A Boolean algebra and a Banach space obtained by push-out iteration

A BOOLEAN ALGEBRA AND A BANACH SP A CE OBT AINED BY PUSH-OUT ITERA TION ANTONIO A VIL ´ ES AND CHRISTI NA BRECH Abstract. Under the assumption that c is a regular cardinal, we pro ve the existence and uniqueness of a Boolean algebra B of size c defined by shari ng the main struct ural properties that P ( ω ) /f in has under CH and in the ℵ 2 -Cohen model. W e prov e a simil ar result in the category of Banach spaces. 1. Introduction In this pa pe r tw o lines of re search con verge, one related to the theor y of Bo o lean algebras and the other one to Banach spaces. In the context of Boolea n algebras, the topic go es back to P aroviˇ cenko’s theo- rem [12], whic h es tablishes that, u nder CH, P ( ω ) /f in is the unique Bo olea n a lgebra of size c with the prop erty that given any diagram of em b eddings of Bo olea n alg e- bras lik e S x   R − − − − → P ( ω ) /f i n, where R a nd S are countable, there exis ts an embedding S − → P ( ω ) /f i n whic h makes the diagra m commutativ e. This c hara cterization is indeed equiv alent to CH [4]. Ther e has been a line of research [5, 6 , 7, 8, 9, 1 4] showing that ma n y r e- sults abo ut P ( ω ) /f in under CH can b e gener alized to t he ℵ 2 -Cohen mo del 1 (and to a less extent to an y Cohen model). The k ey p o int was pr ov en by Stepr¯ ans [14] and is that in that cas e P ( ω ) /f in is tightly σ -filtered. Later, Do w and Hart [5] in troduce d the notion of Co hen-Parovi ˇ cenko Boolean algebra s. On the one hand, P ( ω ) /f in is Cohen-Parovi ˇ cenko in Cohen models, and on the other hand there exists a unique Cohen-Parovi ˇ cenko Bo olean algebr a of size c whenever c ≤ ℵ 2 . Ther efore this prop- erty characterizes P ( ω ) / f in in the ℵ 2 -Cohen mo del in the spirit o f P aroviˇ cenko’s theorem. 1991 Mathematics Subje ct Classific ation. Primary 06E05; Secondary 0 3E35, 03G05, 46B26, 54G05. A. Avil´ es was supported by MEC and FEDER (Pro j ect MTM 2008-05396), F undaci´ on S´ eneca (Pro j ect 08848/PI/08), Ram´ on y Ca jal contract (R YC- 2008-02051) and an FP7-PEOPLE-E RG- 2008 action. C. Brech was supp orted by F APESP grant (2007/08213-2), which is part of Thematic Pro ject F APESP (2006/02 378-7). 1 By ( κ -)Cohen m o del, we mean a model obtained by adding ( κ many) C ohen r eals to a model of CH 1 2 ANTONIO A VIL ´ ES AND CHRISTINA BRECH Our main re sult states that whenever c is a regular car dinal, there exists a unique tightly σ - filtered Cohe n-Parovi ˇ cenko Bo olean a lgebra. Essentially , we are proving tha t whenever c is regular , there exists a ca nonical Bo olean a lgebra that behaves like P ( ω ) /f in do es under CH a nd in the ℵ 2 -Cohen mo del. By a result of Ges chk e [9], this Bo olean algebra cannot be P ( ω ) /f in when c > ℵ 2 . Hence, in the κ -Cohen mo del ( κ > ℵ 2 ), P ( ω ) /f in and our Bo olean a lgebra are tw o no n- isomorphic Cohen-Parovi ˇ cenk o B o olean algebra s, and this solves a question o f Dow and Hart [5, Questions 2 and 3] ab out uniqueness for this prop erty . W e found that many notions b eing used in the literature o n this to pic can b e reformulated using the concept of push-o ut from category theory . Apart fro m aes- thetic considerations, this appro ach has an o b jectiv e adv an tage: it allows us to exp ort ideas to other categor ies like w e do with Banach spa ces, where w e hav e push-outs with similar prop er ties. In the con text o f Banach spaces, an ana logue of Parovi ˇ cenko’s theorem ha s b een established b y Kubi ´ s [1 1]: Under CH, there exists a unique Bana ch s pace X of density c with the pro per ty that given any diagram of isometric embeddings like S x   R − − − − → X where R a nd S are s eparable, ther e ex ists a n embedding S − → X whic h mak es the diagram co mm utative. The e xistence o f such a space can b e proven in ZFC but not its uniqueness [2]. In this pap er we sha ll consider a stro nger prop er t y than the one stated ab ove whic h will imply exis tence and uniqueness o f the s pace X under the weak er as sumption that c is a regula r ca rdinal. In bo th situations the metho d of construction is the same as in [5] and [2 ] resp ec- tively , by a long chain of push-o uts. The difficulty lies in pr oving that the alg ebra or the space cons tructed in this way is indeed unique. It would b e nice to have a unified approach for Banach spaces and Bo ole an algebras in the con text of category theory , but o ur a ttempts to do so seemed to b ecome to o technical and to obs cure bo th sub jects rather tha n to enlighten them. The re ader w ill find, nevertheless, a n obvious para llelism. The pap er is str uctured as follows: In Section 2 we prov e the existence and uniqueness of our Boo lean algebra B . In Section 3 we do the same with our Banach space X . The tw o sections run para llel but they can b e rea d indep endently , and for the conv enience of the reader they include a self-co n tained account of the facts that we need ab out push-outs in each categ ory . In Section 4 we consider the c ompact space K , Stone dual of B , we establish that X is isometric to a subspace of the space of con tinuous functions C ( K ) and we also pro ve that K is homogeneous with resp ect to P -p o in ts, generalizing the re sults of Rudin [1 3] under CH, Stepr¯ ans [1 4] in the ℵ 2 -Cohen mo del, and Geschk e [9]. In Section 5 we state several op en pr oblems a nd in Section 6 w e point out more general v ersions of our results for diff erent cardinals. 3 W e would like to thank Wies law Kubi ´ s for some r elev an t rema rks that help ed to improv e this pap er. 2. Boolean algebras 2.1. Preliminary definiti ons. The push-out is a general notion of ca tegory the- ory , and the one that we shall use here refers to the ca tegory of Bo olean algebras. W e sha ll co nsider only push-o uts made of embeddings (one-to-one morphisms), al- though it is a more gener al conc ept. F or this reason, w e pr esent the sub ject in a different wa y tha n usual, more convenien t for us and equiv alen t for the case of embeddings. The join a nd meet op er ations in a Bo olean a lgebra B ar e denoted by ∨ a nd ∧ , and the complement o f r ∈ B by r . The subalgebra generated by H is h H i . Definition 1. Let B b e a B o olean algebr a and let S and A b e subalg ebras of B . W e say that B is the internal push-out of S a nd A if the follo wing co nditions ho ld: (1) B = h S ∪ A i . (2) F or every a ∈ A and e very s ∈ S , if a ∧ s = 0 , then there exists r ∈ A ∩ S such that a ≤ r and s ≤ r . W e notice th at condition (2) abov e can b e substitu ted b y th e f ollowing equiv alent one: (2’) F or e very a ∈ A and every s ∈ S , if a ≤ s , then there exists r ∈ A ∩ S such that a ≤ r ≤ s . The same definition can b e found in [9] a s “ A and S commute”. Suppo se that we have a diagra m of embeddings o f Boo lean algebra s: S − − − − → B x   x   R − − − − → A. W e say that it is a push-out diagr am if, when a ll a lgebras are viewed as subal- gebras of B , we hav e tha t R = S ∩ A and B is the internal push-out of A and S . If B is the push-out of S and A , then B is isomorphic to ( S ⊗ A ) /V where S ⊗ A denotes the free s um of S and A , and V is the ideal gener ated by the formal int ersec tions r ∧ r , where r ∈ A ∩ S (vie wing r ∈ S , r ∈ A ). Also, g iven a diagr am of em b eddings S x   R − − − − → A, there is a unique wa y (up to isomorphism) to complete it into a push o ut diag