On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts

ON UNIVERSAL SP A CES F OR THE CLASS OF BANACH SP A CES WHOSE DUAL BALLS ARE UNIF ORM EBERLEIN COMP A CTS CHRISTINA BRECH AND PIOTR K OSZMIDER Abstract. F or κ being the first uncou ntab le cardinal ω 1 or κ being the cardi- nality of the contin uum c , we pro v e that it is consisten t that there is no Banac h space of densit y κ in whic h it is p ossible to isomorphically em bed every Ba- nac h space of th e same densit y whic h has a uniform ly Gˆ ateaux differen tiable renorming or, equiv alen tly , whose dua l unit ball with the w eak ∗ topology is a subspace of a Hilb ert space (a uniform Eberlein compact space). This com- plemen ts a consequence of results of M. Bell and of M. F abian, G. Godefroy , V. Zizler that assuming the con tin uum h ypothesis, there is a unive rsal space for all Banac h spaces of density κ = c = ω 1 which hav e a uniformly Gˆ ate aux differen tiable renorming. Our result im plies, in particular, that β N \ N ma y not map contin uously onto a compact subset of a Hil b ert space with the we ak topology of density κ = ω 1 or κ = c and that a C ( K ) space for some uniform Eberl ein compact space K may not embed isomorphically into ℓ ∞ /c 0 . 1. intr oduction A classical r e s ult of Banach and Mazur states that e v ery s eparable Banach space can b e isometrically embedded into the Banach space C ([0 , 1]). In this pa p er we deal with the pro blem o f embedding nonsepa rable Bana c h spaces of a g iv en class int o a single nonseparable space. W e are interested in the tw o most imp ortant uncountable dens ities , ω 1 and c . Since a contin uous on to map φ : K → L (for K, L compact Hausdorff spaces) gives an isometric em b edding T ( f ) = f ◦ φ of the Banach spa c e C ( L ) in to C ( K ), we hav e the following strictly related notions o f universalit y: Definition 1. 1. L et X b e a class of Ba nach sp ac es. We say tha t X ∈ X is (isometric al ly) universal sp ac e for the class X if for every Y ∈ X ther e is an (isometric) isomorphism onto its r ange T : Y → X . L et K b e a class of c omp act Haus dorff sp ac es. We say that K ∈ K is a universal sp ac e for the class K if for every L ∈ K ther e is a c ont inuous onto map T : K → L . The follo wing pro positio n shows that universal compact Hausdorff space s and universal Ba nac h spaces are c losely related: Prop osition 1.2 . Supp ose K is a class of c omp act Hausdorff sp ac es and X is a class of Banach sp ac es such that for e ach K ∈ K , C ( K ) ∈ X and for e ach X ∈ X , 2010 M athematics Subje ct Classific ation. Pri mary 46B26; Seconda ry 03E35, 46B03. The first author was partially supp or ted b y F APESP (2010/126 39-1) and Pr´ o-reitoria de Pe squisa USP (10.1.24497.1.2). The second aut hor w as partially supported by Po lish M inistry of Science and Higher Education researc h grant N N201 386234. 1 2 CHRISTINA BRECH AND PIOTR KOSZMIDER the dual unit b al l B X ∗ with the we ak ∗ top olo gy is in K . If K is a universal c omp act sp ac e for K , then C ( K ) is an isometric al ly universal Banach sp ac e for X . Pr o of. Let X ∈ X . It is well known tha t X isometrically em b eds into C ( B X ∗ ). By the hypothesis B X ∗ ∈ K and s o K contin uously maps on to B X ∗ , whic h means that C ( B X ∗ ) isometrica lly e mbeds into C ( K ). Comp osing this isometric embedding with the isometric e mbedding of X into C ( B X ∗ ), w e obtain the desired em bedding of X into C ( K ).  Let κ b e an infinite cardinal. Examples of pairs of class es K and X satisfying the h yp othesis of the abov e propo sition are: • the class o f all compac t spaces of weigh t ≤ κ and the class of a ll Banach spaces of densit y ≤ κ ; • the class of all E berlein co mpa ct spa ces (compact subsets of Ba na c h spaces with the weak top ology ) of weigh t ≤ κ a nd the class of all weakly c o mpactly generated Banach s pa ces (W C G) o f densit y ≤ κ ; • the class of all uniform Eb erlein compact s paces (compact subsets of Hilbert spaces with the weak to p ology ) of weight ≤ κ and the class of all Banach spaces which hav e a unifor mly Gˆ ateaux differentiable renorming (UG) of density ≤ κ . The nonexistence of a universal Ba nac h space implies, by the a b ov e, the nonexis- tence of a corresp onding universal compact s pace, but the oppos ite direction is not immediate. There are some reasons for this. Fir st, C ( K ) ma y b e universal without K b e ing universal, as in the ca s e o f K = [0 , 1] for the cla ss of separa ble Bana c h spaces and metriza ble compact spaces (ho wev er C ( K ) is isomor phic to C (∆) where ∆ is the Cantor set, which is universal for metrizable compact spaces ). Sec o ndly , there are universal Ba nac h spaces which are not isometrically universal. F or exam- ple, c onsider a strictly co n vex renorming ( || x + y || = 2 for || x || = || y || = 1 implies x = y ) of C ([0 , 1]) (which e xists by Theo rem 9 of [5]) a nd notice that this space contin ues to be a n is o morphically univ ersal space but cannot isometrically include spaces whose norm is not strictly con vex. In the case of nonsepara ble Banach spaces we e ncoun ter ma inly negative results concerning univ ersality or results sho wing that to obtain a univ ersal s pace o ne has to assume additiona