Projectivity of the free Banach lattice generated by a lattice
aa r X i v : . [ m a t h . F A ] F e b PROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BYA LATTICE
ANTONIO AVIL´ES AND JOS´E DAVID RODR´IGUEZ ABELL ´AN
Abstract.
We study the projectivity of the free Banach lattice generated by a lattice L in two cases: when the lattice is finite, and when the lattice is an infinite linearly orderedset. We prove that in the first case it is projective while in the second case it is not., if thelinear order contains either an increasing sequence without upper bounds or a decreasingsequence without lower bounds, then it is not projective. We found a mistake in this paper. At some point in the proof of the second part of Lemma4.2 we considered functions ϕ i that had to be continuous but they were not. So Lemma 4.2and Theorem 4.4 must be stated in a weaker form. We indicate how to fix the error: Redtext indicates what should be removed and blue text what should be added to the publishedversion in Archiv der Mathematik 113 (2019), 515–524 .1. Introduction
Free and projective Banach lattices were introduced in [4]. The free Banach lattice
F BL ( A ) generated by a set A is a Banach lattice characterized by the property that everybounded map T : A −→ X into a Banach lattice X extends to a unique Banach latticehomomorphism ˆ T : F BL ( A ) −→ X with the same norm. This idea was generalized in [2]and [1], where the free Banach lattice generated by a Banach space E and by a lattice L arerespectively studied. By a lattice we mean here a set L together with two operations ∧ and ∨ that are the infimum and supremum of some partial order relation on L , and a latticehomomorphism is a function between lattices that commutes with those two operations. Definition 1.1.
Given a lattice L , the free Banach lattice generated by L is a Banachlattice F together with a lattice homomorphism φ : L −→ F such that for every Banachlattice X and every bounded lattice homomorphism T : L −→ X , there exists a uniqueBanach lattice homomorphism ˆ T : F −→ X such that || ˆ T || = || T || and T = ˆ T ◦ φ . Mathematics Subject Classification.
Key words and phrases.
Free Banach lattice; lattice; linear order; projectivity.Authors supported by project MTM2017-86182-P (Government of Spain, AEI/FEDER, EU) and project20797/PI/18 by Fundaci´on S´eneca, ACyT Regi´on de Murcia. Second author supported by FPI contract ofFundaci´on S´eneca, ACyT Regi´on de Murcia.
Here, the norm of T is k T k := sup {k T ( x ) k X : x ∈ L } , while the norm of ˆ T is the usualnorm of an operator acting between Banach spaces. This definition determines a Banachlattice that we denote by F BL h L i in an essentially unique way. When L is a distributivelattice the function φ is injective and we can view F BL h L i as a Banach lattice whichcontains a subset lattice-isomorphic to L in a way that its elements work as free generatorsmodulo the lattice relations on L , cf. [1]. To see that, it is well known that a lattice L isdistributive if, and only if, L is lattice-isomorphic to a bounded subset of a Banach lattice.Thus, it is clear that if φ is inyective then L is distributive. On the other hand, if L isdistributive, we have a bounded injective lattice homomorphism T : L −→ X for someBanach lattice X . Using the definition of being the free Banach lattice generated by thelattice L , there is ˆ T such that ˆ T ◦ φ = T . Since T is inyective, φ is also inyective.The notions of free and projective objects are closely related in the general theory ofcategories. In the context of Banach lattices, de Pagter and Wickstead [4] introducedprojectivity in the following form: Definition 1.2.
