed in [23] using the following terminology: Definition 2.2. A metric space X we akly c ontains a family { G n } ∞ n =1 of expanders with vertex sets { V n } ∞ n =1 if there are maps f n : V n → Y satisfying (i) sup n Lip ( f n ) < ∞ , (ii) lim n →∞ sup v ∈ V n | f − 1 n ( v ) | | V n | = 0. Pr ob l e m 2.3 . [23, p. 26 1] Let Γ b e a finitely presen ted group whose Ca yley graph G with its natural metric is not coarsely em b eddable in to ℓ 2 . Do es it follow t hat G w eakly contains a family of expanders? The follow ing theorem can help to finding an expander-lik e structure in metric spaces with b ounded g eometry whic h are not coarsely em b eddable in to a Hilb ert space. In the theorem w e consider coa rse em b eddabilit y into L 1 = L 1 (0 , 1 ) . F or tec h- nical reasons it is more conv enien t to w ork with L 1 . As w e shall see in section 4 coa rse embeddability into L 1 is equiv alent to coarse em b eddabilit y in to a Hilb ert space. Theorem 2.4. L et M b e a lo c al ly finite metric sp ac e which is not c o arsel y emb e dda ble into L 1 . Then ther e exis ts a c onstant D , de p e n ding on M only, such that for e ach n ∈ N ther e exists a fin i te s e t B n ⊂ M × M and a pr ob ab ility me asur e µ on B n such that 6 M.I. OSTRO VSKII • d M ( u, v ) ≥ n for e ach ( u , v ) ∈ B n . • F or e ach Lipsch itz function f : M → L 1 the ine quality (2.1) Z B n || f ( u ) − f ( v ) || L 1 dµ ( u, v ) ≤ D Lip ( f ) holds. Lemma 2.5. Ther e exists a c onstant C dep ending on M only such that for e ach Lips c hitz function f : M → L 1 ther e exists a subset B f ⊂ M × M s uch that sup ( x,y ) ∈ B f d M ( x, y ) = ∞ , but sup ( x,y ) ∈ B f || f ( x ) − f ( y ) || L 1 ≤ C Lip ( f ) . Pr o of. Assume the con trary . Then, for eac h n ∈ N , the n umber n 3 cannot serv e as C . This means, t hat for eac h n ∈ N there exists a L ipschitz mapping f n : M → L 1 suc h tha t for eac h subset U ⊂ M × M with sup ( x,y ) ∈ U d M ( x, y ) = ∞ , w e hav e sup ( x,y ) ∈ U || f n ( x ) − f n ( y ) || > n 3 Lip ( f n ) . W e choose a p oin t in M and denote it b y O . Without loss of generalit y we ma y assume that f n ( O ) = 0. Consider the mapping f : M → ∞ X n =1 ⊕ L 1 ! 1 ⊂ L 1 giv en by f ( x ) = ∞ X n =1 1 K n 2 · f n ( x ) Lip( f n ) , where K = P ∞ n =1 1 n 2 . It is clear that the series conv erges and Lip ( f ) ≤ 1. Let us sho w that f is a coarse em b edding. W e need an estimate from below only (the estimate from ab o ve is satisfied b ecause f is Lipsc hitz). COARSE EMBEDDABILITY INTO BANA CH SP ACES 7 The a ssumption implies t ha t f or eac h n ∈ N there is N ∈ N suc h that d M ( x, y ) ≥ N ⇒ || f n ( x ) − f n ( y ) || > n 3 Lip ( f n ) . On the ot her hand || f n ( x ) − f n ( y ) || > n 3 Lip ( f n ) ⇒ || f ( x ) − f ( y ) || > n K Hence f : M → L 1 is a coa r se em b edding and w e get a contra- diction.  