ram, putting B = ( S ⊗ A ) /V wher e V is as ab ov e. 4 ANTONIO A VIL ´ ES AND CHRISTINA BRECH If B is the push-out of S and A we will write B = PO [ S, A ]. Sometimes we wan t to mak e ex plicit the intersection spac e R = S ∩ A , and then w e wr ite B = PO R [ S, A ], meaning that B is the push- out of S and A and R = S ∩ A . Definition 2. An embedding of Bo olean a lgebras A − → B is s aid to b e a p ose x (push-out separable extension) if there exists a push-out diagram of embeddings A − − − − → B x   x   R − − − − → S such that S and R are countable. If A ⊂ B , we ca n rephra se this definition saying that B is a p osex of A if there exists a countable subalg ebra S ⊂ B such that B = PO [ S, A ]. Definition 3. W e say that a Bo ole an algebr a B is tightly σ -filtered if there exists an ordinal λ and a family { B α : α ≤ λ } of subalgebras of B such that (1) B α ⊂ B β whenever α < β ≤ λ , (2) B 0 = { 0 , 1 } a nd B λ = B , (3) B α +1 is a p osex of B α for every α < λ , (4) B β = S α<β B α for every limit ordina l β ≤ λ . W e shall see later that the de finition of Kopp elb erg [1 0] of tightly σ -filtered Bo olean algebra is equiv a lent to this o ne. 2.2. The main result. W e are now ready to state the main res ult of this s ection. Theorem 4 ( c is a r egular car dinal) . Ther e exists a unique (up t o isomorphism) Bo ole an algebr a B with the fol lowing pr op erties: (1) | B | = c , (2) B is tightly σ -fi lt er e d, (3) F or any diagr am of emb e ddings of t he form B x   A − − − − → B , if | A | < c and A − → B is p osex, then ther e exists an emb e dding B − → B which makes the diagr am c ommutative. 2.3. Basic prop erties of push-outs. Befor e going to the pr o of of Theorem 4, we collect some elementary prop er ties of push-outs of Bo olean algebr as. Prop ositi o n 5. The fol lowing facts ab out push-outs hold: (1) If B = PO [ S α , A ] for every α , wher e { S α : α < ξ } is an incr e asing chain of Bo ole an sub algebr as of B , S 0 ⊂ S 1 ⊂ · · · , t hen B = PO [ S α S α , A ] . (2) If B = PO [ S, A ] and we h ave S ′ ⊂ S and A ′ ⊂ A , then B = PO [ h S ∪ A ′ i , h A ∪ S ′ i ] . (3) If B 1 = PO S 0 [ S 1 , B 0 ] and B 2 = PO S 1 [ S 2 , B 1 ] , t hen B 2 = PO S 0 [ S 2 , B 0 ] . 5 (4) Supp ose t hat we have two incr e asing se quenc es of su b algebr as of B , S 0 ⊂ S 1 ⊂ · · · and B 0 ⊂ B 1 ⊂ · · · , and B n = PO S 0 [ S n , B 0 ] for every n . Then S n B n = PO S 0 [ S n S n , B 0 ] . (5) Supp ose t hat we have two incr e asing se quenc es of su b algebr as of B , S 0 ⊂ S 1 ⊂ · · · and B 0 ⊂ B 1 ⊂ · · · , and B n +1 = PO S n [ S n +1 , B n ] for every n . Then S n B n = PO S 0 [ S n S n , B 0 ] . (6) Supp ose that A = S i ∈ I A i ⊂ B , S = S j ∈ J S j ⊂ B and h A i ∪ S j i = PO [ S j , A i ] for every i and every j . Then h A ∪ S i = PO [ S, A ] . Pro of: (1) is trivial. F or item (2) it is enough to consider the case where S ′ = ∅ , that is to pr ov e B = PO [ h S ∪ A ′ i , A ], the general case b eing obtained b y the successive application of the cases S ′ = ∅ and its symmetric one A ′ = ∅ . Clearly B = h S ∪ A i = h ( S ∪ A ′ ) ∪ A i . Let s ∈ S ∪ A ′ and a ∈ A b e such that a ≤ s . If s ∈ S , our hyp othesis guarantees that there is r ∈ A ∩ S ⊆ A ∩ ( S ∪ A ′ ) such that a ≤ r ≤ s . If s ∈ A ′ ⊆ A ∩ ( S ∪ A ′ ), take r = s and we are done. F or item (3), it is clear that S 2 ∩ B 0 = S 0 and B 2 = h S 2 ∪ B 0 i . O n the other hand as sume that a ≤ s 2 for some a ∈ B 0 and s ∈ S 2 . Then, since a ∈ B 1 and B 2 = PO S 1 [ S 2 , B 1 ], there exists r 1 ∈ S 1 such that a ≤ r 1 ≤ s . Since we als o hav e B 1 = PO S 0 [ S 1 , B 0 ], we find r 0 ∈ S 0 with a ≤ r 0 ≤ r 1 ≤ s . Item (4) is eviden t (it holds even f or transfinite sequences though we do not need to use that) and item (5) follows fr om co mbin ing (3) and (4). Item (6) is also easy to see.  W e r emark that some of the prope rties of push-outs of Ba nach spaces in Pro po - sition 1 7 do not hold in the context of Bo o lean alge bras. Namely , supp os e that A is free ly genera ted by { x i : i ∈ I } and B is freely generated by { x i : i ∈ I } ∪ { y } . If S = h y i , then B = PO [ S, A ]. Ho w ever, if we co nsider D = h x i , x i ∧ y : i ∈ I i , then A ⊂ D ⊂ B but B 6 = PO [ S, D ] and D 6 = PO [ S ∩ D , A ]. So, we don’t hav e a n analogue of items (1) and (2) of Pro po sition 1 7. Prop ositi o n 6. B is a p osex of A if and only if the fol lowing two c onditions hold: (1) B is c ountably gener ate d over A (t hat is, B = h A ∪ Q i for some c ountable set Q ), and (2) F or every b ∈ B , the set { a ∈ A : a ≤ b } is a c ountably gener ate d id e al of A . Pro of: W e suppos e first that B is a p osex of A , B = PO R [ S, A ] for some countable S . It is ob vious that B is co un tably generated ov er A so we co ncentrate in proving the other prop er t y . It is clear that prop erty (2) holds whenever b ∈ A (in that ca se the idea l is just genera ted by b ) and also when b ∈ S , since by the push-out pr op erty the ideal I b = { a ∈ A : a ≤ b } is g enerated by I b ∩ R , which is countable. Let us consider now a n ar bitrary element b ∈ B . W e can express it in the for m b = ( s 1 ∧ a 1 ) ∨ · · · ∨ ( s n ∧ a n ) where a 1 , . . . , a n form a pa rtition in A (that is, they are dis joint and their join is 1) and s 1 , . . . , s n ∈ S . Pick now a ∈ A such that a ≤ b . Then a ∧ a i ≤ s i for every i , so since a ∧ a i ∈ A and s i ∈ S , there exists r i ∈ R with a ∧ a i ≤ r i ≤ s i . W e have that a = ( a ∧ a 1 ) ∨ · · · ∨ ( a ∧ a n ) ≤ ( r 1 ∧ a 1 ) ∨ · · · ∨ ( r n ∧ a n ) ≤ ( s 1 ∧ a 1 ) ∨ · · · ∨ ( s n ∧ a n ) = b. 6 ANTONIO A VIL ´ ES AND CHRISTINA BRECH Notice that ( r 1 ∧ a 1 ) ∨ · · · ∨ ( r n ∧ a n ) ∈ A . It follows tha t the idea l { a ∈ A : a ≤ b } is generated by the co un table set { r = ( r 1 ∧ a 1 ) ∨ · · · ∨ ( r n ∧ a n ) : r i ∈ R, r ≤ b } W e prov e now the c onv erse implication. So we assume tha t B = h A ∪ Q i with Q countable, and for every b ∈ B , we fix a countable set G b ⊂ A that gener ates the ideal { a ∈ A : a ≤ b } . W e define an incr easing sequence of subalgebr as of B making S 0 = h Q i and S n +1 = h S n ∪ S b ∈ S n G b i . All these are countably generated -hence countable- Bo olean algebras. T aking S = S n<ω S n we get that B = PO [ S, A ]. Namely , if a ≤ s with s ∈ S and a ∈ A , then s ∈ S n for s ome n , and then there exists r ∈ G s with a ≤ r ≤ s . Just observe that r ∈ G s ⊂ A ∩ S n +1 ⊂ A ∩ S .  