A Banach lattice P is projective if whenever X is a Banach lattice, J aclosed ideal in X and Q : X −→ X/J the quotient map, then for every Banach latticehomomorphism T : P −→ X/J and ε >
0, there is a Banach lattice homomorphismˆ T : P −→ X such that T = Q ◦ ˆ T and k ˆ T k ≤ (1 + ε ) k T k .Some examples of projective Banach lattices given in [4] include F BL ( A ), ℓ , all finitedimensional Banach lattices and Banach lattices of the form C ( K ), where K is a compactneighborhood retract of R n . But we are still far from understanding what the projectiveBanach lattices are. Such basic questions as whether c , ℓ or C ([0 , N ) are projective wereleft open in [4].Since the canonical projective Banach lattice is the free Banach lattice F BL ( A ), it isnatural to think that its variants F BL [ E ] (the free Banach lattice generated by the Banachspace E ) and F BL h L i may also be projective at least in some cases. In this paper we focuson the case of F BL h L i . We prove that F BL h L i is projective whenever L is a finite lattice,while it is not projective when L is an infinite linearly ordered set containining either anincreasing sequence without upper bounds or a decreasing sequence without lower bounds.If L is a finite lattice, F BL h L i is a renorming of a Banach lattice of continuous functions C ( K ) on a compact neighborhood retract K of R n , which is projective [4]. Projectiv-ity, however, is not preserved under renorming, because of the (1 + ε ) bound required inDefinition 1.2. Getting this bound will be the key point in the proof.In the infinite case, we considered only linearly ordered sets, as they are easier to handlethan general lattices. We do not know if there is some infinite lattice L such that F BL h L i is projective. ROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BY A LATTICE 3 Preliminaries
Absolute neighborhood retracts.
An absolute neighborhood retract (ANR) is atopological space X with the property that whenever X is a subspace of Y , then there isan open subset V of Y such that X ⊂ V ⊂ Y and X is a retract of V , meaning that thereis a continuous function r : V −→ X such that r ( x ) = x for all x ∈ X .The following are two basic facts of the theory that can be found in [3] as Theorems1.5.1 and 1.5.9: • Every closed convex subset of R n is ANR. • If X , X are closed subsets of X , and X , X and X ∩ X are ANR, then X ∪ X is also ANR.From the two facts above, one can easily prove that every finite union of closed convexsubsets of R n is ANR, by induction on the number of convex sets in that union.2.2. Free Banach lattices.
We collect the necessary facts and definitions about freeBanach lattices from [4, 2, 1].An explicit construction of the free Banach lattice
F BL ( A ) generated by a set A isas follows. For x ∈ A , let δ x : [ − , A −→ [ − ,
1] be the evaluation function given by δ x ( x ∗ ) = x ∗ ( x ) for every x ∗ ∈ [ − , A , and for f : [ − , A −→ R we define k f k = sup ( n X i =1 | r i f ( x ∗ i ) | : r i ∈ R , x ∗ i ∈ [ − , A , sup x ∈ A n X i =1 | r i x ∗ i ( x ) | ≤ ) , which we will denote by k f k or k f k F BL ( A ) . The Banach lattice F BL ( A ) is the Banachlattice generated by the evaluation functions δ x inside the Banach lattice of all functions f : [ − , A −→ R with finite norm. The natural identification of A inside F BL ( A ) isgiven by the map u : A −→ F BL ( A ) where u ( x ) = δ x . Since every function in F BL ( A )is an uniform limit of such functions, they are all continuous and positively homogeneous(they commute with multiplication by positive scalars). When A is finite, then F BL ( A )consists of all continuous and positively homogeneous functions on [ − , A , or equivalentlyin this case, all positively homogeneous functions on [ − , A that are continuous on theboundary ∂ [ − , A . Thus, when A is finite, F BL ( A ) is a renorming of the Banach latticeof continuous functions on ∂ [ − , A .We can describe of F BL h L i as the quotient of F BL ( L ) (the free Banach lattice generatedby the underlying set of the lattice L ) by the closed ideal I of F BL ( L ) generated by theset { δ x ∨ δ y − δ x ∨ y , δ x ∧ δ y − δ x ∧ y : x, y ∈ L } . A. AVIL´ES AND J.D. RODR´IGUEZ ABELL ´AN
In [1] we prove that,
F BL ( L ) / I , together with the map φ : L −→ F BL ( L ) / I given by φ ( x ) = δ x + I is the free Banach lattice generated by the lattice L .Also in [1] there is a different description of F BL h L i as a space of functions. Theconstruction is analogous to that of F BL ( A ) but taking into account the lattice structure.Namely, if we see [ − ,
1] as a lattice, define L ∗ = { x ∗ : L −→ [ − ,
1] : x ∗ is a lattice-homomorphism } . For every x ∈ L consider the evaluation function ˙ δ x : L ∗ −→ [ − ,
1] given by ˙ δ x ( x ∗ ) = x ∗ ( x ),and for f ∈ R L ∗ , define k f k ∗ = sup ( n X i =1 | r i f ( x ∗ i ) | : r i ∈ R , x ∗ i ∈ L ∗ , sup x ∈ L n X i =1 | r i x ∗ i ( x ) | ≤ ) . Let
F BL ∗ h L i be the Banach lattice generated by the evaluations n ˙ δ x : x ∈ L o inside theBanach lattice of all functions f ∈ R L ∗ with k f k ∗ < ∞ , endowed with the norm k · k ∗ andthe pointwise operations. This, together with the assignment φ ( x ) = ˙ δ x is the free Banachlattice generated by L .Thus, we have two alternative constructions of the free Banach lattice generated by L that we are denoting as F BL h L i = F BL ( L ) / I and F BL ∗ h L i , respectively. There is anatural Banach lattice homomorphism R : F BL ( L ) −→ F BL ∗ h L i given by restriction R ( f ) = f | L ∗ . This is surjective and its kernel is the ideal I , and thus R induces thecanonical isometric Banach lattice isomorphism between F BL h L i and F BL ∗ h L i .2.3. Projective Banach lattices.