Lemma 2.6. L et C b e the c onstant who s e existenc e is p r ov e d in L emma 2.5 and let ε b e an a rb itr ary p ositive numb er. F or e a ch n ∈ N we c an find a finite subset M n ⊂ M such that for e a ch Lipsc h itz mapping f : M → L 1 ther e is a p air ( u f ,n , v f ,n ) ∈ M n × M n such that • d M ( u f ,n , v f ,n ) ≥ n . • || f ( u f ,n ) − f ( v f ,n ) || ≤ ( C + ε ) Lip ( f ) . Pr o of. W e choose a p oint in M and denote it by O . The ba ll in M of radius R cen tered a t O will b e denoted b y B ( R ). It is clear that it suffices to pro ve the result for 1-Lipsc hitz mappings satisfying f ( O ) = 0. Assume the con trary . Since M is lo cally finite, this implies that for eac h R ∈ N there is a 1 - Lipsc hitz mapping f R : M → L 1 suc h that f R ( O ) = 0 and, for u, v ∈ B ( R ), the inequalit y d M ( u, v ) ≥ n implies || f R ( u ) − f R ( v ) | | L 1 > C + ε . W e r efer to [9 ], [24], or [12, Chapter 8] for results on ultra- pro ducts, our terminology and notation follow s [12]. W e form an ultrapro duct of the mappings { f R } ∞ R =1 , that is, a mapping f : M → ( L 1 ) U , giv en by f ( m ) = { f R ( m ) } ∞ R =1 , where U is a non-trivial ultrafilter on N and ( L 1 ) U is the corresp onding ul- trap ow er. Each ultrap o w er of L 1 is isometric to an L 1 space on some measure space ( see [12, Theorem 8.7 ], [9], [24]), and its separable subspaces are isometric to subspaces of L 1 (0 , 1 ) (see [14, p. 168], [27, pp. 14–15], and [53, pp. 399 & 416]). There- fore we can consider f a s a mapping into L 1 (0 , 1 ) . It is easy 8 M.I. OSTRO VSKII to v erif y that Lip ( f ) ≤ 1 and that f satisfies the condition d M ( u, v ) ≥ n ⇒ || f ( u ) − f ( v ) || L 1 ≥ ( C + ε ) . W e get a contradiction with the definition of C .  Pr o of of The or em 2 .4 . Let D be a n umber satisfying D > C , and let B b e a num b er satisfying C < B < D . According to Lemma 2.6, there is a finite subset M n ⊂ M suc h that for each 1-Lipsc hitz function f on M there is a pair ( u, v ) in M n suc h that d M ( u, v ) ≥ n and || f ( u ) − f ( v ) || ≤ B . Let α n b e the cardinalit y of M n , w e c ho ose a p oint in M n and denote it b y O . Proving the theorem it is enough to consider 1-Lipsc hitz functions f : M n → L 1 satisfying f ( O ) = 0. Eac h α n -elemen t subset of L 1 is isometric to a subset in ℓ α n ( α n − 1) / 2 1 (see [57], [3]) . Therefore it suffice s to pro v e the result for 1- Lipsc hitz em b eddings in to ℓ α n ( α n − 1) / 2 1 . It is clear that it suffices to pro v e the inequalit y Z B n || f ( u ) − f ( v ) || dµ ( u , v ) ≤ B for a  D − B 2  -net in the set of all functions satisfying the con- ditions men tioned ab ov e, endow ed with the metric τ ( f , g ) = max m ∈ M n || f ( m ) − g ( m ) || By compactness there exists a finite net satisfying the con- dition. Let N be suc h a net. W e are going to use the minimax theorem, see, e.g. [56 , p. 344]. In particular, we use the nota- tion similar to the one use d in [56 ]. Let A b e the matrix whose columns are lab elled b y func- tions from N , whose ro ws are lab elled b y pairs ( u, v ) of ele- men ts of M n satisfying d M ( u, v ) ≥ n , and whose entry on the in tersection of the column corresp o nding to f , and the ro w corresp onding to ( u, v ) is || f ( u ) − f ( v ) || . Then, for each column v ector x = { x f } f ∈ N with x f ≥ 0 and P f ∈ N x f = 1, the entries of the pro duct Ax are the differences COARSE EMBEDDABILITY INTO BANA CH SP ACES 9 || F ( u ) − F ( v ) || , where F : M → X f ∈ N ⊕ ℓ α n ( α n − 1) / 2 1 ! 