Condition (2) of Pro po sition 6 is found in the liter ature under the following names: A is a go od subalgebr a o f B [1 4]; A is an ℵ 0 -ideal subalgebra of B [5]; A is a σ -subalg ebra o f B [8, 9]. W e keep the latter terminolo gy . So , B is a p osex of A if and only if A is a σ -s ubalgebra o f B and B is countably genera ted ov er A . Co n versely , A is a σ -subalg ebra of B if and only if every intermediate algebr a A ⊂ C ⊂ B which is countably generated o ver A is a p osex of A . After these equiv- alences, it b ecomes o bvious that Kopp elb erg’s definition [10] of tightly σ - filtered algebra and our own ar e the s ame, cf. also [9, Theorem 2.5]. Prop ositi o n 7. The fol lowing facts hold: (1) If A ⊂ B and B is c ountable, then B is a p osex of A . (2) If B is a p osex of A , A ⊂ B ′ ⊂ B , and B ′ is c ountably gener ate d over A , then B ′ is a p osex of A . (3) If B 1 is a p osex of B 0 and B 2 is a p osex of B 1 , then B 2 is a p osex of B 0 . (4) If we have B n ⊂ B n +1 and B n is a p osex of B 0 for every n < ω , then S n B n is a p osex of B 0 . (5) If B n +1 is a p osex of B n for every n ∈ N , t hen S n B n is a p osex of B 0 . (6) If B is a p osex of A and S 0 ⊂ B is c ountable, then ther e exists a c ountable sub algebr a S 0 ⊂ S ⊂ B with B = PO [ S, A ] . (7) If we have B 0 ⊂ B 1 ⊂ B 2 , B 2 is a p osex of B 0 and B 1 is c ount ably gener ate d over B 0 , t hen B 2 is a p osex of B 1 . Pro of: (1) is trivial, (2 ) follows from Prop os ition 6. W e prove now (3). Suppose that B 1 = PO R 0 [ S 0 , B 0 ] and B 2 = PO T 0 [ U 0 , B 1 ]. Our o b jectiv e is to a pply Prop o- sition 5(3), so we need to o vercome the difficult y that S 0 6 = T 0 . Inductively on n , we will define countable subalge bras R n , S n , T n , U n forming four increasing sequence s so that B 1 = PO R n [ S n , B 0 ] and B 2 = PO T n [ U n , B 1 ] for every n . The inductive pro cedure is as follows: pick a countable set Q n ⊂ B 0 such that T n ⊂ h S n ∪ Q n i , then define S n +1 = h S n ∪ Q n i , R n +1 = S n +1 ∩ B 0 , U n +1 = h U n ∪ S n +1 i and T n +1 = U n +1 ∩ B 1 . The push-out relatio ns B 1 = PO R n [ S n , B 0 ] and B 2 = PO T n [ U n , B 1 ] follow fro m Prop osition 5(2). Also notice that T n ⊂ S n +1 ⊂ T n +1 for ev ery n . Hence S n S n = S n T n . On the other hand, by Pro po sition 5(4) we hav e that B 1 = PO S n R n [ S n S n , B 0 ] and B 2 = PO S n T n [ S n U n , B 1 ] . By P rop osition 5(3), B 2 = PO S n R n [ S n U n , B 0 ], which prov es that B 2 is a p osex of B 0 , since S n U n is countable. Item (4) is proven eas ily using Pro po sition 6, and (5) is a cons equence of (3 ) and (4). F or (6), consider first a countable subalgebra S 1 ⊂ B such that B = PO [ S 1 , A ]. 7 Then find a countable set Q ⊂ A such that S 0 ⊂ h Q ∪ S 1 i . F rom Pr op osition 5(2) we get that B = PO [ h Q ∪ S 1 i , A ]. T o prov e (7), use (2) to g et that B 1 is a p osex of B 0 hence B 1 = PO [ S 1 , B 0 ] for some countable subalgebra S 1 . By (6) ther e exis ts a countable s ubalgebra S 2 such that B 2 = PO [ S 2 , B 0 ] and S 1 ⊂ S 2 . By Prop osi- tion 5(2), B 2 = PO [ S 2 , h B 0 ∪ S 1 i ] but h B 0 ∪ S 1 i = B 1 .  Consider ag ain the ex ample where A is fr eely ge nerated by { x i : i ∈ I } with I uncountable, B is freely generated by { x i : i ∈ I } ∪ { y } and D = h x i , x i ∧ y : i ∈ I i . Then B = PO [ h y i , A ], so B is a po sex o f A . On the other hand, A ⊂ D ⊂ B but neither D is a p osex of A nor B is a p osex of D . Indeed D is not countably generated ov er A , a nd { a ∈ D : a ≤ y } is not a countably gener ated idea l of D . 2.4. Pro of of Theo rem 4. The following definition, as well as Lemma 9 ar e due to Geschke [9]. A pro of of Lemma 9 can also b e obtained by imitating the pro of of Lemma 20. Definition 8. Let B b e a (uncountable) Bo olean a lgebra. An additive σ - skeleton of B is a family F of suba lgebras of B with the following prop erties: (1) { 0 , 1 } ∈ F (2) F or every subfamily G ⊂ F , we have h S G i ∈ F . (3) F or every infinite subalgebr a A ⊂ B , there exis ts A ′ ∈ F such that with A ⊂ A ′ ⊂ B and | A | = | A ′ | . (4) Every A ∈ F is a σ - subalgebra of B . W e will often use the following pro p erty o f a n additive σ -skeleton: If A ∈ F , A ⊂ A ′ ⊂ B and A ′ is countably genera ted over A , then there exists B ∈ F such that A ′ ⊂ B and B is countably ge nerated over A ′ . This is a direct c onsequence of pro per ties (2) and (3): Supp os e tha t A ′ = h A ∪ S i with S countable B o olean algebra. Then there ex ists a countable S 1 ∈ F with S ⊂ S 1 , and we can take B = h A ∪ S 1 i . Lemma 9 (Geschke) . F or a Bo ole an algebr a B t he fol lowing ar e e quivalent: (1) B is tightly σ -fi lt er e d. (2) Ther e exists an additive σ -skeleton F of B . W e can now prove Theorem 4. First we prove existence. W e consider c = S α< c Φ α a decomp os ition of the contin uum into c man y subs ets of cardinality c such that α ≤ min(Φ α ) fo r every α . W e define recursively an increa sing chain o f Bo olean a lgebras { B α : α < c } , so that at the end B = S α< c B α . W e sta rt with B 0 = { 0 , 1 } . After B α is defined, we co nsider a family { ( R γ , S γ ) : γ ∈ Φ α } such that • F or every γ ∈ Φ α , S γ is a countable Bo olea n algebra and R γ = B α ∩ S γ . • F or every countable subalgebra R ⊂ B α and every countable super algebra S ⊃ R there ex ists γ ∈ Φ α and a Bo o lean isomor phism j : S − → S γ such that R = R γ and j ( x ) = x for x ∈ R . F or limit or dinals β we define B β = S α<β B α . At succe ssor stages we define B α +1 = PO R α [ S α , B α ]. By construction, it is clear that B is a tightly σ - filtered Bo olean algebra of car dinality c . W e chec k prop er ty (3) in the statement o f the 8 ANTONIO A VIL ´ ES AND CHRISTINA BRECH theorem. Supp os e that we hav e A ⊂ B with | A | < c , and B = PO R [ S, A ] with S countable. By the r egularity of c , we can find α < c such that A ⊂ B α . Then, there exists γ ∈ Φ α such that R = R γ and (mo dulo an isomor phism) S = S γ , s o that B γ +1 = PO R [ S, B γ ]. Consider ˜ B = h S ∪ A i ⊂ B γ +1 , so that ˜ B = PO R [ S, A ]. Since push-out is unique up to isomor phism, we can find a n isomor phism ˜ u : B − → ˜ B ⊂ B such that ˜ u ( a ) = a for all a ∈ A . W e prove no w uniqueness. Supp ose that w e have t w o Bo olean algebra s like this, B and B ′ . W e consider their resp ective additive σ -skeletons F a nd F ′ that witness that they a re tightly σ -filtered. Let us supp o se that B = { x α : α < c } and B ′ = { y α : α < c } . W e shall cons truct recursively tw o incr easing chains of subalgebra s { B α : α < c } a nd { B ′ α : α < c } a nd a family o f Bo o lean isomorphisms f α : B α − → B ′ α with the following prop er ties: (1) The isomor phisms a re coherent, that is f β | B α = f α whenever α < β . (2) F or every α , B α ∈ F and B ′ α ∈ F ′ . (3) x α ∈ B α +1 and y α ∈ B ′ α +1 . In this way w e make sure that B = S α< c B α and B ′ = S α< c B ′ α . (4) B α +1 is countably gener ated ov er B α for every α . This implies that | B α | = | α | for every α ≥ ω . After this, the isomorphis ms f α induce a global iso morphism f : B − → B ′ . W e pro ceed to the inductive construc tion. W e sta rt with B 0 = { 0 , 1 } a nd B ′ 0 = { 0 , 1 } . If β is a limit o rdinal, we simply put B β = S α<β B α , B ′ β = S α<β B ′ α an the isomorphism f β is induced by the pr evious ones. Now we s how how to co nstruct B α +1 , B ′ α +1 and f α +1 from the previous ones. W e construct inductively on n , sequences of subalgebras B α [ n ] ⊂ B , B ′ α [ n ] ⊂ B ′ and coher ent is omorphisms f α [ n ] : B α [ n ] − → B ′ α [ n ] as in the picture: B α = B α [0] ⊂ B α [1] ⊂ B α [2] ⊂ B α [3] · · · ⊂ B ↓ ↓ ↓ ↓ B ′ α = B ′ α [0] ⊂ B ′ α [1] ⊂ B ′ α [2] ⊂ B ′ α [3] · · · ⊂ B ′ and we will make B α +1 = S n<ω B α [ n ], B ′ α +1 = S n<ω B ′ α [ n ] and f α +1 induced by the isomor phisms f α [ n ]. All the algebr as B α [ n + 1] and B ′ α [ n + 1] will b e countably generated ov er B α [ n ] and B ′ α [ n ] res pec tively . The inductive pro cedure is a s follo ws. There are t wo ca ses: Case 1: n is even. Then, we define B α [ n + 1] to be such that x α ∈ B α [ n + 1], B α [ n + 1] is co untably g enerated ov er B α [ n ], and B α [ n + 1] ∈ F . Since B α ∈ F which is an additiv e σ -skeleton, B α [ n + 1] is a p osex of B α , and b y Prop osition 7(7), also B α [ n + 1] is a p ose x of B α [ n ]. Hence, since B ′ satisfies the statement of o ur theorem, we can find a Bo olean embedding f α [ n + 1] : B α [ n + 1 ] − → B ′ such that