We state here a variation of [4, Theorem 10.3]:
Proposition 2.1.
Let P be a projective Banach lattice, I an ideal of P and π : P −→ P/ I the quotient map. The quotient P/ I is projective if and only if for every ε > there existsa Banach lattice homomorphism u ε : P/ I −→ P such that π ◦ u ε = id P/ I and k u ε k ≤ ε .Proof. If P/ I is projective, then we can just apply Definition 1.2. On the other hand, ifwe have the above property and we want to check Definition 1.2, take ε >
0, a quotientmap Q : X X/J and a Banach lattice homomorphism T : P/ I −→
X/J . Take ε with(1 + ε ) ≤ ε . Since P is projective we can find S : P −→ X with Q ◦ S = T ◦ π and k S k ≤ (1 + ε ) k T ◦ π k = (1 + ε ) k T k . If we take ˆ T = S ◦ u ε , then Q ◦ ˆ T = Q ◦ S ◦ u ε = T ◦ π ◦ u ε = T and k ˆ T k ≤ (1 + ε ) k T k ≤ (1 + ε ) k T k as desired. (cid:3) Since
F BL ( L ) is projective [4, Proposition 10.2], and the restriction map describedabove R : F BL ( L ) −→ F BL ∗ h L i is a quotient map [1], we get, as a particular instance ofProposition 2.1, ROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BY A LATTICE 5
Proposition 2.2.
Let L be a lattice and let R : F BL ( L ) −→ F BL ∗ h L i be the restrictionmap R ( f ) = f | L ∗ . The Banach lattice F BL ∗ h L i is projective if, and only if, for every ε > there exists a Banach lattice homomorphism u ε : F BL ∗ h L i −→ F BL ( L ) such that k u ε k ≤ ε and R ◦ u ε = id F BL ∗ h L i . Projectivity of the free Banach lattice generated by a finite lattice
We are going to prove that if L is a finite lattice, then F BL ∗ h L i is a projective Banachlattice. Proposition 3.1. If L = { , . . . , n − } is a finite lattice, then L ∗ ∩ ∂ [ − , n is AN R .Proof.
Clearly, ∂ [ − , n is a finite union of closed convex subsets of R n .On the other hand, let A ijk = { ( x , . . . , x n ) ∈ R n : x i ∨ x j = x k } and B ijk = { ( x , . . . , x n ) ∈ R n : x i ∧ x j = x k } . It is clear that A ijk = { ( x , . . . , x n ) ∈ R n : x i = x k , x j ≤ x i } ∪ { ( x , . . . , x n ) ∈ R n : x j = x k , x i ≤ x j } , and B ijk = { ( x , . . . , x n ) ∈ R n : x i = x k , x j ≥ x i } ∪ { ( x , . . . , x n ) ∈ R n : x j = x k , x i ≥ x j } are union of two closed convex sets.Since L ∗ = \ i ∨ j = k A ijk ! \ \ i ∧ j = k B ijk ! , we have that L ∗ is also a finite union of closed convex subsets of R n . We conclude that L ∗ ∩ ∂ [ − , n is a finite union of closed convex subsets of R n and thus ANR. (cid:3) A. AVIL´ES AND J.D. RODR´IGUEZ ABELL ´AN
In the context of compact metric spaces, the retractions in the definition of ANR can betaken arbitrarily close to the identity. We state this fact as a lemma in the particular casethat we need:
Lemma 3.2.