l set-theor etic axio ms. F or example, it was proved in [2] that there are no universal W CG Bana c h spaces of densit y ω 1 or c nor universal Eb erlein compact spaces of w eight ω 1 or c . Assuming the cont inuum h yp othesis, the compact spa ce β N \ N is a univ ersal compact space of w eight c and b y the ab o ve pro position ℓ ∞ /c 0 ≡ C ( β N \ N ) is an isometric ally universal Banach spac e for the class of s paces of density c = ω 1 . How ev er, it was shown in [7] that it is co nsisten t that there is no iso morphically universal space of dens ity c . K . Thompson and second author noted that a v ersion of the pro of from [7] g ives the consistency of the nonexistence of a universal Banach space of densit y ω 1 . In [3] M. Bell show ed that assuming κ ω = κ there is a universal uniform Eb erlein compact s pace of w eight κ . M. F abia n, G. Go de fr o y a nd V. Zizler showed that the class o f Banach spaces sa tisfying the hypothesis of P rop o sition 1 .2 corr esponding to the class of uniform E berlein co mpact spaces is the class of UG Banach spaces. So, these tw o results imply b y Propo sition 1.2 that there are universal U G Banach spaces of densit y c = ω 1 . M. Bell also showed in [3] that it is consistent that there ON ISOMORP HI C EMBEDDINGS 3 is no universal Eberlein compact of densit y ω 1 . The main result o f this pap er is to complement the ab o ve result by its B a nac h space version. Actually the full result is considerably stronger: Theorem 1.3. F or κ = ω 1 and for κ = c , it is c onsistent that ther e is n o Banach sp ac e X of density κ su ch that every UG Banach s p ac e of density κ emb e ds into X . In p articular, t her e is no u n iversal Banach s p ac e for the class of UG Banach sp ac es of density κ . As in the case of the class of separ able s paces the notions of univ ersality and isometrical univ ersality do not coincide fo r the class of UG spa c es: a universal UG Banach spa c e ma y not b e an isometrica lly univ ersal UG Bana c h space and this follows from the fact that WCG Ba nac h spa ces can b e renormed to b e strictly conv ex (see [1 ]). The above theorem implies in par ticular tha t, co nsisten tly , ℓ ∞ /c 0 do es not con- tain an isomor phic copy of some UG Banach spaces of densit y c . This is r elated to some recent results of M. Dzamonja and L. Soukup [8] as well as to recent results of S. T o dorcevic ([13]) concerning the co nsisten t existence of C ( K )s for K Cor son compact whic h do not embed into ℓ ∞ /c 0 . Also , the arguments lik e in the pro of of Prop osition 1.2 give that β N \ N cannot b e ma pped onto some co mpact subset of a Hilbert spa ce with dens it y K with the weak top ology . The nota tion is fairly standar d, the Bana c h s paces terminolog y follows [9] and the s et-theoretic terminolog y follows [1 1]. Given a function f , we deno te by G f the g raph o f f . F n <ω ( n, D ) stands for all finite partial functions from { 0 , ...n − 1 } int o D and F n <ω ( ω , D ) (resp. F n ≤ ω ( ω , D )) for all finite (resp. co un table) partial functions fro m ω in to D . I f X is a set, then [ X ] 1 denotes the collection of one- element subsets of X . If A is a Bo olea n a lgebra and a ∈ A , then [ a ] denotes the basic clop en set of the Stone space of A determined by a , that is the set o f a ll ultrafilters of A co ntaining a . A partial o rder has precalibe r ω 1 if and only if every uncountable subset of it includes an uncountable subset wher e every finite subs e t has a lo wer b ound. Definition 1.4. A Bo ole an algebr a A is a (c)-algebr a if i t is gener ate d by elements { A ξ, i : ξ < κ , i < ω } of c ountably many p airwise disjoint antichai ns A i = { A ξ, i : ξ < κ } (c ol le ctions of p airwise disjoint elements of A ) such that (c) A ξ 1 ,i 1 ∨ ... ∨ A ξ m ,i m 6 = 1 A for any distinct p airs ( i 1 , ξ 1 ) , ..., ( i m , ξ m ) ∈ ω × κ . W e use the fact proved in [3] (Theorems 2.1 and 2.2) based on the results of [6] that the Stone space of (c)-alg ebra A is a uniform E b erlein co mpa ct space. The link be t w een uniform Eb erlein compact spaces and UG Bana c h spaces is based on the following: Theorem 1.5 (Theorem 2, [10]) . (1) A Banach sp ac e X admits an e qu ivalent uniformly Gˆ ate aux differ entiable norm if and only i f the dual unit b al l B X ∗ e quipp e d with the we ak ∗ top olo gy is a u niform Eb erlein c omp act. (2) A c omp act sp ac e K is a un ifo rm Eb erlein c omp act if and only if C ( K ) admits an e quivalent uniformly Gˆ ate aux differ entiable norm, if and only if ther e is a Hilb ert sp ac e H and a b oun de d line ar op er ator fr om H onto a dense set in C ( K ) . 4 CHRISTINA BRECH AND PIOTR KOSZMIDER The main idea of the pro ofs is to cons truct a large family of generic c -alg e bras and prove that g iv en a UG Banach space X o ne of the a lgebras is suficiently generic ov er X to pre vent a p ossibility o f an isomor phic e mbedding of the B anac h spa ce C ( K ) where K is the Stone s pace of the a lgebra. The c -a lgebras are c onstructed by par tial order of approximations (see definitions 2 .1 and 3.1) and the method o f forcing is used to mak e conclusions abo ut their consisten t existence (see [11]). 