f α [ n + 1 ] | B α [ n ] = f α [ n ]. W e define finally B ′ α [ n + 1 ] = f α [ n + 1]( B α [ n + 1 ]). Case 2: n is o dd. Then, we define B ′ α [ n + 1] to be such that y α ∈ B ′ α [ n + 1], B ′ α [ n + 1] is countably gener ated ov er B ′ α [ n ] and B ′ α [ n + 1] ∈ F ′ . Since B ′ α ∈ F ′ which is an a dditiv e σ -skeleton, B ′ α [ n + 1] is a p osex of B α , hence also a p o sex of B ′ α [ n ], so since B satisfies the s tatement of our theo rem, we can find an embedding g α [ n + 1] : B ′ α [ n + 1] − → B such that g α [ n + 1] | B α [ n ] = f − 1 α [ n ]. W e define finally 9 B α [ n + 1 ] = g α [ n + 1]( B α [ n + 1 ]) and f α [ n + 1] = g − 1 α [ n + 1 ]. Pro ceeding this way , we hav e that B α [ n ] ∈ F for n o dd, while B ′ α [ n ] ∈ F ′ for n even. At the end, B α +1 = S n<ω B α [2 n + 1] ∈ F and B ′ α +1 = S n<ω B ′ α [2 n ] ∈ F ′ , which c oncludes the pro of.  2.5. Remarks. Dow and Har t [5] sa y that B is ( ∗ , ℵ 0 )-ideal, if for ev ery κ < c there exists κ - cub of σ -subalg ebras o f B . If we have an additive σ -skeleton F for B , then the algebras in F of cardinality κ form a κ -cub of σ - subalgebra s of B . Hence, ev ery tightly σ -filtered Bo olea n alg ebra of cardinality c is ( ∗ , ℵ 0 )-ideal. Also, they say that a subalg ebra A ⊂ B is ℵ 0 -ideal complete if for every tw o orthogo nal countably g enerated ideals I , J o f A , there e xists c ∈ B such that I = { a ∈ A : a ≤ c } and J = { a ∈ A : a ≤ ¯ c } . W e hav e the following fact: Prop ositi o n 10. A is an ℵ 0 -ide al c omplete sub algebr a of B if and only if for every p osex A − → C ther e ex ists an emb e dding g : C − → B with g | A = 1 A . Pro of: If the s tatement in the right holds, and we tak e t wo orthogonal coun tably generated ideals I , J ⊂ A , we can consider R a countable subalgebra of A genera ted by the unio n o f c ountable sets of generators of I and J . T ake a sup era lgebra o f the form S = h R ∪ { c }i wher e x ≤ c for every x ∈ I and x ≤ ¯ c for every x ∈ J , and let C = PO R [ S, A ]. Our ass umption provides an embedding g : C − → B and the element g ( c ) is the desir ed one. Assume now that A is ℵ 0 -ideal complete and consider f : A − → C p os ex. It is eno ugh to consider the ca se when C is finitely gener ated ov er f ( A ), so supp ose that C = h f ( A ) ∪ { c 1 , c 2 , . . . , c n }i where { c 1 , . . . , c n } form a partition. F or every i < n find d i ∈ B such that { a ∈ A : f ( a ) ≤ c i } = { a ∈ A : a ≤ d i } { a ∈ A : f ( a ) ≤ c i } = { a ∈ A : a ≤ d i } Define d n = 1. Con vert the d i ’s into a par tition by setting d ′ k = d k \ ( W i ℵ 2 : 10 ANTONIO A VIL ´ ES AND CHRISTINA BRECH Theorem 11 . If B = P ( ω ) /f in is the algebr a of The or em 4, then c ≤ ℵ 2 . Pro of: Geschk e [9] pr ov es that a complete Bo olean algebr a of size gr eater than ℵ 2 cannot be tightly σ -filtered. Hence, if c > ℵ 2 , P ( ω ) is not tightly σ -filtered. This implies that neither P ( ω ) /f in is such, b eca use a tow er of subalg ebras witnes sing tight σ -filtration could b e lifted to P ( ω ).  Another prop er t y of the algebra B of Theo rem 4 is: Prop ositi o n 12. If A is a tightly σ - fi lter e d Bo ole an algebr a with | A | ≤ c , then A is isomorphi c to a sub algebr a of B . Pro of: Let { A α : α < κ } b e s ubalgebras that witness tha t A is tigh tly σ -filtered. W e can suppose that κ is the cardinality of A , cf.[9]. Then, inductively we can extend a given embedding A α − → B to A α +1 − → B , b y the prop er ties of B .  3. Banach sp aces 3.1. Preliminary definiti ons. The push-out is a general notion of ca tegory the- ory , and the one that we shall use here refers to the categor y B an 1 of Banach spaces, toge ther with op er ators o f no rm at most 1. W e shall co nsider only push- outs made of iso metric embeddings o f B anach spaces, although it is a more gener al concept. F or this reaso n, we present the sub ject in a differen t wa y than usual, more conv enient for us but equiv alent for the cas e of isometric em b eddings. Definition 13. Let Y b e a Bana ch space and let S and X b e s ubspaces of Y . W e say tha t Y is the internal push-o ut of S and X if the following co nditions hold: (1) Y = S + X (2) k x + s k = inf { k x + r k + k s − r k : r ∈ S ∩ X } for every x ∈ X and ev ery s ∈ S . Suppo se that we have a diagram of isometric embeddings of Banach spaces like: S − − − − → Y x   x   R − − − − → X . W e say that it is a push-out diag ram if, when all spaces a re se en a s subspace s of Y , we have that R = S ∩ X and Y is the int erna l push-out of X and S . Note that Y b eing the push-out of S and X means tha t Y is is ometric to the quotient s pace ( S ⊕ ℓ 1 X ) / V where V = { ( r, − r ) : r ∈ X ∩ S } . In particular, Y = S + X = { s + x : s ∈ S, x ∈ X } . A lso, giv en a diagra m o f isometric embeddings S u x   R v − − − − → X , there is a unique wa y (up to isometr ies) to complete it into a push o ut diagr am, precisely by making Y = ( S ⊕ ℓ 1 X ) / ˜ R a nd putting the obvious a rrows, wher e 11 ˜ R = { ( u ( r ) , v ( − r )) : r ∈ R } . If Y is the push-out of S and X as ab ov e, we will write Y = PO [ S, X ]. Sometimes we wan t to mak e explicit the intersection space R = S ∩ X , and then we write Y = PO R [ S, X ]. Definition 14. An isometric embedding o f Banach spaces X − → Y is said to b e a po sex if there exists a push-out diagr am of isometric embeddings X − − − − → Y x   x   R − − − − → S such that S and R are separable. If X ⊂ Y , we ca n rephrase this definition saying that there exists a separ able subspace S ⊂ Y such that Y = PO [ S, X ]. Definition 15. W e say tha t a Banach space X is tightly σ -filtered if ther e exists an ordinal λ and a family { X α : α ≤ λ } of subs paces of X suc h that (1) X α ⊂ X β whenever α < β ≤ λ , (2) X 0 = 0 and X λ = X , (3) X α +1 is a p osex of X α for every α < λ , (4) X β = S α<β X α for every limit o rdinal β ≤ λ . 3.2. The main result. Theorem 16 ( c is a reg ular cardinal) . Ther e exists a unique (up t o isometry) Banach sp ac e X with the fol lowing pr op erties: (1) dens ( X ) = c , (2) X is tightly σ -filter e d, (3) F or any diagr am of isometric emb e ddings of the form Y x   X − − − − → X , if dens ( X ) < c and X − → Y is p osex, then ther e exists an isometric emb e dding Y − → X which makes the diagr am c ommutative. 3.3. Basic prop erties of push-outs. Before entering the pr o of o f T heorem 16, we collec t some elemen tary prop erties of push-outs that w e shall use. F or the sake of completeness, we include their pr o ofs. Prop ositi o n 17. The fol lowing facts ab out push-out s hold: (1) Supp ose Y = PO [ S, X ] , S ⊂ S ′ ⊂ Y and X ⊂ X ′ ⊂ Y . Then Y = PO [ S ′ , X ′ ] . (2) Supp ose Y = PO [ S, X ] , and X ⊂ Y ′ ⊂ Y . Then Y ′ = PO [ S ∩ Y ′ , X ] . (3) Supp ose Y = PO [ S, X ] , and S ⊂ Y ′ ⊂ Y . Then Y ′ = PO [ S, X ∩ Y ′ ] . (4) L et X ⊂ Y ⊂ Z and S 0 ⊂ S 1 ⊂ S 2 ⊂ Z b e such t hat Y = PO S 0 [ S 1 , X ] and Z = PO S 1 [ S 2 , Y ] . Then Z = PO S 0 [ S 2 , X ] . 