Let L = { , . . . , n − } be a finite lattice. Then, given ε > , there existan open set V ε = V ε ( L ∗ ) with L ∗ ∩ ∂ [ − , n ⊂ V ε ⊂ R n and a continuous map ϕ : V ε −→ L ∗ ∩ ∂ [ − , n such that ϕ | L ∗ ∩ ∂ [ − , n = id L ∗ ∩ ∂ [ − , n and d ( x ∗ , ϕ ( x ∗ )) < ε for every x ∗ ∈ V ε ,where d is the square metric in R n .Proof. As L ∗ ∩ ∂ [ − , n is an ANR by Proposition 3.1, we cand find a bounded neighbor-hood V of L ∗ ∩ ∂ [ − , n in R n and a retraction ψ : V −→ L ∗ ∩ ∂ [ − , n . Let us take an openset W such that L ∗ ∩ ∂ [ − , n ⊂ W ⊂ W ⊂ V ⊂ R n . Now, ψ | W : W −→ L ∗ ∩ ∂ [ − , n is a continuous map between compact metric spaces, so it is uniformly continuous. Given ε >
0, there exists δ > d ( ψ ( x ∗ ) , ψ ( y ∗ )) < ε/ x ∗ , y ∗ ∈ W and d ( x ∗ , y ∗ ) < δ .Put η = min( ε/ , δ ) and take V ε = { x ∗ ∈ W : there exists y ∗ ∈ L ∗ ∩ ∂ [ − , n with d ( x ∗ , y ∗ ) < η } , and ϕ = ψ | V ε : V ε −→ L ∗ ∩ ∂ [ − , n . Clearly, ϕ is continuous and ϕ | L ∗ ∩ ∂ [ − , n = id L ∗ ∩ ∂ [ − , n . Let x ∗ ∈ V ε , and let y ∗ ∈ L ∗ ∩ ∂ [ − , n such that d ( x ∗ , y ∗ ) < η . Then, d ( x ∗ , ϕ ( x ∗ )) ≤ d ( x ∗ , y ∗ ) + d ( y ∗ , ϕ ( x ∗ )) = d ( x ∗ , y ∗ ) + d ( ϕ ( y ∗ ) , ϕ ( x ∗ )) < ε ε ε. (cid:3) Theorem 3.3. If L is a finite lattice, then F BL ∗ h L i is a projective Banach lattice.Proof. Let n be the cardinality of L . We may suppose that L = { , . . . , n − } with somelattice operations, and in this way we identity [ − , L with [ − , n . We fix ε >
0, and wewill construct the map u ε : F BL ∗ h L i −→ F BL ( L ) of Lemma 2.2. Let V ε and ϕ be given byLemma 3.2. By Urysohn’s lemma, we can find a continuous function 1 ε : ∂ [ − , n −→ [0 , ε ( x ∗ ) = 1 if x ∗ ∈ L ∗ ∩ ∂ [ − , n , and 1 ε ( x ∗ ) = 0 if x ∗ V ε . We define u ε ( f )( x ∗ ) = 1 ε ( x ∗ ) · f ( ϕ ( x ∗ )) if x ∗ ∈ V ε , and u ε ( f )( x ∗ ) = 0 if x ∗ / ∈ V ε , for every f ∈ F BL ∗ h L i and x ∗ ∈ ∂ [ − , n . We extend the definition for elements x ∗ ∈ [ − , n \ ∂ [ − , n insuch a way that u ε ( f ) is positively homogeneous. Since L is finite, the fact that u ε ( f ) iscontinuous on ∂ [ − , n and positively homogeneous guarantees that u ε ( f ) ∈ F BL ( L ). Itis easy to check that u ε is a Banach lattice homomorphism and that R ◦ u ε = id F BL ∗ h L i . Itwould remain to check that k u ε k ≤ ε . We will prove instead that for this u ε we have k u ε k ≤ nε , which is still good enough. We know that k u ε k = sup {k u ε ( f ) k : f ∈ F BL ∗ h L i , k f k ∗ ≤ } , ROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BY A LATTICE 7 where k u ε ( f ) k = sup ( m X i =1 | r i u ε ( f )( x ∗ i ) | : x ∗ i ∈ ∂ [ − , n , r i ∈ R , sup x ∈ L m X i =1 | r i x ∗ i ( x ) | ≤ ) . So we fix f ∈ F BL ∗ h L i with k f k ∗ ≤