2. Density ω 1 This section is devoted to the pro of of the main result (Theor em 1.3) for the case κ = ω 1 . The follo wing partial order will appr o ximate the generic c -algebras of cardinality ω 1 , one of which will induce a Banach space UG without an isomorphic embedding in to a given B anac h space UG. Definition 2.1. P is the for cing notion c onsisting of c onditions p = ( n p , D p , F p ) wher e n p ∈ ω , D p ∈ [ ω 1 ] <ω and F p ⊆ F n <ω ( n p , D p ) ar e such t hat | F p | < ω and [ n p × D p ] 1 ⊆ F p , or der e d by p ≤ q if n p ≥ n q , D p ⊇ D q , F p ⊇ F q and (P) given f ∈ F p , ther e is g ∈ F q such t hat G f ∩ ( n q × D q ) ⊆ G g . Given a mode l V and a P -gener ic filter G ov er V , f or each ξ ∈ ω 1 and each i ∈ ω , we define in V [ G ] the following set: A ξ, i = { f ∈ F n <ω ( ω , ω 1 ) : ∃ p ∈ G suc h that f ∈ F p and f ( i ) = ξ } . Let B b e the subalgebra o f the Bo olean a lgebra ℘ ( F n <ω ( ω , ω 1 )) ge nerated b y the sets { A ξ, i : ( i , ξ ) ∈ ω × ω 1 } . Lemma 2. 2 . F or every ξ ∈ ω 1 , i ∈ ω and every q ∈ P ther e is p ≤ q such that ξ ∈ D p and i < n p . Pr o of. Let n p = max( i + 1 , n q ) and D q = D p ∪ { ξ } , F p = F q ∪ [ n p × D p ] 1 . Cle a rly p ∈ P . As the gra ph of every new partial function in F p has empty intersection with ( n q × D q ), w e conclude that p ≤ q .  Prop osition 2. 3. In V [ G ] , we have that for every i ∈ ω , ( A ξ, i ) ξ<ω 1 ar e p air- wise disjoint nonempty sets such t hat whenever ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ) ar e di stinct p airs, then A i,ξ \ ( A i 1 ,ξ 1 ∪ ... ∪ A i k ,ξ k ) 6 = ∅ . Ther efor e, the Bo ole an algebr a B gener ate d by them is a c-algebr a of c ar dinality ω 1 and its Stone sp ac e K is a uniform Eb erlein c omp act sp ac e of weight ω 1 . Pr o of. F or all distinct ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ), by 2.2, w e ha ve { ( i, ξ ) } ∈ A i,ξ \ ( A i 1 ,ξ 1 ∪ ... ∪ A i k ,ξ k ) . So each A ξ, i is nonempty a nd the co ndition (c) of Definition 1.4 is satisfied. It also follo ws directly from the fact that f ’s are functions and that ( i, ξ ) ∈ A ξ, i that ( A ξ, i ) ξ<ω 2 are pairwise disjoin t and nonempty .  F or each ( i, ξ ) ∈ ω × ω 1 , let ˙ A ξ, i be a P - na me f or A ξ, i . Definition 2.4. Given p 1 = ( n 1 , D 1 , F 1 ) , p 2 = ( n 2 , D 2 , F 2 ) ∈ P , we say that they ar e isomorphic if n 1 = n 2 and t her e is an or der-pr eserving bije ction e : D 1 → D 2 such t hat e | D 1 ∩ D 2 = id and f or al l f ∈ F n <ω ( ω , ω 1 ) , f ∈ F 1 if and only i f e [ f ] ∈ F 2 , wher e e [ f ]( i ) = e ( f ( i )) . ON ISOMORP HI C EMBEDDINGS 5 Lemma 2. 5 . L et p k = ( n, D k , F k ) in P , for 1 ≤ k ≤ m , b e p airwise isomorphic c onditions su ch t hat ( D k ) 1 ≤ k ≤ m is a ∆ -system with r o ot D . Then, t her e is p ∈ P , p ≤ p 1 , . . . , p m such that for any distinct 1 ≤ i, i ′ ≤ n , any distinct 1 ≤ k, k ′ ≤ m and any ξ ∈ D k \ D and ξ ′ ∈ D k ′ \ D we have p  ˙ A ˇ ξ, i ∩ ˙ A ˇ ξ ′ ,i ′ = ∅ . Pr o of. W e define p = ( n p , D p , F p ) b y letting n p = n , D p = D 1 ∪ · · · ∪ D m and F p = F 1 ∪ · · · ∪ F m . Let us prove that p ∈ P and that p ≤ p k for every 1 ≤ k ≤ m . As fin ite unions of finite sets, D p and F p are countable. Notice that [ n × D p ] 1 = [ n × D 1 ] 1 ∪ · · · ∪ [ n × D m ] 1 ⊆ F 1 ∪ · · · ∪ F m = F p and that F p ⊆ F n <ω ( n, D 1 ) ∪ · · · ∪ F n <ω ( n, D m ) ⊆ F n <ω ( n, D p ), so that p ∈ P . Fix 1 ≤ k ≤ m and let us now show that p ≤ p k . By the definit ion of p , n p = n , D p ⊇ D k and F p ⊇ F k . Given f ∈ F p , (P) of Definition 2 .1 is trivially satisfied if f ∈ F k . So, suppos e f ∈ F k ′ \ F k for some k ′ 6 = k and let e b e th e order- pr eserving bijection fr om D k ′ onto D k . Since p k and p k ′ are isomorphic and f ∈ F k ′ , we get that g = e [ f ] ∈ F k . Let us prov e that G f ∩ ( n k × D k ) = G g ∩ ( n p × D ) ⊆ G g : giv en ( i, ξ ) ∈ G f ∩ ( n k × D k ), we have that f ( i ) = ξ and since f ∈ F k ′ ⊆ n p × D k ′ , we get that ξ ∈ D k ∩ D k ′ = D . Then g ( i ) = e [ f ]( i ) = e ( f ( i )) = e ( ξ ) = ξ , since ξ ∈ D . This prov es t hat G f ∩ ( n k × D k ) ⊆ G g and concludes the pro of that p ≤ p k for all 1 ≤ k ≤ m . Now let 1 ≤ i, i ′ ≤ n b e distinct, 1 ≤ k < k ′ ≤ m and let ξ ∈ D k \ D and ξ ′ ∈ D k ′ \ D . Note that in F p there is no function f suc h that f ( i ) = ξ and f ( i ′ ) = ξ ′ . Let G be a P -g eneric filter suc h that p ∈ G . Given any p ′ ∈ G , there is p ′′ ∈ G suc h that p ′′ ≤ p, p ′ . B y (P) of Definition 2.1, no function g ∈ F p ′′ satisfies g ( i ) = ξ and g ( i ′ ) = ξ ′ , hence there is no such function in p ′ . It follows from the definition of A ξ, i ’s that A ξ, i ∩ A ξ ′ ,i ′ = ∅ in V [ G ]. Since G was an arbitrar y P - generic filter cont aining p , we conclude that p  ˙ A ˇ ξ, i ∩ ˙ A ˇ ξ ′ ,i ′ = ∅ .  Theorem 2.6 . P has pr e c alib er ω 1 and henc e sat isfies the c.c.c. Pr o of. Let ( p α ) α<ω 1 ⊆ P , where each p α = ( n α , D α , F α ). By the ∆ -sys tem lemma, we ma y assume that ( D α ) α<ω 1 is a ∆-sys tem with ro ot D . By standar d co un ting arguments, we may ass ume w itho ut loss of gener alit y that n α = n for every α < ω 1 and some fixed n ∈ ω . By thinning o ut using further c o un ting arg umen ts, we can assume that for ev ery α < β < ω 1 , p α and p β are isomorphic. Now, given α 1 < ... < α n < ω 1 , b y Lemma 2.5 there is p ≤ p α 1 , ...p α n .  