12 ANTONIO A VIL ´ ES AND CHRISTINA BRECH (5) L et X 0 ⊂ X 1 ⊂ · · · and S 0 ⊂ S 1 ⊂ · · · b e Banach sp ac es s u ch that for every n ∈ N we have that X n +1 = PO S n [ S n +1 , X n ] . L et X = S n X n and S = S n S n . Then X = PO S 0 [ S, X 0 ] . (6) Supp ose X = S i ∈ I X i ⊂ Y , S = S j ∈ J S j ⊂ Y and X i + S j = PO [ S j , X i ] for every i and every j . Then X + S = PO [ S, X ] . Pro of: F o r (1) it is enough to consider the case when X = X ′ . The cas e S = S ′ is just the same, and the general fact follows from the application o f those tw o. So we have to prove that g iven x ∈ X , s ′ ∈ S ′ and ε > 0 there exists t ∈ X ∩ S ′ such that k x + s ′ k + ε > k x + t k + k s ′ − t k , W rite s ′ = s + x ′ where s ∈ S and x ′ ∈ X . Then there exists r ∈ X ∩ S such that k x + s ′ k = k x + x ′ + s k > k x + x ′ + r k + k s − r k − ε = k x + ( x ′ + r ) k + k s ′ − ( x ′ + r ) k − ε so just take t = x ′ + r . Item (2) is immediate, simply notice that Y ′ = X + ( Y ′ ∩ S ), b ecause if y ∈ Y ′ and y = x + s w ith x ∈ X and s ∈ S , then s = y − x ∈ Y ′ ∩ S . Item (3) is symmetric to item (2). F or (4), consider ε > 0, x ∈ X and s 2 ∈ S 2 . Then since Z = PO S 1 [ S 2 , Y ], there exists s 1 ∈ S 1 such that k x + s 2 k > k x + s 1 k + k s 2 − s 1 k − ε/ 2 and since Y = PO S 0 [ S 1 , X ] there exists s 0 ∈ S 0 with k x + s 1 k > k x + s 0 k + k s 1 − s 0 k − ε/ 2 so finally k x + s 2 k > k x + s 0 k + k s 1 − s 0 k + k s 2 − s 1 k − ε ≥ k x + s 0 k + k s 2 − s 0 k − ε F or (5), w e know by r ep eated application of item ( 4) that X n +1 = PO S 0 [ S n , X 0 ]. Consider ε > 0, x 0 ∈ X 0 and s ∈ S . Then there exists n and s n ∈ S n such that k s n − s k < ε/ 3. Then, we can find r ∈ S 0 such that k x 0 + s k > k x 0 + s n k − ε/ 3 > k x 0 + r k + k s n − r k − 2 ε/ 3 > k x 0 + r k + k s − r k − ε Item (6) is prov en similarly b y approximation.  Corollary 18. The fol lowing facts ab out p osexes hold: (1) If Y is a p osex of X , t hen Y /X is sep ar able. (2) X ⊂ Y and Y is sep ar able, then Y is a p osex of X . (3) If Y is a p osex of X and X ⊂ Y ′ ⊂ Y , then Y ′ is a p osex of X . (4) If Y is a p osex of X and X ⊂ X ′ ⊂ Y , then Y is a p osex of X ′ . (5) If Y is a p osex of X and Z is a p osex of Y , then Z is a p osex of X . (6) If X n +1 is a p osex of X n for every n ∈ N , then S n X n is a p osex of X 0 . (7) If we h ave X n ⊂ X n +1 and X n is a p osex of X 0 for every n ∈ N , then S n X n is a p osex of X 0 . A subspace X of a Bana ch space X will b e called a σ -subs pace o f X , if for ev ery X ⊂ Y ⊂ X , if Y /X is separ able, then Y is a pos ex of X . 13 Definition 19. Let X b e a (nonsepar able) Banach space. An additive σ -skeleton of X is a family F of subspaces of X with the following pr op erties: (1) 0 ∈ F (2) F or every subfamily G ⊂ F , we have span ( S G ) ∈ F . (3) F or every infinite-dimensio nal X ⊂ X there e xists Y ∈ F with X ⊂ Y ⊂ X and dens ( Y ) = dens ( X ). (4) Each X ∈ F is a σ -subspace of X . W e will often use the following pro per ty of a n a dditive σ - skeleton: If X ∈ F , X ⊂ Y ⊂ X and Y / X is sepa rable, then there exists Z ∈ F such that Y ⊂ Z and Z/ Y is separable. The pro o f is a direct conseq uence of prope rties (2) a nd (3): Suppo se that Y = X + S with S separa ble. Then there exists a separ able S 1 ∈ F with S ⊂ S 1 , and we ca n take Z = X + S 1 . W e prov e now a result for Banach space s corresp onding to Lemma 9. Lemma 20 . F or a Banach sp ac e X the fol lowing ar e e quivalent: (1) X is tightly σ -filter e d. (2) X has an additive σ -skeleton. Pro of: That (2) implies (1) is evident: it is enough to define the subspace s X α inductively just taking care that X α ∈ F for every α . No w, we supp ose that w e hav e a tower of subs paces { X α : α ≤ λ } like in (2). F or every α < λ we consider separable subspaces R α ⊂ S α such that X α +1 = PO R α [ S α , X α ]. Given a set of or dinals Γ ⊂ λ , we define E (Γ) = span  S γ ∈ Γ S γ  . W e say that the s et Γ is sa turated if for every α ∈ Γ we have that R α ⊂ E (Γ ∩ α ). W e shall prov e that the family F = { E (Γ) : Γ ⊂ λ is s aturated } is the additive σ -skeleton that w e ar e lo o king for. It is clear that 0 = E ( ∅ ). Also, if we hav e a family { Γ i : i ∈ I } of saturated sets, then span ( S i E (Γ i )) = E ( S i Γ i ), so the union of s aturated sets is saturated. Given a n y countable set Γ ⊆ λ , it is p ossible to find a countable saturated set ∆ s uch that Γ ⊂ ∆: W e define ∆ = { δ s : s ∈ ω <ω } where ( δ s ) s ∈ ω <ω is defined inductively on the leng th of s ∈ ω <ω as follows. Let Γ = { δ ( n ) : n ∈ ω } and g iven s ∈ ω <ω , let { δ s ⌢ n : n ∈ ω } be such tha t δ s ⌢ n < δ s and R δ s ⊂ E ( { δ s ⌢ n : n ∈ ω } ). Notice that ∆ is a countable and satura ted set whic h con tains Γ and also sup( δ + 1 : δ ∈ Γ) = sup( δ + 1 : δ ∈ ∆). Now, supp ose X ⊆ X and let us find a saturated set ∆ such that X ⊆ E (∆) and dens ( X ) = dens ( E (∆)). Since the union o f sa turated se ts is sa turated, w e can assume without loss o f generality that X is sepa rable and find a countable set Γ such that X ⊂ E (Γ). T ake ∆ as in the previous paragr aph and notice that X ⊂ E (∆) ∈ F a nd E (∆) is separa ble since ∆ is countable. It r emains to prove that if X ∈ F and we hav e X ⊂ Y ⊂ X with Y /X separable, then Y is a p osex o f X . It is enough to prove the following statement, and we sha ll do it b y induction on δ 1 = sup( δ + 1 : δ ∈ ∆): 14 ANTONIO A VIL ´ ES AND CHRISTINA BRECH “F or e very sa turated set Γ ⊂ λ and every co unt able set ∆ ⊂ λ , E (Γ ∪ ∆) is a po sex of E (Γ).” W e fix δ 1 < λ a nd we as sume that the statement ab ov e holds for a ll saturated sets Γ ⊂ λ and for all countable s ets ∆ ′ ⊂ λ with sup( δ + 1 : δ ∈ ∆ ′ ) < δ 1 . Case 1: δ 1 is a limit o rdinal. T his follows immediately fr om the inductive hy- po thesis using Corolla ry 18(7). Case 2: δ 1 = δ 0 + 1 for some δ 0 and ∆ = { δ 0 } . W e distinguish t wo sub ca ses: Case 2a: sup(Γ) ≤ δ 0 . Consider the ch ain of subspa ces E (Γ) ⊂ E (Γ) + R δ 0 ⊂ E (Γ ∪ { δ 0 } ) . The left hand extension is a po sex extension by the inductive hypothesis, b ecause there exists a countable set ∆ ′ such that R δ 0 ⊂ E (∆ ′ ) and sup( δ + 1 , δ ∈ ∆ ′ ) ≤ δ 0 < δ 1 . The righ t hand extension is also p os ex, becaus e we hav e the push-out diagram S δ 0 − − − − → X δ 1 x   x   R δ 0 − − − − → X δ 0 and since sup(Γ) ≤ δ 0 (and δ 0 6∈ Γ, otherwise it is trivial), we can interpola te S δ 0 − − − − → E (Γ ∪ { δ 0 } ) − − − − → X δ 1 x   x   x   R δ 0 − − − − → E (Γ) + R δ 0 − − − − → X δ 0 where cle arly E (Γ ∪ { δ 0 } ) = E (Γ) + R δ 0 + S δ 0 , so that the left hand squar e is a push-out diagram. Case 2 b: sup(Γ) > δ 0 . F or every ξ ≤ λ , we call Γ ξ = Γ ∩ ξ . By Case 2a , there exists a separable space S such that E (Γ δ 1 ∪ { δ 0 } ) = PO [ S, E (Γ δ 1 )] . W e can suppo se that S δ 0 ⊂ S . W e shall prov e b y induction on ξ that E (Γ ξ ∪ { δ 0 } ) = PO [ S, E (Γ ξ )] for δ 1 ≤ ξ ≤ λ. If ξ is a limit ordinal, then E (Γ ξ ) = S η<ξ E (Γ η ), and w e just need Propo si- tion 17(6). Other wise supp os e that ξ = η + 1. In the nontrivial case, η ∈ Γ a nd Γ ξ = Γ η ∪ { η } . By the inductive hypothesis we hav e that E (Γ η ∪ { δ 0 } ) = PO [ S, E (Γ η )] and on the other hand X ξ = X η +1 = PO [ S η , X η ]. Suppo se that we are g iven x ∈ E (Γ ξ ), s ∈ S and ε > 0 and w e hav e to find r ∈ E (Γ ξ ) ∩ S such that k x + s k ≥ k x + r k + k s − r k − ε . W e write x = s η + x η where s η ∈ S η and x η ∈ E (Γ η ). Notice that x η + s ∈ E (Γ η ) + S ⊂ E (Γ η ∪ { δ 0 } ) ⊂ X η 15 since S δ 0 ⊂ X η ( δ 0 < η as δ 1 < ξ ). Hence, using that X ξ = PO [ S η , X η ] we find r η ∈ S η ∩ X η such that k s η + x η + s k ≥ k x η + s + r η k + k s η − r η k − ε/ 2 . Since η ∈ Γ and Γ is satura ted, we hav e that R η ⊂ E (Γ η ), therefore x η + r η ∈ E (Γ η ) a nd s ∈ S . Hence, using that E (Γ η ∪ { δ 0 } ) = PO [ S, E (Γ η )] we g et r ∈ S ∩ E (Γ η ) such tha t k x η + s + r η k ≥ k s + r k + k x η + r η − r k − ε/ 2 . Combining b oth inequa lities, k s η + x η + s k ≥ k s + r k + k x η + r η − r k + k s η − r η k − ε ≥ k s + r k + k x η + s η − r k − ε, as desired. Case 3: δ 1 = δ 0 + 1 for some δ 0 ∈ ∆ and | ∆ | > 1. W e consider the set ∆ \ { δ 0 } . W e have prov en few par agraphs ab ove in this pro of that we can find a countable saturated set ∆ ′ ⊃ ∆ \ { δ 0 } s uch that sup( δ + 1 : δ ∈ ∆ ′ ) ≤ δ 0 . By the inductive hypothesis E (Γ ∪ ∆ ′ ) is a po sex o f E (Γ). Since Γ ∪ ∆ ′ is sa turated, the already prov en ca se 2 pr ovides that E (Γ ∪ ∆ ′ ∪ { δ 0 } ) is a po sex of E (Γ ∪ ∆ ′ ). Comp osing bo th po sex extensions w e get that E (Γ ∪ ∆ ′ ∪ { δ 0 } ) is a posex of E (Γ). Since E (Γ) ⊂ E (Γ ∪ ∆) ⊂ E (Γ ∪ ∆ ′ ∪ { δ 0 } ) we fina lly get that E (Γ ∪ ∆) is a po sex o f E (Γ).  