1, where k f k ∗ = sup ( m X i =1 | s i f ( y ∗ i ) | : y ∗ i ∈ L ∗ , s i ∈ R , sup x ∈ L m X i =1 | s i y ∗ i ( x ) | ≤ ) , and we want to prove that k u ε ( f ) k ≤ nε . Using the expression of k u ε ( f ) k as a supremum,we pick x ∗ , . . . , x ∗ m ∈ ∂ [ − , n , r , . . . , r m ∈ R such that sup x ∈ L P mi =1 | r i x ∗ i ( x ) | ≤
1, and wewant to prove that m X i =1 | r i u ε ( f )( x ∗ i ) | ≤ nε. The first estimation is that m X i =1 | r i u ε ( f )( x ∗ i ) | = X x ∗ i ∈ V ε | r i ε ( x ∗ i ) f ( ϕ ( x ∗ i )) | ≤ X x ∗ i ∈ V ε | r i f ( ϕ ( x ∗ i )) | . If we write y i ∗ := ϕ ( x ∗ i ), the inequality above becomes( ⋆ ) m X i =1 | r i u ε ( f )( x ∗ i ) | ≤ X x ∗ i ∈ V ε | r i f ( y i ∗ ) | . On the other hand, if x ∈ L then X x ∗ i ∈ V ε | r i y ∗ i ( x ) | = X x ∗ i ∈ V ε | r i ϕ ( x ∗ i )( x ) |≤ X x ∗ i ∈ V ε | r i x ∗ i ( x ) | + X x ∗ i ∈ V ε | r i | | ϕ ( x ∗ i )( x ) − x ∗ i ( x ) |≤ ε X x ∗ i ∈ V ε | r i | ≤ εn. The last inequality is because x ∗ i ∈ ∂ [ − , n , and therefore m X i =1 | r i | = m X i =1 | r i | sup x ∈ L | x ∗ i ( x ) | ≤ X x ∈ L m X i =1 | r i || x ∗ i ( x ) | ≤ | L | · n. A. AVIL´ES AND J.D. RODR´IGUEZ ABELL ´AN
Taking s i = r i nε , we have that, for all x ∈ L , X x ∗ i ∈ V ε | s i y ∗ i ( x ) | = X x ∗ i ∈ V ε (cid:12)(cid:12)(cid:12)(cid:12) r i nε y ∗ i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ . Thus, the s i and the y i are as in the supremum that defines k f k ∗ ≤
1. Therefore X x ∗ i ∈ V ε | s i f ( y ∗ i ) | ≤ , and getting back to our initial estimation ( ⋆ ), we get m X i =1 | r i u ε ( f )( x ∗ i ) | ≤ X x ∗ i ∈ V ε | r i f ( y ∗ i ) | ≤ nε. (cid:3) Projectivity of the free Banach lattice generated by an infinitelinear order
Now, we are going to prove that if L is an infinite linear order containing either anincreasing sequence without upper bounds or a decreasing sequence without lower bounds,then F BL ∗ h L i is not projective. This will be a direct consequence of the fact that the freeBanach lattice generated by the set of the natural numbers is not projective. In the proof,we will use the following: Lemma 4.1.