Lemma 2. 7 . L et p k = ( n, D k , F k ) in P , for 1 ≤ k ≤ m , b e p airwise isomorphic c onditions such that ( D k ) 1 ≤ k ≤ m is a ∆ -system with r o ot D . Given, for al l 1 ≤ k ≤ m , ξ k ∈ D k \ D and distinct i k < n , ther e is p ∈ P , p ≤ p 1 , . . . , p m such t hat p  ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m 6 = ∅ . Pr o of. W e define p = ( n p , D p , F p ) b y letting n p = n , D p = D 1 ∪ · · · ∪ D m and F p = F 1 ∪ · · · ∪ F m ∪ { f 0 } , where f 0 = { ( i 1 , ξ 1 ) , . . . , ( i m , ξ m ) } . L e t us chec k that p ∈ P and that p ≤ p k for every 1 ≤ k ≤ m . As finite unions of finite sets, D p and F p are finite. Notice that [ n × D p ] 1 = [ n × D 1 ] 1 ∪ · · · ∪ [ n × D m ] 1 ⊆ F 1 ∪ · · · ∪ F m ⊆ F p and that F p \ { f 0 } ⊆ F n <ω ( n, D 1 ) ∪ · · · ∪ 6 CHRISTINA BRECH AND PIOTR KOSZMIDER F n <ω ( n, D m ) ⊆ F n <ω ( n, D p ). Since f 0 ∈ F n <ω ( n, D 1 ∪ · · · ∪ D m ) ⊆ F n <ω ( n, D p ), we get that p ∈ P . Fix 1 ≤ k ≤ m and let us no w verify that p ≤ p k . By the definition of p , n p = n , D p ⊇ D k and F p ⊇ F k . T o c heck (P) of Definition 2.1, given f ∈ F p , let us consider three cases: Case 1 . If f ∈ F k , then (P) is trivially satisfied. Case 2. If f ∈ F k ′ \ F k for some k ′ 6 = k , let e b e the order -preserving bijection from D k ′ onto D k . Since p k and p k ′ are isomorphic and f ∈ F k ′ , w e g e t that g = e [ f ] ∈ F k . Let us prov e that G f ∩ ( n k × D k ) = G g ∩ ( n p × D ) ⊆ G g : given ( i, ξ ) ∈ G f ∩ ( n k × D k ), we have that f ( i ) = ξ and since f ∈ F k ′ ⊆ n p × D k ′ , we get that ξ ∈ D k ∩ D k ′ = D . Then g ( i ) = e [ f ]( i ) = e ( f ( i )) = e ( ξ ) = ξ , since ξ ∈ D . This prov es t hat G f ∩ ( n k × D k ) ⊆ G g , as w e wan ted. Case 3. If f = f 0 = { ( i 1 , ξ 1 ) , . . . , ( i m , ξ m ) } , then G f ∩ ( n k × D k ) = { ( i k , ξ k ) } ⊆ F k by Definition 2.1. Finally , since f 0 ∈ F p , p forces that ˇ f 0 ∈ ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m , so that p  ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m 6 = ∅ , which concludes the pro of.  Definition 2.8. Σ is the pr o duct of ω 2 c opies of P , with fin ite supp orts, t hat is, Σ = { σ : dom ( σ ) → P : dom ( σ ) ∈ [ ω 2 ] <ω } , or der e d by σ 1 ≤ σ 2 if dom ( σ 1 ) ⊇ dom ( σ 2 ) and for every α ∈ dom ( σ 2 ) , σ 1 ( α ) ≤ σ 2 ( α ) . Given A ⊆ ω 2 , let Σ A = { σ ∈ Σ : dom ( σ ) ⊆ A } . Prop osition 2.9. Given γ 0 ∈ ω 2 , if A = ω 2 \ { γ 0 } , then the for cing Σ A is isomor- phic to Σ . Pr o of. Lift to P a bijection betw een ω 2 and ω 2 \ { γ 0 } .  Theorem 2.10. V Σ  “ther e is n o Banach sp ac e X of density ω 1 < c such that for every uniform Eb erlein c omp act sp ac e K of weight at m ost ω 1 , C ( K ) c an b e isomorphi c al ly emb e dde d into X ”. Pr o of. W e work in V . By co n tradiction, s uppose that s uc h a Bana c h space exists in V Σ and let ˙ X be a Σ-name for it. Since Σ forces that ˙ X ha s density ω 1 , let ( ˙ v η ) η<ω 1 be a family of Σ-names such tha t Σ  { ˙ v η : η < ω 1 } is a dense subset 1 of ˙ X . F or each F ∈ [ ω 1 ] <ω , let A F ⊆ Σ be a maxima l antic hain in Σ such that for every σ ∈ A F , either σ  k X η ∈ F ˙ v η k > 2 or σ  k X η ∈ F ˙ v η k ≤ 2 . Since Σ is ccc, each A F is co un table a nd for ea c h σ ∈ A F , | dom ( σ ) | < ω . Then, the set [ { dom ( σ ) : σ ∈ A F for some F ∈ [ ω 1 ] <ω } 1 Unfortunately , th e Banac h space ˙ X will not b elong to any i ntermediate model, so we cannot mak e use of an y factorization of Σ unless we ar e willing to deal wi th normed non-complete spaces o ver the rationals and approximations (possibly nonlinear) of linear oper ators int o its completion. W e hav e choosen a simpler approac h, in our opini on, and work in V wi th the ful l pro duct Σ. ON ISOMORP HI C EMBEDDINGS 7 has cardina lit y at most ω 1 . Fix γ 0 ∈ ω 2 such that for every F ∈ [ ω 1 ] <ω and every σ ∈ A F , γ 0 / ∈ dom ( σ ). Since Σ ∼ Σ A × Σ { γ 0 } ∼ Σ A × P , let us consider in V Σ A × Σ { γ 0 } , the Stone space K γ 0 of the a lgebra generated by the family ( A ξ, i ( γ 0 )) ξ ∈ ω 1 ,i ∈ ω added by Σ { γ 0 } ∼ P ov er V Σ A . By Prop osition 2.3, K γ 0 is a uniform Eber lein compa ct of weight ω 1 and let us show that C ( K γ 0 ) do es not embed isomorphica lly in to ˙ X . F or each ( i, ξ ) ∈ ω × ω 1 , let ˙ A ξ, i be a Σ-name for A ξ, i ( γ 0 ). Suppo se there is ˙ T : C ( K γ 0 ) → ˙ X a n isomor phic embedding and without loss o f generality , assume that Σ  k ˙ T k = 1. Let p 0 ∈ Σ, now find p 1 ≤ p 0 and m ∈ ω such that p 1  3 · k ˙ T − 1 k < ˇ m . F or ea c h α < ω 1 , let { ξ k ( α ) : 1 ≤ k ≤ m } ⊆ ω 1 \ α have cardinalit y m and take σ α ∈ Σ with σ α ≤ p 1 and η 1 ( α ) , . . . η m ( α ) ∈ ω 1 such that σ α  ∀ 1 ≤ k ≤ m k ˙ T ( χ ˙ A ˇ ξ k ( α ) ,k ( γ 0 ) ) − ˙ v ˇ η k ( α ) k < 1 m . Let s α = σ α | ω 2 \{ γ 0 } ∈ Σ ω 2 \{ γ 0 } and σ α ( γ 0 ) = ( n α , D α , F α ). Without loss of gener - ality , w e may assume that ξ 1 ( α ) , . . . , ξ m ( α ) ∈ D α . Using the ∆- s ystem lemma, w e may assume that ( D α ) α<ω 1 is a ∆-system with ro ot D and that ξ 1 ( α ) , . . . , ξ m ( α ) ∈ D α \ D for each α < ω 1 and for all α < ω 1 the conditions σ α ( γ 0 ) are pa irwise isomorphic. By Prop osition 2 .9, Σ ω 2 \{ γ 0 } is isomorphic to Σ a nd since, by the pro ductivit