W e fina lly prove Theorem 16. First we prove existence. W e co nsider c = S α< c Φ α a decomp osition of the contin uum into c many subsets of cardinality c such tha t α ≤ min(Φ α ) for every α . W e define r ecursively a n incr easing chain of Bana ch spaces { X α : α < c } , s o that at the end X = S α< c X α . W e start with X 0 = 0. After X α is defined, we co nsider a family { ( R γ , S γ ) : γ ∈ Φ α } where • F or every γ ∈ Φ α , S γ is a separa ble Ba nach spa ce and R γ = X α ∩ S γ . • F or every sepa rable subspace R ⊂ X α and every s eparable supers pace S ⊃ R there exists γ ∈ Γ and an isometry j : S − → S γ such that R = R γ and j ( x ) = x for x ∈ R . F or limit ordinals β we define X β = S α<β X α . A t succ essor stages we define X α +1 = PO R α [ S α , X α ]. By co nstruction, it is c lear that X is a tightly σ -filtered Banach space o f density c . W e c heck prop er t y (3) in the statement of the theorem. Suppo se that we ha ve X ⊂ X with dens ( X ) < c , a nd Y = PO R [ S, X ] with S separable. By the reg ularity of c , w e c an find α < c such that X ⊂ X α . Then, there ex ists γ ∈ Φ α such that R = R γ and (mo dulo an isometry) S = S γ . Then X γ +1 = PO R [ S, X γ ]. Consider ˜ Y = S + X ⊂ X γ +1 , so that ˜ Y = PO R [ S, X ]. Since push-out is unique up to isometry , we c an find an isometry ˜ u : Y − → ˜ Y ⊂ X such that ˜ u ( x ) = x for all x ∈ X . W e prov e now uniqueness. Supp ose that we hav e tw o spaces lik e this, X a nd X ′ . W e consider their r esp ective additive σ -skeletons F and F ′ that witness that they are tightly σ -filtered. Le t us suppo se that X = span { x α : α < c } and X ′ = span { y α : α < c } . W e shall construct recurs ively t wo increasing c hains of subspaces { X α : α < c } and { X ′ α : α < c } and a family of bijective iso metries f α : X α − → X ′ α with the following prop er ties: 16 ANTONIO A VIL ´ ES AND CHRISTINA BRECH (1) The isometries a re c oherent, that is f β | X α = f α whenever α < β . (2) F or every α , X α ∈ F and X ′ α ∈ F ′ . (3) x α ∈ X α +1 and y α ∈ X ′ α +1 . In this way we make sure that X = S α< c X α and X ′ = S α< c X ′ α . (4) Each quotient X α +1 /X α is separable. This implies that dens ( X α ) = | α | for every α ≥ ω . After this, the iso metries f α induce a g lobal isometry f : X − → X ′ . W e pro c eed to the inductive constructio n. W e start with X 0 = 0 and X ′ 0 = 0 . If β is a limit ordinal, w e simply put X β = S α<β X α , X ′ β = S α<β X ′ α and the iso metry f β is induced by the pr evious isometries. No w, we show how to co nstruct X α +1 , X ′ α +1 and f α +1 from the previous ones. W e construct inductively on n , sequences of subspaces X α [ n ] ⊂ X , X ′ α [ n ] ⊂ X ′ and coherent isometries f α [ n ] : X α [ n ] − → X ′ α [ n ] as in the picture: X α = X α [0] ⊂ X α [1] ⊂ X α [2] ⊂ X α [3] · · · ⊂ X ↓ ↓ ↓ ↓ X ′ α = X ′ α [0] ⊂ X ′ α [1] ⊂ X ′ α [2] ⊂ X ′ α [3] · · · ⊂ X ′ and w e will make X α +1 = S n<ω X α [ n ], X ′ α +1 = S n<ω X ′ α [ n ] and f α +1 induced by the isometries f α [ n ]. All the quo tien t spa ces X α [ n + 1] / X α [ n ] and X ′ α [ n + 1] / X ′ α [ n ] will be s eparable. The inductive pro cedure is as follows. T here are tw o cases: Case 1: n is even. Then, we define X α [ n + 1] to b e such tha t x α ∈ X α [ n + 1], X α [ n + 1] /X α [ n ] is separable, and X α [ n + 1] ∈ F . Since X α ∈ F which is an additive σ -skeleton, X α [ n + 1] is a p osex of X α [ n ], so since X ′ satisfies the state- men t o f o ur theorem, we can find a n into isometry f α [ n + 1] : X α [ n + 1] − → X ′ such tha t f α [ n + 1] | X α [ n ] = f α [ n ]. W e define finally X ′ α [ n + 1] = f α [ n + 1]( X α [ n + 1]). Case 2: n is o dd. Then, we define X ′ α [ n + 1] to b e such that y α ∈ X ′ α [ n + 1], X ′ α [ n + 1] / X ′ α [ n ] is s eparable, and X ′ α [ n + 1] ∈ F ′ . Since X ′ α ∈ F ′ which is an additive σ -skeleton, X ′ α [ n + 1] is a po sex of X ′ α [ n ], so since X satisfies the statement of o ur theo rem, we ca n find an into iso metry g α [ n + 1] : X ′ α [ n + 1] − → X such that g α [ n + 1] | X α [ n ] = f − 1 α [ n ]. W e define finally X α [ n + 1] = g α [ n + 1]( X α [ n + 1]) and f α [ n + 1 ] = g − 1 α [ n + 1 ]. Pro ceeding this wa y , we have that X α [ n ] ∈ F for n o dd, while X ′ α [ n ] ∈ F ′ for n even. At the end, X α +1 = S n<ω X α [2 n + 1] ∈ F and X ′ α +1 = S n<ω X ′ α [2 n ] ∈ F ′ .  3.4. Univ ersality prop erty. Let X denote the space in Theorem 16. Theorem 21 . If X is a t ightly σ -fi lter e d Banach sp ac e with dens ( X ) ≤ c , then X is isometric to a subsp ac e of t he sp ac e X . Pro of: Let { X α : α ≤ κ } b e subspaces that witness that X is tightly σ -filtered. By the pro of of Lemma 2 0, we ca n suppos e that κ is the car dinal dens ( X ). Then, inductively we can extend a g iven iso metric embedding X α − → X to X α +1 − → X , by the prop er ties of X .  17 4. Comp act sp aces Along this section and the subseq uent o nes, X will alw ays denote the B anach space in Theorem 16 and B the B o olean algebr a in Theor em 4. 4.1. The compact space K . Definition 22. Suppo se that we hav e a commutativ e diagra m of contin uous sur- jections betw een co mpact spaces, K f − − − − → L g   y v   y S u − − − − → R. W e say that this is a pull-back diag ram if the following conditions hold: (1) F or every x, y ∈ K , if x 6 = y , then either f ( x ) 6 = f ( y ) or g ( x ) 6 = g ( y ). (2) If we are g iven x ∈ S and y ∈ L such that u ( x ) = v ( y ), then ther e exists z ∈ K such that f ( z ) = y and g ( z ) = x . Again, the notion o f pull-back is mo re gener al in categor y theor y , a nd in par- ticular pull-back diagrams of co nt inuous functions which are not s urjective c an b e defined, but for our purp oses we restrict to the case defined ab ov e. If we a re given tw o contin uous sur jections u : S − → R and v : L − → R b etw een compact spaces, we c an alwa ys construct a pull-back dia gram as ab ov e making K = { ( x, y ) ∈ S × L : u ( x ) = v ( y ) } and taking f and g to b e the co ordinate pro jections. Mo reov er a ny other pull-back diagram K ′ f ′ − − − − → L g ′   y v   y S u − − − − → R . is homeomorphic to the canonical one by a homeomorphism h : K − → K ′ with f ′ h = f and g ′ h = g . Prop ositi o n 23. A diagr am of emb e ddi ngs of Bo ole an algebr as A − − − − → B x   x   R − − − − → S is a push-out diagr am if and only if and only if the diagr am of c omp act sp ac es obtaine d by S tone duality, S t ( A ) ← − − − − S t ( B )   y   y S t ( R ) ← − − − − S t ( S ) , is a pul l-b ack diagr am. 18 ANTONIO A VIL ´ ES AND CHRISTINA BRECH Pro of: Left to the reader.  