1 is giv en b y F ( m ) = X f ∈ N x f f ( m ). The function F can b e considered as a function in to L 1 . It satisfies Lip ( F ) ≤ 1 . Hence there is a pair ( u, v ) in M n satisfying d M ( u, v ) ≥ n and || F ( u ) − F ( v ) || ≤ B . Therefore w e ha ve max x min µ µAx ≤ B , where the minimum is tak en ov er all v ectors µ = { µ ( u , v ) } , indexed b y u, v ∈ M n , d M ( u, v ) ≥ n , and satisfying the condi- tions µ ( u, v ) ≥ 0 and P µ ( u, v ) = 1. By the v on Neumann minimax theorem [56, p. 344], w e hav e min µ max x µAx ≤ B , whic h is exactly the inequalit y we need to pro v e b ecause µ can b e regarded as a probability measure o n the set o f pairs from M n with distance ≥ n .  3. Coarse embeddability into reflexive Banach sp aces The first r esult of this nature was obtained b y N. Brown and E. G uentner [8, Theorem 1]. They prov ed tha t for eac h metric space A ha ving b ounded geometry there is a sequence { p n } , p n > 1, lim n →∞ p n = ∞ suc h that A em b eds coarsely in to the Banac h space ( P ∞ n =1 ⊕ ℓ p n ) 2 , which is, obviously , reflexiv e. This result was strengthened in [5], [31], and [45]. (Observ e that the space ( P ∞ n =1 ⊕ ℓ p n ) 2 has no cotype.) Theorem 3.1. [45] L et X b e a B anach sp ac e with no c otyp e and let A b e a lo c a l ly fin ite metric sp ac e. Then A emb e ds c o a rsely into X . 10 M.I. OSTRO VSKII Theorem 3.2. [5 ] L et X b e a B a n ach sp ac e with no c otyp e and let A b e a lo c al ly finite metric sp ac e. Then ther e exists a Lipschitz emb e dding of A into X . R em ark 3.3 . In terested readers can reconstruct the pro of f rom [5] b y applying Prop osition 6.1 (see b elo w) to Z = c 0 in com- bination with the result of I. Aharo ni men t io ned in Section 4.1. Definition 3.4. A metric space ( X , d ) is called stable if fo r an y t w o b ounded sequences { x n } and { y n } in X and for any t wo non- trivial ultrafilters U and V on N the condition lim n, U lim m, V d ( x n , y m ) = lim m, V lim n, U d ( x n , y m ) holds. Theorem 3.5. [31] L et A b e a stabl e metric sp ac e. Then A emb e ds c o arsely into a r eflexive B a nach sp ac e. R em ark 3.6 . It is easy to see that lo cally finite metric spaces are stable. N.J. Kalton [31] found examples o f Bana c h spaces whic h are not coarsely em b eddable into reflexiv e Banach spaces, c 0 is one of the examples of suc h spaces. Apparen t ly his result provides the first example of a metric spaces whic h is not coarsely em- b eddable into reflexiv e Banach spaces. (See the Problem (3) in the list of op en problems in [48].) 4. Coarse classifica tion of Banach sp aces As w e a lready men tioned the result of G. K asparo v a nd G. Y u [32] mak es it very in teresting to compare the conditions of coarse em b eddabilit y in to a Banac h space X for differen t spaces X . Since comp ositions of coarse em b eddings are coarse em b eddings, one can approach this problem b y studying coarse em b eddability of Banac h space in to each o t her. In this subsec- tion w e describe the existing kno wledge on this mat t er. COARSE EMBEDDABILITY INTO BANA CH SP ACES 11 4.1. Essential ly nonlinear coarse em b eddings. There a re man y examples of pairs ( X , Y ) of Bana c h spaces suc h that X is coarsely em b eddable in to Y , but the Banac h-space-theoretical structure of X is quite differen t fr o m the Banac h-space-theore- tical structure of eac h subs pace of Y : • A result whic h g o es bac k to I.J. Sc ho en b erg [54 ] (see [38, p. 385] for a simple pro of ) states that L 1 with the metric p || x − y || 1 is isometric to a subset of L 2 . Hence L 1 and all of its subspaces, in