y of precalib er ω 1 and Lemma 2 .6 Σ ha s preca liber ω 1 , so do es Σ ω 2 \{ γ 0 } . So, given α 1 < · · · < α m , there is s ∈ Σ ω 2 \{ γ 0 } such that s ≤ s α 1 , . . . , s α m . Let F = { η 1 ( α 1 ) , . . . , η m ( α m ) } and since A F is a maximal an tichain in Σ whic h is contained in Σ ω 2 \{ γ 0 } , A F is also a maximal antic hain in Σ ω 2 \{ γ 0 } . Then, there is s ′ ∈ A F and s ′′ ∈ Σ ω 2 \{ γ 0 } such that s ′′ ≤ s, s ′ . Since s ′′ ≤ s ′ and s ′ ∈ A F , either s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ 2 or s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 2 . Let us no w get a con tradiction. Case 1 . s ′′ forces that k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 2. In this case, let p ∈ P be such that p ≤ σ α 1 ( γ 0 ) , . . . , σ α m ( γ 0 ) o btained b y Lemma 2.5 and let σ = ( s ′′ , p ) ∈ Σ. Then, σ ≤ s ′′ , p and σ ≤ σ α 1 , . . . , σ α m . So, σ  ∀ 1 ≤ k < k ′ ≤ m ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 ) ∩ ˙ A ˇ ξ k ′ ( α k ′ ) ,i k ′ ( γ 0 ) = ∅ so that σ  k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k = 1 . Since σ ≤ σ α 1 , . . . , σ α m , σ  ∀ 1 ≤ k ≤ m k ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) − ˙ v ˇ η k ( α k ) k < 1 m , so that, using the fact that Σ  k ˙ T k = 1, w e get that σ  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ k m X k =1 ( ˙ v ˇ η k ( α k ) − ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] )) + m X k =1 ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) k ≤ k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k + m · 1 m ≤ 2 , 8 CHRISTINA BRECH AND PIOTR KOSZMIDER contradicting the fact that σ ≤ s ′′ and s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 2 . Case 2 . s ′′ forces that k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ 2. In this case, let p ∈ P be such that p ≤ σ α 1 ( γ 0 ) , . . . , σ α m ( γ 0 ) o btained b y Lemma 2.7 and let σ = ( s ′′ , p ) ∈ Σ. Then, σ ≤ s ′′ , p and σ ≤ σ α 1 , . . . , σ α m . So, σ  ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 ) ∩ · · · ∩ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 ) 6 = ∅ so that σ  k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k = m. Since σ ≤ σ α 1 , . . . , σ α m , σ  ∀ 1 ≤ k ≤ m k ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) − ˙ v ˇ η k ( α k ) k < 1 m , so that, using the fact that σ ≤ s ′′ , we get that σ  k ˙ T ( χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ] ) k ≤ k m X k =1 ( ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) − ˙ v ˇ η k ( α k ) ) + m X k =1 ˙ v ˇ η k ( α k ) k ≤ k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k + m · 1 m ≤ 3 , which contradicts the fact that Σ  3 · k ˙ T − 1 k < m . Since the co ndition p 0 ∈ Σ was ar bitrary , we s howed that a dense subset o f Σ forces the nonexistence of the em b edding ˙ T , hence it do es not exist in V [ G ].  3. Density c This section is dev oted to the pro of o f the main result 1.3 fo r the c ase κ = c . The following par tial order will approximate the generic c -alge br as of cardinality ω 2 = c , one of which will induce a Banach space UG without an iso morphic embedding in to a given Bana c h space UG . Definition 3.1 . Q is t he for cing notion of c onditions q = ( D q , F q ) wher e D q ∈ [ ω 2 ] ≤ ω and F q ⊆ F n ≤ ω ( ω , D q ) ar e such t hat | F q | ≤ ω and [ ω × D q ] 1 ⊆ F q , or der e d by p ≤ q if D p ⊇ D q , F p ⊇ F q and (Q) given f ∈ F p , ther e is g ∈ F q such t hat G f ∩ ( ω × D q ) ⊆ G g . Prop osition 3.2. Q is σ -close d. Pr o of. Let ( q n ) n ∈ ω ⊆ Q be suc h that q n +1 ≤ q n for all n ∈ ω , where each q n = ( D n , F n ). Define q = ( D q , F q ) by D q = S n ∈ ω D n and F q = S n ∈ ω F n . As countable unions of countable sets, D q and F q are countable. Also , [ ω × D q ] 1 = S n ∈ ω ([ ω × D n ] 1 ) ⊆ S n ∈ ω F n = F q and F q ⊆ S n ∈ ω F n ≤ ω ( ω , D n ) ⊆ F n ≤ ω ( ω , D q ), so that q ∈ Q . Fix n ∈ ω and let us now prove that q ≤ q n . Clearly D q ⊇ D n and F q ⊇ D n . T o prov e (Q) of Definition 3.1, given f ∈ F q , there is k ∈ ω such that f ∈ F k . If k ≤ n , then q n ≤ q k , so that f ∈ F k ⊆ F n and we are done. If k > n , since q k ≤ q n , there is g ∈ F n such that G f ∩ ( ω × D n ) ⊆ G g . If suffices now to notice that g ∈ F n ⊆ F q .  Lemma 3.3. F or every ξ ∈ ω 2 and every q ∈ Q ther e is p ≤ q su ch that ξ ∈ D p . ON ISOMORP HI C EMBEDDINGS 9 Pr o of. Let D q = D p ∪ { ξ } , F p = F q ∪ [ ω × { ξ } ] 1 . Clearly p ∈ P . As the gra phs of all new pa rtial functions in F p hav e empt y in ter sections with ( ω × D q ) we conclude that p ≤ q .  Given a mo del V and a Q -g eneric filter G ov er V , for each ξ ∈ ω 2 and each i ∈ ω , we define in V [ G ] the following set: A ξ, i = { f ∈ F n ≤ ω ( ω , ω 2 ) : ∃ q ∈ G suc h that f ∈ F q and f ( i ) = ξ } . Let B b e the Bo olean algebra generated b y { A ξ, i : ( i, ξ ) ∈ ω × ω 2 } . Prop osition 3. 