Prop ositi o n 24. Le t Y b e a Banach sp ac e and X , S, R su bsp ac es of Y such that Y = PO R [ S, X ] . Then the diagr am obtaine d by duality b etwe en the dual b al ls endowe d with t he we ak ∗ top olo gy, B Y ∗ − − − − → B X ∗   y   y B S ∗ − − − − → B R ∗ is a pul l-b ack diagr am. Pro of: W e c an supp ose that Y = ( X ⊕ ℓ 1 S ) /V wher e V = { ( r, − r ) : r ∈ R } , a nd then B Y ∗ ⊂ B ( X ⊕ ℓ 1 S ) ∗ = B Y ∗ × B S ∗ and it is pr ecisely the set of pa irs which agr ee on R .  Definition 25. A c ontin uous s urjection b etw een compact s paces f : K − → L is called po sex if there exists a pull-back diagram of cont inuous s urjections K f − − − − → L   y   y S − − − − → R. with R and S metriza ble compact spaces. Definition 26. A compa ct space K is called pull-back genera ted if there exists a family { K α : α ≤ ξ } of compact spaces and contin uous s urjections { f β α : K β − → K α : α ≤ β ≤ ξ } suc h that (1) K 0 is a singleton and K ξ = K , (2) f α α is the iden tity map on K α , (3) f β α f γ β = f γ α for α ≤ β ≤ γ ≤ ξ , (4) f α +1 α : K α +1 − → K α is po sex for every α < ξ , (5) If γ ≤ ξ is a limit or dinal and x, y ∈ K γ with x 6 = y , then there ex ists β < γ such that f γ β ( x ) 6 = f γ β ( y ) (this means that K γ is the inv erse limit of the system b elow γ ). W e can pro ve aga in a theore m about existence and uniqueness of a co mpact space in a similar wa y as w e did for Bana ch spaces and Bo olean alg ebras. But it is not worth to repea t the pro cedur e becaus e we w ould obtain just the Sto ne compact space of the Bo olean alg ebra B of Theo rem 4. Hence, we just deno te this Stone space b y K = S t ( B ). Lemma 27. L et K b e a pul l-b ack gener ate d c omp act sp ac e. Then ther e exists a zer o-dimensio nal pul l-b ack gener ate d c omp act sp ac e L of t he same weight as K and such that ther e is a c ontinuous su rje ct ion fr om L onto K . Pro of: Assume that we hav e an inv erse system { f β α : K β − → K α } α ≤ β ≤ ξ as above witnessing that K is pull-back gener ated. W e pro duce o ur compact spac e L and the contin uous sur jection h : L − → K by co nstructing inductiv ely a similar inverse system { g β α : L β − → L α } α ≤ β ≤ ξ together with contin uous surjections h α : L α − → 19 K α satisfying h α g β α = f β α h β for α ≤ β . W e need that e ach L α is zer o-dimensiona l and the weigh t of L α equals the weight of K α . The key step is providing L α +1 , f α +1 α and h α +1 from L α and h α . Since f α +1 α is p osex there exist metrizable compact spaces S and R and a pull ba ck diagr am K α +1 − − − − → K α   y   y S − − − − → R Consider S ′ a metrizable zero- dimensional compact space and u : S ′ − → S a contin uous sur jection. W e have a lar ger diagram L α h α   y K α +1 − − − − → K α   y   y S ′ u − − − − → S − − − − → R. W e can define L α +1 and g α +1 α by making the pull bac k of the larg er square abov e, L α +1 g α +1 α − − − − → L α   y   y S ′ − − − − → R . The contin uous surjection h α +1 : L α +1 − → K α +1 can b e o btained by applying the so calle d universal prope rty of pull-ba ck. In this case, the pull-back K α +1 can be seen as a subs pace of S × K α and similar ly L α +1 ⊂ S ′ × L α . O ne can define simply h α +1 ( s, x ) = ( u ( s ) , h α ( x )).  Prop ositi o n 28 . F or any diag r am of c ontinuous surje ctions b etwe en c omp act sp ac es K   y L ← − − − − K , if f : K − → L is p osex and w eig ht ( L ) < c , then ther e exists a c ontinuous s u rje ction K − → K that makes the diagr am c ommutative. Pro of: Without los s o f generality , we suppos e that we have a pull-back diagr am K − − − − → L   y   y S − − − − → R where S is metrizable and zer o-dimensional. W e can factorize into cont inuous surjections K − → L ′ − → L suc h that L ′ is zero-dimensio nal and w eig ht ( L ′ ) = 20 ANTONIO A VIL ´ ES AND CHRISTINA BRECH wei g ht ( L ). Consider then K ′ the pull-ba ck of K , L and L ′ , which is zero -dimensional bec ause it is also the pull-back of S , R and L ′ , S ← − − − − K ← − − − − K ′   y   y   y R ← − − − − L ← − − − − L ′ ← − − − − K . Then we ha ve a similar diag ram as in the statement of the theorem but all compact spaces are zer o-dimensional. By Stone duality a nd pro per ty (3) of B in Theorem 4 there is a contin uous surjection K − → K ′ that completes the diagr am, and this provides the desir ed K − → K .  4.2. The relation b etw een X and C ( K ) . Theorem 29 . X is isometric to a subsp ac e of C ( K ) . Pro of: Let { X α : α ≤ ξ } b e an increasing chain of subspaces of X witnes sing the fact that X is a push-genera ted Banach space. Then, by P rop osition 24 the dual balls { B X ∗ α : α ≤ ξ } for m an inv erse system that witnes s the fact that B X ∗ is a pull-back gene rated compac t space. B y Lemma 27 there exis ts a zer o-dimensional pull-back generated compact space L of weight c tha t maps onto B X ∗ . B y Stone duality , using Pr op osition 2 3, we g et that the B o olean algebra B of clo p ens o f L is a tightly σ -filtered Bo olea n algebr a of car dinality c . Hence, by Prop o sition 12, we can write B ⊂ B . Therefore K maps con tinuously onto L , whic h maps contin uously onto B X ∗ . And this implies that X ⊂ C ( B X ∗ ) ⊂ C ( L ) ⊂ C ( K ).  4.3. P-p oints. Remem b er that a p oint p of a to po logical s pace K is called a P - po int if the int erse ction of countably many neighbor ho o ds of p c ontains a neigh- bo rho o d of p . The fo llowing result w as proven by Rudin [1 3] under CH and by Stepr¯ ans [14] in the ℵ 2 -Cohen mo del: for ev ery tw o P -p oints p, q ∈ ω ∗ there exists a homeomo rphism f : ω ∗ − → ω ∗ such that f ( p ) = q . Ges chk e [9] pr ov es that it is sufficient to as sume that P ( ω ) is tightly σ -filtered. In this section we pr ov e that this is a prop erty o f our compact space K . Theorem 30. L et p, q ∈ K b e P-p oints. Th en ther e exists a home omorphism f : K − → K su ch that f ( p ) = q . In wha t follows, p oints of the Stone space of a Bo olean algebra B are co nsidered as ultrafilters on B . Given Q 1 , Q 2 ⊂ B w e write Q 1 ≤ Q 2 if q 1 ≤ q 2 for every q 1 ∈ Q 1 and q 2 ∈ Q 2 . If b ∈ B and Q ⊂ B , then b ≤ Q mea ns { b } ≤ Q and Q ≤ b means Q ≤ { b } . Definition 31. If A is a subalgebr a of B , Q ⊂ A and b ∈ B , then we write b ≤ A Q if { a ∈ A : b ≤ a } equa ls the filter generated by Q in A . Similarly , we wr ite Q ≤ A b if { a ∈ A : a ≤ b } equals the idea l generated by Q in A . With this no tation, the no tion of push- out of Bo olea n algebra s can b e rephr ased as follows: g iven Bo olea n alg ebras R ⊂ A, S ⊂ B , then B = PO R [ S, A ] if and only if for every b ∈ S , { a ∈ R : a ≤ b } ≤ A b ≤ A { a ∈ R : b ≤ a } . Lemma 32. L et p ∈ K b e a P-p oint, let A ⊂ B b e a sub algebr a with | A | < c and let Q ⊂ A ∩ p b e c ountable. Then, ther e ex ists b ∈ p such that { 0 } ≤ A b ≤ A Q . 21 Pro of: Beca use p is a P -p oint and Q is countable, there exis ts b 0 ∈ p such that b 0 ≤ Q . Since | A | < c , an y p o sex extension of A ca n b e represented inside B . In particular, we can find b 1 ∈ B such that { 0 } ≤ A b 1 ≤ A Q in such a wa y that B 1 = h A ∪ { b 1 }i is a p osex of A . Observe that b 0 ∪ b 1 ≤ A Q a nd b 0 ∪ b 1 ∈ p . By the same reason as b efore, we can find b 2 ∈ B such that { 0 } ≤ B 1 b 2 ≤ B 1 { 1 } . Notice that w e ha ve { 0 } ≤ A ( b 1 ∪ b 0 ) ∩ b 2 ≤ A Q a nd { 0 } ≤ A ( b 1 ∪ b 0 ) \ b 2 ≤ A Q a nd since b 1 ∪ b 0 ∈ p and p is an ultra filter, either ( b 1 ∪ b 0 ) ∩ b 2 or ( b 1 ∪ b 0 ) \ b 2 belo ng to p .  