particular, L p and ℓ p (1 ≤ p ≤ 2) (see [2 9 ] and [7]) em b ed coarsely into L 2 = ℓ 2 . • This res ult was generalized b y M. Mendel and A. Nao r [39, Remark 5.10]: F or ev ery 1 ≤ q < p the metric space ( L q , || x − y || q /p L q ) is isometric to a subspace of L p . • The w ell-kno wn result of I. Aharoni [1] implies that eac h separable Banac h space is coarsely embeddable in to c 0 (although its Banach space theoretical prop erties can b e quite differen t from those of an y subspace of c 0 ). A simpler pro of of this result w as obta ined in [2], see, a lso, [6, p. 176]. • N.J. Kalton [30] pro ved that c 0 em b eds coarsely in to a Banac h space with the Sch ur pro p ert y . • P . Now a k [42] pro ved that ℓ 2 is coarsely em b eddable in to ℓ p for all 1 ≤ p ≤ ∞ . 4.2. Obstructions to coarse em b eddabilit y of B anach spaces. The list of disco v ered obstructions to coarse embed- dabilit y also constan tly increases: • Only minor adjustmen ts o f the argument of Y. Ra ynaud [51] (see, also [6, pp. 212–215]) are needed to prov e the follo wing results : (1) Let A b e a Ba na c h space with a spreading basis whic h is no t an unconditional basis. Then A do es not em b ed coarsely in to a stable metric space. (See [6, p. 429] for the definition of a spreading basis and 12 M.I. OSTRO VSKII [33] for examples of stable Banach spaces. Exam- ples of stable Banac h spaces include L p (1 ≤ p < ∞ ).) (2) Let A b e a nonreflexiv e Banac h space with non- trivial t yp e. Then A do es not em b ed coarsely in t o a stable metric space. (Examples of nonreflexiv e Ba- nac h spaces with non-t rivial type w ere constructed in [25], [26], [49].) • A.N. Dra nishnik o v, G. Go ng, V. Lafforgue, and G. Y u [13] adjusted the argumen t o f P . Enflo [16] to pro v e that Banac h spaces with no cot yp e ar e not coarsely em b ed- dable in to ℓ 2 . • W. B. Johnson and L. Randrianariv on y [2 8] pro v ed that ℓ p ( p > 2 ) is not coarsely em b eddable in to ℓ 2 . • M. Mendel and A. Naor [40] pro v ed (f or K -con vex spa- ces) that coty p e of a Banac h space is an o bstruction to coarse em b eddabilit y , in particular, ℓ p is not coa rsely em b eddable into ℓ q when p > q ≥ 2. • L. Randrianariv ony [50] strengthened the result from [28] to a characterization of quasi-Banac h spaces whic h em b ed coarsely in to a Hilb ert space, and pro v ed: a separable Banac h space is coar sely em b eddable in to a Hilb ert space if and only if it is isomorphic to a subspace of L 0 ( µ ). • N.J. Kalton [31] found some more o bstructions to coarse em b eddability . In particular, N.J. Kalton discov ered a n in v ariant, whic h he named the Q -prop erty , whic h is nec- essary for coarse em b eddability in to reflexiv e Banac h spaces. 4.3. T o what exten t is ℓ 2 the most difficult space to em b ed in t o? Because ℓ 2 is, in man y resp ects, the ‘b est’ space, and because of Dvoretzk y’s theorem (see [15] and [41]) it is natural to expect that ℓ 2 is among the most difficult spaces to em b ed in to. The strongest p ossible result in this direction w ould b e a p ositiv e solution to the f ollo wing problem. COARSE EMBEDDABILITY INTO BANA CH SP ACES 13 Pr ob l e m 4.1 . Do es ℓ 2 em b ed coarsely into an arbitrary infinite dimensional Banac h space? This problem is still op en, but the coa rse em b eddability of ℓ 2 is kno wn