4. In V [ G ] , we have that for every i ∈ ω , ( A ξ, i ) ξ<ω 2 ar e p air- wise disjoint nonempty sets such t hat whenever ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ) ar e di stinct p airs, then A i,ξ \ ( A i 1 ,ξ 1 ∪ ... ∪ A i k ,ξ k ) 6 = ∅ . Ther efor e, the Bo ole an algebr a B gener ate d by them is a c-algebr a of c ar dinality ω 2 and its Stone sp ac e K is a uniform Eb erlein c omp act sp ac e of weight ω 2 . Pr o of. By 3.3, for all distinct ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ) we ha ve { ( i, ξ ) } ∈ A i,ξ \ ( A i 1 ,ξ 1 ∪ ... ∪ A i k ,ξ k ) . So each A ξ, i is nonempty a nd the co ndition (c) of Definition 1.4 is satisfied. It also follo ws directly from the fact that f ’s are functions and that ( i, ξ ) ∈ A ξ, i that ( A ξ, i ) ξ<ω 2 are pairwise disjoin t.  F or each ( i, ξ ) ∈ ω × ω 2 , let ˙ A ξ, i be a Q - name for A ξ, i . Definition 3.5. Given q 1 = ( D 1 , F 1 ) , q 2 = ( D 2 , F 2 ) ∈ Q , we say that t hey ar e iso- morphic if ther e is an or der-pr eserving bije ction e : D 1 → D 2 such that e | D 1 ∩ D 2 = id and for al l f ∈ F n ≤ ω ( ω , ω 2 ) , f ∈ F 1 if and only if e [ f ] ∈ F 2 , wher e e [ f ]( i ) = e ( f ( i )) . Lemma 3.6. L et q k = ( D k , F k ) in Q , for 1 ≤ k ≤ m , b e p airwise isomorphic c onditions such that ( D k ) 1 ≤ k ≤ m is a ∆ -syst em with a c ountable r o ot D . Then, ther e is q ∈ Q , q ≤ q 1 , . . . , q m such t hat for any distinct i, i ′ ∈ ω , any distinct 1 ≤ k , k ′ ≤ m and any ξ ∈ D k \ D and ξ ′ ∈ D k ′ \ D we have q  ˙ A ˇ ξ, i ∩ ˙ A ˇ ξ ′ ,i ′ = ∅ . Pr o of. W e define q = ( D q , F q ) by letting D q = D 1 ∪ · · · ∪ D m and F q = F 1 ∪ · · · ∪ F m . Let us prov e that q ∈ Q and that q ≤ q k for every 1 ≤ k ≤ m . As finite unions of coun table sets, D q and F q are coun table. Notice that ω × D q = ( ω × D 1 ) ∪ · · · ∪ ( ω × D m ) ⊆ F 1 ∪ · · · ∪ F m = F q and that F q ⊆ F n ≤ ω ( ω , D 1 ) ∪ · · · ∪ F n ≤ ω ( ω , D m ) ⊆ F n ≤ ω ( ω , D q ), so that q ∈ Q . Fix 1 ≤ k ≤ m and let us now show that q ≤ q k . By the definition of q , D q ⊇ D k and F q ⊇ F k . Giv en f ∈ F q , (Q) of Definition 3.1 is trivia lly satisfied if f ∈ F k . So, supp ose f ∈ F k ′ \ F k for some k ′ 6 = k and let e b e the orde r -preserving bijection from D k ′ onto D k . Since q k and q k ′ are isomor phic and f ∈ F k ′ , w e get that g = e [ f ] ∈ F k . Let us prove that G f ∩ ( ω × D k ) = G g ∩ ( ω × D ) ⊆ G g : given ( i, ξ ) ∈ G f ∩ ( ω × D k ), w e hav e that f ( i ) = ξ and since f ∈ F k ′ ⊆ ω × D k ′ , we get that ξ ∈ D k ∩ D k ′ = D . Then g ( i ) = e [ f ]( i ) = e ( f ( i )) = e ( ξ ) = ξ , since ξ ∈ D . This prov es t hat G f ∩ ( ω × D k ) ⊆ G g and concludes the proo f. Now let i, i ′ ∈ ω be distinct, 1 ≤ k < k ′ ≤ m and let ξ ∈ D k \ D and ξ ′ ∈ D k ′ \ D . Note that in F q there is no function f such that f ( i ) = ξ and f ( i ′ ) = ξ ′ . Let G be a Q -gener ic filter such that q ∈ G . Giv en any q ′ ∈ G , there is q ′′ ∈ G suc h 10 CHRISTINA BRECH AND PIOTR KOSZMIDER that q ′′ ≤ q , q ′ . By Definition 3.1 (Q), no function g ∈ F q ′′ satisfies g ( i ) = ξ and g ( i ′ ) = ξ ′ , hence there is no such function in F q ′ . It follows from the definition of A ξ, i ’s that A ξ, i ∩ A ξ ′ ,i ′ = ∅ in V [ G ]. Since G was an ar bitrary Q -generic filter containing q , we conclude that q  ˙ A ˇ ξ, i ∩ ˙ A ˇ ξ ′ ,i ′ = ∅ .  Theorem 3.7. Assuming CH, given ( q α ) α<ω 2 ⊆ Q , ther e is A ∈ [ ω 2 ] ω 2 such that for every α 1 < · · · < α m ∈ A , ther e is q ∈ Q , q ≤ q α 1 , . . . q α m . In p articular, Q satisfies the ω 2 -c.c. Pr o of. Let ( q α ) α<ω 2 ⊆ Q , where each q α = ( D α , F α ). Using CH, by the ∆-sys tem lemma for countable s ets (see [11]), there is A ′ ∈ [ ω 2 ] ω 2 such that ( D α ) α ∈ A is a ∆- system with countable ro ot D . By thinning out using C H and c o un ting arguments, in pa rticular the fact that there ar e c = ω 1 countable sets of the set of D ω of cardinality c = ω 1 , there is A ∈ [ A ′ ] ω 2 such that for ev ery α, β ∈ A , q α and q β are isomorphic. No w, giv en α 1 < · · · < α m ∈ A , by Lemma 2.5 there is q ∈ Q , q ≤ q α 1 , . . . , q α m , which concludes the pro of.  Lemma 3.8. L et q k = ( D k , F k ) in Q , for 1 ≤ k ≤ m , b e p airwise isomorphic c onditions such that ( D k ) 1 ≤ k ≤ m is a ∆ -system with a c ountable r o ot D . Given, for al l 1 ≤ k ≤ m , ξ k ∈ D k \ D and distinct i k ∈ ω , ther e is q ∈ Q , q ≤ q 1 , . . . , q m such that q  ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m 6 = ∅ . Pr o of. W e define q = ( D q , F q ) b y letting D q = D 1 ∪ · · · ∪ D m and F q = F 1 ∪ · · · ∪ F m ∪ { f 0 } , where f 0 = { ( i 1 , ξ 1 ) , . . . , ( i m , ξ m ) } . Let us chec k tha t q ∈ Q and tha t q ≤ q k for every 1 ≤ k ≤ m . As finite unions of countable sets, D q and F q are co un table. Notice that [ ω × D q ] 1 = [ ω × D 1 ] 1 ∪ · · · ∪ [ ω × D m ] 1 ⊆ F 1 ∪ · · · ∪ F m ⊆ F q and that F q \ { f 0 } ⊆ F n ≤ ω ( ω , D 1 ) ∪ · · · ∪ F n ≤ ω ( ω , D m ) ⊆ F n ≤ ω ( ω , D q ). Since f 0 ∈ F n ≤ ω ( ω , D 1 ∪ · · · ∪ D m ) ⊆ F n ≤ ω ( ω , D q ), w e get that q ∈ Q . Fix 1 ≤ k ≤ m and let us now v erify that q ≤ q k . By the definition of q , D q ⊇ D k and F q ⊇ F k . T o chec k (Q) o f Definition 3.1, g iv en f ∈ F q , let us consider three cases: Case 1 . If f ∈ F k , then (Q) is trivially satisfied. Case 2. If f ∈ F k ′ \ F k for some k ′ 6 = k , let e b e the order -preserving bijection from D k ′ onto D k . Since q k and q k ′ are isomor phic and f ∈ F k ′ , w e get that g = e [ f ] ∈ F k . Let us prove that G f ∩ ( ω × D k ) = G g ∩ ( ω × D ) ⊆ G g : given ( i, ξ ) ∈ G f ∩ ( ω × D k ), w e hav e that f ( i ) = ξ and since f ∈ F k ′ ⊆ ω × D k ′ , we get that ξ ∈ D k ∩ D k ′ = D . Then g ( i ) = e [ f ]( i ) = e ( f ( i )) = e ( ξ ) = ξ , since ξ ∈ D . This prov es t hat G f ∩ ( ω × D k ) ⊆ G g , as w e wan ted. Case 3. If f = f 0 = { ( i 1 , ξ 1 ) , . . . , ( i m , ξ m ) } , then G f ∩ ( ω × D k ) = { ( i k , ξ k ) } ⊆ F k by Definition 3.1. Finally , since f 0 ∈ F q , q fo r ces that ˇ f 0 ∈ ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m and w e get that q  ˙ A ˇ ξ 1 ,i 1 ∩ · · · ∩ ˙ A ˇ ξ m ,i m 6 = ∅ , which concludes the pro of.  