Lemma 33. L et A b e a Bo ole an algebr a with | A | < c , and u : A − → B a nd v : A − → B b e Bo ole an emb e ddings with v b eing p osex. Fix P -p oints, p 0 , p and q of A , B and B r esp e ctively such that u ( p 0 ) ⊂ p and v ( p 0 ) ⊂ q . Then ther e ex ists an emb e dding ˜ u : B − → B such that ˜ uv = u and ˜ u ( q ) ⊂ p . Pro of: Supp ose that B = PO R [ S, v ( A )] where S is a countable subalg ebra of B . W e can pro duce a further po sex s upe ralgebra B 0 ⊃ B of the form B 0 = h B ∪ { b 0 }i such tha t { 0 } ≤ B b 0 ≤ B S ∩ q . By Lemma 3 2 we can find b 1 ∈ p s uch that { 0 } ≤ u ( A ) b 1 ≤ u ( A ) u ( R ∩ p 0 ). No tice that { 0 } ≤ v ( A ) b 0 ≤ v ( A ) v ( R ∩ p 0 ), and this allows to define a Bo olea n embedding w : h v ( A ) ∪ { b 0 }i − → B such that w v = u and w ( b 0 ) = b 1 . Since B 0 is a p osex of v ( A ) and h v ( A ) ∪ { b 0 }i is co unt ably g en- erated ov er v ( A ), we hav e that B 0 is a p osex of h v ( A ) ∪ { b 0 }i , so using the seco nd prop erty stated in Theorem 4, we find ˜ w : B 0 − → B such that ˜ w | h v ( A ) ∪{ b 0 }i = w . W e cons ider finally ˜ u = ˜ w | B . It is clear that ˜ uv = u . On the other ha nd, s ince ˜ u ( b 0 ) = w ( b 0 ) = b 1 ∈ p a nd b 0 ≤ S ∩ q , we hav e that ˜ u ( S ∩ q ) ⊂ p . It is also clear that ˜ u ( q ∩ v ( A )) = u ( p 0 ) ⊂ p . Since B = h S ∪ v ( A ) i the ultr afilter q is the filter generated b y ( q ∩ S ) ∪ ( q ∩ v ( A )), so we finally get that ˜ u ( q ) ⊂ p .  W e can rephrase the s tatement of Theorem 30 as f ollows: “ If B and B ′ are Bo olean algebr as satisfying the conditions of Theor em 4 and p and q are P -p o int s in B and B ′ resp ectively , then there e xists an isomorphism f : B − → B ′ such that f ( p ) = q ”. In or der to prov e this, we just hav e to follow the pro o f o f uniqueness in Theorem 4 a nd make sur e that at each step it is p oss ible to cho ose the par tial isomorphisms f α : B α − → B ′ α in such a wa y that f ( p ∩ B α ) = q ∩ B ′ α . And what we need for that is exa ctly Lemma 33 applied to B and B ′ . 5. Open problems 5.1. When c is singul ar. Pr oblem 1 . Do Theo rem 16 and Theorem 4 hold when c is singular? Here the p oint is that regular it y lo ok s essential to co nt rol all substructures of size le ss than c in the ex istenc e part o f the pro of. Perhaps thes e theorems do not hold for singular c as they are stated, but it w ould be satisfactory an y v aria tion that would allow us to sp ea k a bo ut unique ob jects X and B defined b y some prop er ties in ZF C. W e remark that if we r estrict below a given car dinal, existenc e ca n b e prov en. W e state it for Bo o lean algebr as, and leav e the Bana ch spa ce version to the reader. Prop ositi o n 34. Fix a c ar dinal λ < c . Ther e ex ists a Bo ole an algebr a B with the fol lowing pr op erties: 22 ANTONIO A VIL ´ ES AND CHRISTINA BRECH (1) | B | = c , (2) B is tightly σ -fi lt er e d, (3) F or any diagr am of emb e ddings of t he form B x   A − − − − → B , if | A | ≤ λ and A − → B i s p osex, t hen ther e exists an isometric emb e dding B − → B which m akes the diagr am c ommutative. Pro of: Let λ + be the successor cardinal of κ . Construct B in the same wa y as in the pro o f of Theorem 4 or Theorem 1 6, but do it with a tow er o f length c · λ + instead of length c .  5.2. Prop erties of B refle cted on Banac h spaces. W e know m uch more a bo ut B than ab out its Banach space r elative X . So it is natura l to a sk whether certain facts that hold for B in so me mo dels ar e reflected by analog ous prop er ties for X or for C ( K ). F or example, we mentioned that in the ℵ 2 -Cohen mo del, B = P ( ω ) /f in [5]. In this model, therefo re, B has some additional properties of extensions of morphisms : given a ny diagr am o f arbitrar y morphisms S x   R − − − − → B where R is co un table and S is arbitrar y , there exists a mor phism S − → B which makes the diagra m co mm utative. The analogo us prop er t y for Banach s paces is called (1-)universally se parably injectivity (we add a 1 if the op er ator S − → X can be found with the same norm as R − → X ), cf. [2]. This pro p erty implies that the space cont ains ℓ ∞ . Pr oblem 2 . In the ℵ 2 -Cohen model, is the space X univ ersally separa bly injective? Do es it contain ℓ ∞ ? Another observ ation is that in this mo del, P ( ω ) /f in do es no t contain any ω 2 - chain. Indeed, Dow and Har t [5] prove in ZFC that no Cohe n-Parovi ˇ cenko Bo o lean algebra (in particular B ) can con tain ω 2 -chains. On the o ther hand, Brech and Koszmider [3] pr ov e that in the ℵ 2 -Cohen mo del, ℓ ∞ /c 0 do es not contain the space C [0 , ω 2 ] of co n tinuous functions on the or dinal interv al [0 , ω 2 ]. So the natura l question is: Pr oblem 3 . Is it true in g eneral that C ( K ) (or at lea st X ) do es not contain C [0 , ω 2 ]? Pr oblem 4 . Can spaces like ℓ 2 ( ω 2 ) or c 0 ( ω 2 ) b e subspaces of X ? Under (MA + c = ℵ 2 ), Dow and Hart [5] prove that B do es not contain P ( ω ) as a subalgebra . So we may form ulate Pr oblem 5 (MA + c = ℵ 2 ) . Do es C ( K ) contain a copy of ℓ ∞ ? 23 This question is a lready p osed in [2 ]. A negativ e answer would so lve a prob- lem by Ros enth al b y providing an F -space K such that C ( K ) does not contain ℓ ∞ . In [2] it is pr ov en that, under MA + c = ℵ 2 , there is an isometric embedding c 0 − → C ( K ) which cannot b e extended to an embedding ℓ ∞ − → C ( K ). The space X plays the role of B in the categor y of Banac h spaces. But we do not hav e a Banach space playing the role of P ( ω ) /f in , when it is differen t from B . This is r elated to the general q uestion o f finding intrinsic characterizatio ns of P ( ω ) /f i n out of CH and ℵ 2 -Cohen mo dels, that we could tr anslate into other ca tegories. Pr oblem 6 . Is ther e a Banach spa ce counterpart o f P ( ω ) /f in ? 6. Other cardinals W e co mment that a more gener al version of some of our results can b e stated if we let arbitr ary cardinals to play the role of c and countabilit y . W e state it for Bana ch spaces, and leav e the Bo olean version to the r eader. F or an unco unt able cardina l τ , we say that an isometric embedding X − → Y is τ -p osex if Y = PO [ S, X ] fo r some S with dens ( S ) < τ . W e s ay that X is tightly τ - filtered if X is the union of a contin uous tow er of subspa ces star ting at 0 a nd such that each X α +1 is a τ -po sex of X α (in the case o f B o olean algebra s, such a definition is equiv alent to the one given in [9]). Theorem 35. L et κ b e a r e gular c ar dinal, and τ an unc ountable c ar dinal with κ <τ = κ . Ther e exist s a unique (u p to isometry) Banach sp ac e X = X ( κ, τ ) with the fol lowing pr op erties: (1) dens ( X ) = κ , (2) X is tightly τ -filt er e d, (3) F or any diagr am of isometric emb e ddings of the form Y x   X − − − − → X , if dens ( X ) < κ and X − → Y is τ -p osex, then ther e exist s an isometric emb e dding w : Y − → X which c ompletes the diagr am. The pro of would b e just the same. 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St¯ eprans, T op ological i n v ariant s in the Cohen mo del. T opol ogy Appl. 45 (1992), no. 2, 85-101 Universidad de M urcia, Dep ar t amen to de Ma tem ´ aticas, Campus de Espinardo 3010 0 Murcia, Sp ain E-mail addr ess : avileslo@um. es Dep ar t amento de Ma tem ´ atica, Instituto de Ma tem ´ atica e Est a t ´ ıstica, Universidade de S ˜ ao P aulo, Rua do Ma t ˜ ao, 1010 - 05 508-090, S ˜ ao P aulo, Brazil E-mail addr ess : christina.br ech@gmail.com