for wide classes of Banac h spaces. As was men- tioned ab ov e, P .W. Now ak [42] prov ed that ℓ 2 em b eds coarsely in to ℓ p for eac h 1 ≤ p ≤ ∞ . In Section 5 w e prov e that ℓ 2 em- b eds coarsely in to a Banac h space con ta ining a subspace with an unconditional basis which do es not con ta in ℓ n ∞ uniformly (Theorem 5.1 ). This result is a generalization of P .W. Now ak’s result men tioned a b ov e b ecause the spaces ℓ p (1 ≤ p < ∞ ) sat- isfy the condition of Theorem 5.1, but the spaces satisfying the condition of Theorem 5.1 do not necessarily con tain subspaces isomorphic to ℓ p (see [17], and [36, Section 2 .e]). In all existing applications of coarse em b eddability results the most imp o rtan t is the case when w e em b ed spaces with b ounded geometry into Ba nac h spaces. In this connection the follo wing result from [46] is o f in terest. Theorem 4.2 ([46]) . L et A b e a lo c a l ly finite metric sp ac e which e m b e ds c o arsely into a Hilb ert sp ac e, and let X b e an infinite dim ensional Banach sp ac e. Then ther e exists a c o a rse emb e dding f : A → X . In this pa p er w e use an idea of F. Baudier and G . Lancien [5], and pro ve this result in a stronger f o rm, for Lipsc hitz em- b eddings (see Section 6): Theorem 4.3. L et M b e a lo c a l ly finite subset of a Hilb ert sp ac e. Then M is Lipschitz emb e ddable i nto an arbitr ary infi- nite dimen sional Banach sp ac e. 5. Co ars e embeddings of ℓ 2 Theorem 5.1. L et X b e a Ba n ach sp ac e c ontaining a sub- sp ac e with an unc onditional b asis whi c h do es n o t c ontain ℓ n ∞ uniformly. Then ℓ 2 emb e ds c o ars e ly into X . 14 M.I. OSTRO VSKII Pr o of. W e use the criterion for coarse em b eddabilit y in to a Hilb ert space due to M. Dada r la t a nd E. Guen tner [10, Pro p o- sition 2 .1] (see [35] and [42] for related results). W e stat e it as a lemma (by S ( X ) w e denote the unit sphere of a Banac h space X ). Lemma 5.2 ([10]) . A metric sp a c e A admits a c o arse emb e d- ding into ℓ 2 if and only if for eve ry ε > 0 an d every R > 0 ther e exists a map ζ : A → S ( ℓ 2 ) such that (i) d A ( x, y ) ≤ R implies | | ζ ( x ) − ζ ( y ) || ≤ ε . (ii) lim t →∞ inf {| | ζ ( x ) − ζ ( y ) || : x, y ∈ A, d A ( x, y ) ≥ t } = √ 2 . W e assume without loss of generality that X has an uncon- ditional basis { e i } i ∈ N . Let N = ∪ ∞ i =1 N i b e a partition o f N in to infinitely many infinite subsets . Let X i = cl( span { e i } i ∈ N i ). By the theorem of E. Odell and T. Schlum prec ht [43] (see, also, [6, Theorem 9.4 ]), for each i ∈ N there exists a uniform homeomorphism ϕ i : S ( ℓ 2 ) → S ( X i ). W e apply Lemma 5.2 in the case when A = ℓ 2 . By the uniform con tinuit y of ϕ i and ϕ − 1 i w e g et: for eac h i ∈ N there exists δ i > 0 and a map ζ i : ℓ 2 → S ( X i ) suc h that (5.1) lim t →∞ inf {| | ζ i ( x ) − ζ i ( y ) || X i : || x − y || ℓ 2 ≥ t } ≥ δ i . (5.2) || x − y || ℓ 2 ≤ i implie s || ζ i ( x ) − ζ i ( y ) || X i ≤ δ i i 2 i . Fix x 0 ∈ ℓ 2 . Let f : ℓ 2 → X b e the map defined as the direct sum of the maps i δ i ( ζ i ( x ) − ζ i ( x 0 )). W e claim that it is a coarse em b edding (the fact that it is a we ll-defined map follo ws f rom (5.2)). Let || x − y || = r , then for i ≥ r w e get || i δ i ζ i ( x ) − i δ i ζ i ( y ) || X i ≤ 1 2 i . Hence || f ( x ) − f ( y ) || ≤ P ⌈ r ⌉− 1 i =1 2 i δ i + P ∞ i = ⌈ r ⌉ 1 2 i =: ρ 2 ( r ). W e pro v ed an estimate from ab o v e. COARSE EMBEDDABILITY INTO BANA CH SP ACES 15 T o pro v e an estimate from b elo w, it is enough, fo r a giv en h ∈ R , to find t ∈ R suc h that || x − y | | ℓ 2 ≥ t implies || f ( x ) − f ( y ) || X ≥ h . F or this, b y unconditionalit y (w e assume, for simplicit y , that the basis of X is 1- unconditional), it is enough to find i ∈ N suc h that || x − y || ℓ 2 ≥ t implies || i δ i ζ i ( x ) − i δ i ζ i ( y ) || X i ≥ h . W e c ho o se an arbitrary i > h . The conclusion follo ws f rom the conditio n (5.1).  6. Lipschitz embe ddings of locall y finite metric sp aces The purp ose of this section is to prov e Theorem 4.3. W e pro v e the main step in our argumen t (Prop o sition 6 .1) in a somewhat more general con text than is needed for Theorem 4.3, b ecause it can b e applied in some other situations (see, in this connection, the pap er [4] containing tw o v ersions of Prop osition 6.1). The coarse v ersion of this result w as prov ed in [46], in the pro of of the Lipsc hitz v ersion w e use an idea from [5]. Prop osition 6.1. L et A b e a lo c al ly finite subset of a Banach sp ac e Z . Then ther e ex i s ts a se quenc e of finite dimensional line ar subsp ac es Z i ( i ∈ N ) of Z such that A is Lipsc h itz em- b e d dable into e ach Banach sp ac e Y having a finite dimensional Schauder de c om p osition { Y i } ∞ i =1 with Y i line arly isometric to Z i . See [36, Section 1.g] for infor mation on Sc hauder decomp o- sitions. It is clear that w e may restrict ourselv es to the case when the Schauder decomp o sition satisfies (6.1) || y i || ≤      ∞ X i =1 y i      when y i ∈ Y i for eac h i ∈ N . Pr o of. Let Z i b e the linear subspace of Z spanned b y { a ∈ A : || a || Z ≤ 2 i } and let S i = { a ∈ A : 2 i − 1 ≤ || a | | Z ≤ 2 i } . Let T i : Z i → Y i b e some linear isometries and let E i : Z i → Y b e comp ositions of these linear isometries with the natura l 16 M.I. OSTRO VSKII em b eddings Y i → Y . W e define an em b edding ϕ : A → Y by ϕ ( a ) = 2 i − || a || Z 2 i − 1 E i ( a ) + || a || Z − 2 i − 1 2 i − 1 E i +1 ( a ) for a ∈ S i . One can che c k that t here is no am biguit y for || a || Z = 2 i . R em ark 6.2 . The mapping ϕ is a straightforw ard generalization of the mapping constructed in [5]. It remains to v erify t ha t ϕ is a Lipsc hitz embedding. W e consider three cases. (1) a, b are in the same S i ; (2) a, b are in consecutiv e sets S i , that is, b ∈ S i , a ∈ S i +1 ; (3) a, b are in ‘distan t’ sets S i , that is, b ∈ S i , a ∈ S k , k ≥ i + 2. Ev erywhere in the proof w e assume || a || ≥ | | b || . Case (1). The ineq ualit y (6.1) implie s that the num b er || ϕ ( a ) − ϕ ( b ) || Y is betw een the maxim um and the sum o f t he n um b ers (6.2)     2 i − || a || Z 2 i − 1 E i ( a ) − 2 i − || b || Z 2 i − 1 E i ( b )     , (6.3)     || a || Z − 2 i − 1 2 i − 1 E i +1 ( a ) − || b || Z − 2 i − 1 2 i − 1 E i +1 ( b )     . It is clear that the norm in (6.2) is b et w een the n um b ers 2 i − || a || Z 2 i − 1 || E i ( a ) − E i ( b ) || ∓ || a || Z − || b || Z 2 i − 1 || E i +1 ( b ) || , and the nor m in (6.3) is b et w een the n um b ers || a || Z − 2 i − 1 2 i − 1 || E i +1 ( a ) − E i +1 ( b ) || ∓ || a || Z − || b || Z 2 i − 1 || E i +1 ( b ) || . Therefore 1 2  || a − b || Z − || a || Z − || b || Z 2 i − 2 || b || Z  ≤ || ϕ ( a ) − ϕ ( b ) || Y ≤ || a − b || Z + || a || Z − || b || Z 2 i − 2 || b || Z . COARSE EMBEDDABILITY INTO BANA CH SP ACES 17 This inequalit y implies a suitable estimate from ab o v e for the Lipsc hitz constan t o f ϕ , and an estimate for the Lipsc hitz constan t of its in ve rse in the case when || a − b || Z is m uc h larger than || a | | Z − || b || Z , f or example, if || a − b || Z ≥ 5( || a || Z − || b || Z ). T o complete the pro of in the case (1) it suffice s to estimate || ϕ ( a ) − ϕ ( b ) || from b elo w in the case when || a || Z − || b || Z ≥ || a − b || Z 5 . In this case w e use the observ ation that fo r a, b ∈ S i satisfying || a || Z ≥ || b || Z the sum of (6.2) and (6.3) can b e estimated form b elo w b y  2 i − || a || Z 2 i − 1 || a || Z − 2 i − || b || Z 2 i − 1 || b || Z  +  || a || Z − 2 i − 1 2 i − 1 || a || Z − || b || Z − 2 i − 1 2 i − 1 || b || Z  = || a || Z − || b || Z ≥ || a − b || Z 5 . This completes our pro of in the case (1). Case (2). The ineq ualit y (6.1) implie s that the num b er || ϕ ( a ) − ϕ ( b ) || Y is betw een the maxim um and the sum o f t he n um b ers (6.4)     2 i − || b || Z 2 i − 1 E i ( b )     , (6.5)     2 i +1 − || a || Z 2 i E i +1 ( a ) − || b || Z − 2 i − 1 2 i − 1 E i +1 ( b )     , (6.6)     || a || Z − 2 i 2 i E i +2 ( a )     . Both (6.4) and (6.6) ar e estimated fr om ab ov e b y 2( | | a || Z − || b || Z ). As for (6.5), w e hav e     2 i +1 − || a || Z 2 i E i +1 ( a ) − || b || Z − 2 i − 1 2 i − 1 E i +1 ( b )     18 M.I. OSTRO VSKII (6.7) =     2 i − ( || a || Z − 2 i ) 2 i a + (2 i − || b || Z ) − 2 i − 1 2 i − 1 b     Z ≤ || a − b || Z + 2( || a || Z − 2 i ) + 2(2 i − || b || Z ) ≤ 3 || a − b || Z . W e turn to estimate from b elo w. F rom (6.4) and (6.6) w e get || ϕ ( a ) − ϕ ( b ) || ≥ max { (2 i − || b || Z ) , ( || a || Z − 2 i ) } . Therefore it suffices to find an estimate in the case when (6.8) max { (2 i − || b || Z ) , ( || a || Z − 2 i ) } ≤ || a − b || Z 5 . Rewriting (6.5) in the same wa y as in (6 .7), w e get || ϕ ( a ) − ϕ ( b ) || Y ≥     ( a − b ) + 2 i − || b || Z 2 i − 1 b − || a || Z − 2 i 2 i a     In the case when (6.8) is satisfied, w e can con tinue this chain of ineq ualities with ≥ || a − b || Z − 4 5 || a − b || Z = 1 5 || a − b || Z . Case (3). In this case the num b er || ϕ ( a ) − ϕ ( b ) || Y is b et w een the maxim um and the sum of the four n um b ers: 2 i − || b || Z 2 i − 1 || b || Z , || b || Z − 2 i − 1 2 i − 1 || b || Z , 2 k − || a || Z 2 k − 1 || a || Z , || a || Z − 2 k − 1 2 k − 1 || a || Z . Hence || ϕ ( a ) − ϕ ( b ) || Y is betw een || a || Z 2 (= the a verage of the last t w o n umbers) a nd || a || Z + || b || Z (=the sum of all four n umbers). On the other hand, 1 2 || a || Z ≤ || a || Z − || b || Z ≤ || a − b || Z ≤ || a || Z + || b || Z ≤ 2 || a || Z . These inequalities immediately imply estimates for Lipsc hitz constan ts.  COARSE EMBEDDABILITY INTO BANA CH SP ACES 19 Pr o of of The or em 4 . 3. Eac h finite dimensional subspace of ℓ 2 is isometric to ℓ k 2 for some k ∈ N . By Prop osition 6.1 there ex- ists a sequence { n i } ∞ i =1 suc h that A em b eds coarsely in to eac h Banac h space Y ha ving a Sc hauder decomp osition { Y i } with Y i isometric to ℓ n i 2 . 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