ON ISOMORP HI C EMBEDDINGS 11 Definition 3.9. Π is the pr o duct of ω 3 c opies of Q , with c ountable supp orts, that is: Π = { π : dom ( π ) → Q : dom ( π ) ∈ [ ω 3 ] ≤ ω } , or der e d by π 1 ≤ π 2 if dom ( π 1 ) ⊇ dom ( π 2 ) and for every α ∈ dom ( π 2 ) , π 1 ( α ) ≤ Q π 2 ( α ) . Given A ⊆ ω 3 , let Π A = { π ∈ Π : dom ( π ) ⊆ A } . R is the pr o duct of Cohen for cing adding ω 2 Cohen re als with Π , that is, R = F n <ω ( ω 2 , 2) × Π . Prop osition 3.10. Given γ 0 ∈ ω 3 , if A = ω 3 \ { γ 0 } , then the for cing Π A is isomorphi c to Π and, ther efor e, R = F n <ω ( ω 2 , 2) × Π ∼ F n <ω ( ω 2 , 2) × Π A × Q ∼ R × Q . Pr o of. Lift a bijection b et ween ω 3 and ω 3 \ { γ 0 } to an isomorphis m of Π and Π A .  Given a mo del V of CH and an R -generic filter G ov er V , for each γ 0 ∈ ω 3 , eac h ξ ∈ ω 2 and each i ∈ ω , w e define in V [ G ] the follo wing set: A ξ, i ( γ 0 ) = { f ∈ F n ≤ ω ( ω , ω 2 ) ∩ V : ∃ r = ( C, π ) ∈ G such that f ∈ F π ( γ 0 ) and f ( i ) = ξ } . Let B γ 0 be a subalg ebra of the Boo lean algebra ℘ ( F n ≤ ω ( ω , ω 2 )) g enerated by { A ξ, i ( γ 0 ) : ( i, ξ ) ∈ ω × ω 2 } . Prop osition 3.11. In V [ G ] , we have that for every γ 0 ∈ ω 3 and every i ∈ ω , ( A ξ, i ( γ 0 )) ξ<ω 2 ar e p airwise disjoint nonempty sets such that whenever ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ) ar e distinct p airs, then A i,ξ ( γ 0 ) \ ( A i 1 ,ξ 1 ( γ 0 ) ∪ ... ∪ A i k ,ξ k ( γ 0 )) 6 = ∅ . Ther efor e, the Bo ole an algeb r a B γ 0 gener ate d by them is a c-algebr a of c ar dinality ω 2 and its Stone sp ac e K γ 0 is a u niform Eb erlein c omp act sp ac e of weight ω 2 . Pr o of. Fix γ 0 ∈ ω 3 . By 3.3 for all distinct ( i 1 , ξ 1 ) , ..., ( i k , ξ k ) , ( i , ξ ) we ha ve { ( i, ξ ) } ∈ A i,ξ ( γ 0 ) \ ( A i 1 ,ξ 1 ( γ 0 ) ∪ ... ∪ A i k ,ξ k ( γ 0 )) . So eac h A ξ, i ( γ 0 ) is nonempt y and the condition (c) o f Definition 1.4 is satisfied. It follows directly from the fact that f ’s are functions and that ( i, ξ ) ∈ A ξ, i ( γ 0 ) that ( A ξ, i ( γ 0 )) ξ<ω 2 are pairwise disjoin t.  Since now we work in V . F or each γ 0 ∈ ω 3 and each ( i, ξ ) ∈ ω × ω 2 , let ˙ A ξ, i ( γ 0 ) be a n R -name fo r A ξ, i ( γ 0 ). Lemma 3.12. Ass u ming CH, given ( r α ) α<ω 2 ⊆ R , ther e is A ∈ [ ω 2 ] ω 2 such that for every α 1 < · · · < α m ∈ A , ther e is r ∈ R , r ≤ r α 1 , . . . r α m . In p articular, R is ω 2 -c c. Pr o of. Let ( r α ) α<ω 2 ⊆ R , where each r α = ( C α , π α ). Let B α = dom ( π α ) ∈ [ ω 3 ] ≤ ω . Using CH and the ∆-sy s tem lemma for coun table sets , there is A ′ ∈ [ ω 2 ] ω 2 such that ( B α ) α ∈ A ′ is a ∆-system with coun table ro ot B and that ( dom ( C α )) α ∈ A ′ is a ∆-system with finite ro ot ∆. Using another sta ndard co un ting arg umen t we may as sume that C α | ∆ = C β | ∆ for an y α < β ∈ A ′ . By thinning out using further coun ting argumen ts and CH, there is A ∈ [ A ′ ] ω 2 such that for ev ery ξ ∈ B , ( D r α ( ξ ) )) α ∈ A is a ∆-sys tem with countable ro ot D ξ and ( π α ( ξ )) α ∈ A are pair wise isomorphic. 12 CHRISTINA BRECH AND PIOTR KOSZMIDER Fix α 1 < · · · < α m ∈ A . Let us define r = ( C r , π r ): put C r = C α 1 ∪ · · · ∪ C α m and dom ( π r ) = B α 1 ∪ · · · ∪ B α m . F or ea c h ξ ∈ B , let π r ( ξ ) ∈ Q b e the condition given by Lemma 3.6, such that π r ( ξ ) ≤ π α 1 ( ξ ) , . . . , π α m ( ξ ). If ξ ∈ B α k \ B , let π r ( ξ ) = π α k ( ξ ). Clearly r ∈ R and C r ≤ C α k by the definition, for all 1 ≤ k ≤ m . Also, b oth f or ξ ∈ B a nd fo r ξ ∈ B α k \ B , π r ( ξ ) ≤ π α k ( ξ ), so that r ≤ r α k for all 1 ≤ k ≤ m .  Theorem 3.1 3. If C H holds in V , then V R  “ther e is no Banac h sp ac e X of density c = ω 2 such that for every un ifo rm Eb erlein c omp act sp ac e K of weight at most c , C ( K ) c an b e isomorphic al ly emb e dde d into X ”. Pr o of. First note that R  c = ω 2 . By the pro duct lemma ([11]), for any g eneric G ⊆ R , the extension V [ G ] is of the for m V [ H 1 ][ H 2 ] where H 1 is a Π-gener ic ov er V and H 2 is ˇ F n <ω ( ω 2 , 2)-generic ov er V [ H 1 ]. The fir st forcing Π satisfies the ω 2 -c.c. by Lemma 3.12 a nd is σ -closed by Pro position 3.2, so V [ H 1 ] has the same cardinals as in V and satisfies the CH. The seco nd forcing is equal to F n <ω ( ω 2 , 2) s ince finite functions ar e the same in V and V [ H 1 ]. Hence V [ G ] can b e viewed as an extension of a mo del of CH by the Cohen forcing whic h adds ω 2 Cohen reals. It is well-known (see [11]) that c = ω 2 holds in suc h mo dels. W e work in V . By contradiction, supp ose that in V R there is a Bana c h space which contains an isomorph of ev ery UG Ba nac h spa ce of densit y ω 2 . Let ˙ X b e an R -name for it. Since R  | d ( ˙ X ) | = c = ω 2 , we get that R  | ˙ X | = c ω = ω 2 , so that there is a family of R -names ( ˙ v η ) η<ω 2 such that R  ˙ X = { ˙ v η : η < ω 2 } . F or each F ∈ [ ω 2 ] <ω , let A F ⊆ R be a maxima l antic hain in R such that for every r ∈ A F , either r  k X η ∈ F ˙ v η k > 1 or r  k X η ∈ F ˙ v η k ≤ 1 . Since R is ω 2 -cc, each A F has car dina lit y at mo st ω 1 and for each ( C, π ) ∈ A F , | dom ( π ) | ≤ ω . Then, the set [ { dom ( π ) : ( C, π ) ∈ A F for some F ∈ [ ω 2 ] <ω and C ∈ F n <ω ( ω 2 , 2) } has cardina lit y at most ω 2 . Fix γ 0 ∈ ω 3 such that for every F ∈ [ ω 2 ] <ω and every ( C, π ) ∈ A F , γ 0 / ∈ dom ( π ). In V R , let K γ 0 be the Sto ne space of the c-algebr a generated b y the family ( A ξ, i ( γ 0 )) ξ ∈ ω 2 ,i ∈ ω and let us show that C ( K γ 0 ) do es not embed isomorphically into ˙ X . F or each ( i, ξ ) ∈ ω × ω 2 , let ˙ A ξ, i be a n R -name fo r A ξ, i ( γ 0 ). Suppo se there is ˙ T : C ( K γ 0 ) → ˙ X a n isomor phic embedding and without loss o f generality , assume that R  k ˙ T k = 1. Let r ′ ∈ R and find r ′′ ≤ r ′ and m ∈ ω such that r ′′  k ˙ T − 1 k < ˇ m . F or ea c h α < ω 2 , let { ξ k ( α ) : 1 ≤ k ≤ m } ⊆ ω 2 \ α have cardinalit y m and take r α ∈ R with r α ≤ r ′′ and η 1 ( α ) , . . . η m ( α ) ∈ ω 2 such that r α  ∀ 1 ≤ k ≤ m ˙ T ( χ [ ˙ A ˇ ξ k ( α ) ,k ( γ 0 )] ) = ˙ v ˇ η k ( α ) . Let r α = ( C α , π α ), s α = ( C α , π α | ω 2 \{ γ 0 } ) ∈ F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } and π α ( γ 0 ) = ( D α , F α ). Without loss of generality , by 3 .3 we may assume tha t ξ 1 ( α ) , . . . , ξ m ( α ) ∈ D α . Using CH and the ∆-system lemma for co un table sets, we may a ssume that ( D α ) α<ω 2 is a ∆-system with a countable ro ot D and that ξ 1 ( α ) , . . . , ξ m ( α ) ∈ D α \ D ON ISOMORP HI C EMBEDDINGS 13 for each α < ω 2 and that for e a c h α < ω 2 the conditions π α ( γ 0 ) are is omorphic in the sense of Definition 3.5. By P rop o sition 3 .1 0, F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } is is o morphic to R . So, g iv en α 1 < · · · < α m , w e can apply Lemma 3 .12 to s α 1 , . . . , s α m and get s ∈ F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } such that s ≤ s α 1 , . . . , s α m . Let F = { η 1 ( α 1 ) , . . . , η m ( α m ) } and since A F is a maximal an tichain in R whic h is cont ained in F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } , A F is also a maximal antic hain in F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } . Then, there is s ′ ∈ A F and s ′′ ∈ F n ( ω 2 , 2) × Π ω 2 \{ γ 0 } such that s ′′ ≤ s, s ′ . Since s ′′ ≤ s ′ and s ′ ∈ A F , either s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ 1 or s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 1 . Case 1 . s ′′ forces that k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 1. In this case, let q ∈ Q be s uc h that q ≤ π α 1 ( γ 0 ) , . . . , π α m ( γ 0 ) obtained by Lemma 3.6 and let r = ( s ′′ , q ) ∈ R . Then, r ≤ s ′′ , r ≤ r α 1 , . . . , r α m and r ≤ q ≤ π α 1 ( γ 0 ) , . . . , π α m ( γ 0 ). So, r  ∀ 1 ≤ k < k ′ ≤ m ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 ) ∩ ˙ A ˇ ξ k ′ ( α k ′ ) ,i k ′ ( γ 0 ) = ∅ so that r  k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k = 1 . Since r ≤ r α 1 , . . . , r α m , r  ∀ 1 ≤ k ≤ m ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) = ˙ v ˇ η k ( α k ) , so that, using the fact that R  k ˙ T k = 1, w e get that r  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k = 1 , contradicting the fact that r ≤ s ′′ and s ′′  k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k > 1 . Case 2 . s ′′ forces that k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k ≤ 1. In this case, let q ∈ Q be s uc h that q ≤ π α 1 ( γ 0 ) , . . . , π α m ( γ 0 ) obtained by Lemma 3.8 and let r = ( s ′′ , q ) ∈ R . Then, r ≤ s ′′ , r ≤ r α 1 , . . . , r α m and r ≤ q ≤ π α 1 ( γ 0 ) , . . . , π α m ( γ 0 ). So, r  ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 ) ∩ · · · ∩ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 ) 6 = ∅ so that r  k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ( γ 0 )] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ( γ 0 )] k = m. Since r ≤ r α 1 , . . . , r α m , r  ∀ 1 ≤ k ≤ m ˙ T ( χ [ ˙ A ˇ ξ k ( α k ) ,i k ( γ 0 )] ) = ˙ v ˇ η k ( α k ) , so that, using the fact that R  k ˙ T − 1 k < m and that r ≤ s ′′ , we get that r  k χ [ ˙ A ˇ ξ 1 ( α 1 ) ,i 1 ] + · · · + χ [ ˙ A ˇ ξ m ( α m ) ,i m ] k ≤ k ˙ T − 1 k · k ˙ v ˇ η 1 ( α 1 ) + · · · + ˙ v ˇ η m ( α m ) k < m, which contradicts our assumption. Since the conditio n r ′ ∈ R was a rbitrary , we show ed that a dense subset of R forces the nonexistence of the em b edding ˙ T , hence it do es not exist in V [ G ].  14 CHRISTINA BRECH AND PIOTR KOSZMIDER References 1. D. Amir and J. Lindenstrauss, The Structur e of We akly Comp act Sets in Banach Sp ac es Ann. of Math. 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Dep a r t amento de Ma tem ´ atica, Instituto de M a tem ´ atica e Est a t ´ ıstica, Universid ade de S ˜ ao P aulo, Caixa Post al 66281, 05314-97 0, S ˜ ao P aulo, Brazil E-mail addr ess : christina.brech @gmail.com Institute of Ma thema tics, Polish Acad emy of Sciences, ul. ´ Sniadeckich 8, 00-956 W arsza w a, Poland Institute of Mat hemat ics, Technical Univ ersity of L ´ od ´ z, ul. W ´ olcza ´ nska 2 15, 90-924 L ´ od ´ z, Poland E-mail addr ess : piotr.math@gmai l.com