Suppose that ϕ i : [ − , N −→ R , i = 1 , , . . . , are continuous functions suchthat, for every i ,(1) ϕ i (( x n ) n ∈ N ) = x i whenever x ≤ x ≤ . . . ,(2) ϕ i ( x ) ≤ ϕ i +1 ( x ) for all x ∈ [ − , N .Then, when we view the ϕ i ’s as elements of the free Banach lattice F BL ( N ) , the sequenceof norms k ϕ i k F BL ( N ) is unbounded.Proof. Let π i : [ − , N −→ [ − ,
1] be the projection on the i -th coordinate. Consider theset M := (cid:8) ( x n ) n ∈ N ∈ [ − , N : x ≤ x ≤ . . . (cid:9) ⊂ [ − , N . Since M is closed and [ − , N with the product topology is compact, we have that M is compact. Condition (1) in thelemma means that ϕ i | M = π i | M for all i . Using the compactness of M and the continuityof ϕ i and π i , this implies that there exists a neighborhood U i of M such that d ( ϕ i | U i , π i | U i ) = sup x ∈ U i | ϕ i ( x ) − π i ( x ) | < . ROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BY A LATTICE 9
For an integer k ≥
3, let W k := (cid:8) ( x n ) n ∈ N ∈ [ − , N : x i < x j + k − whenever i < j < k (cid:9) . The family { W k : k ≥ } is a neighborhood basis of M . We define inductively an increasingsequence of natural numbers k < k < k < k < · · · , and a sequence of points y , y , . . . ∈ [ − , N as follows. We take k = 1 as a starting point of the induction. Suppose that wehave defined k j for j < n . We choose k j +1 > k j such that W k j +1 ⊂ U k j , and we define y j +1 : N −→ [ − ,
1] to be the map given by y j +1 ( n ) = if n < k j , if k j ≤ n < k j +1 , if n ≥ k j +1 . We have y j +1 ∈ W k j +1 , so | ϕ k j ( y j +1 ) − π k j ( y j +1 ) | = | ϕ k j ( y j +1 ) − | < and ϕ k j ( y j +1 ) > .When j + 1 ≤ m , using condition (1) of the Lemma, we get that ϕ k m ( y j +1 ) ≥ ϕ k j ( y j +1 ) > . Remember how the norm is defined: k ϕ k F BL ( N ) = sup ( m X j =1 | r j ϕ ( x j ) | : r j ∈ R , x j ∈ [ − , N , sup n ∈ N m X j =1 | r j x j ( n ) | ≤ ) . We have that sup n ∈ N | y ( n ) + · · · + y m ( n ) | = 1, therefore k ϕ k m k F BL ( N ) ≥ | ϕ k m ( y ) | + · · · + | ϕ k m ( y m ) | > m . (cid:3) Now, let N + = N ∪ { + ∞} . Lemma 4.2.
F BL ∗ h N i and F BL ∗ h N + i are is not projective.Proof. First, if If
F BL ∗ h N i was projective, then for ε > u ε : F BL ∗ h N i −→ F BL ( N ) like in Proposition 2.2. Remember that if i ∈ N , ˙ δ i : N ∗ −→ R isthe map given by ˙ δ i ( x ∗ ) = x ∗ ( i ) for every x ∗ ∈ N ∗ , that is an element of F BL ∗ h N i . Weconsider ϕ i = u ε ( ˙ δ i ) ∈ F BL ( N ), that we view as continuous functions ϕ i : [ − , N −→ R .The fact that u ε is a lattice homomorphism gives condition (2) of Lemma 4.1, while thefact that R ◦ u ε = id F BL ∗ h N i gives condition (1) of Lemma 4.1. The fact that k u ε k ≤ ε contradicts the conclusion of Lemma 4.1. Here there was a proof that
F BL ∗ h N + i was not projective, but this was wrong. (cid:3) The following fact can be viewed as a corollary of Proposition 2.1, but we state if forconvenience:
Lemma 4.3.
Let P and P ′ be Banach lattices, and let ˜ π : P −→ P ′ and ˜ ı : P ′ −→ P beBanach lattice homomorphisms such that k ˜ ı k = k ˜ π k = 1 and ˜ π ◦ ˜ ı = id P ′ . If P is projective,then P ′ is projective.Proof. In order to check the projectivity of P ′ , let Q : X −→ X/J , T ′ : P ′ −→ X/J and ε > P considering T = T ′ ◦ ˜ π ,so we get ˆ T : P −→ X such that Q ◦ ˆ T = T ′ ◦ ˜ π and k ˆ T k ≤ (1 + ε ) k T k ≤ (1 + ε ) k T ′ k . Thedesired lift is ˆ T ′ = ˆ T ◦ ˜ ı . On the one hand k ˆ T ′ k ≤ k ˆ T k ≤ (1 + ε ) k T ′ k , and on the otherhand Q ◦ ˆ T ◦ ˜ ı = T ′ ◦ ˜ π ◦ ˜ ı = T ′ . (cid:3) Theorem 4.4.
Let L be an infinite linearly ordered set containing either an increasingsequence without upper bounds or a decreasing sequence without lower bounds. Then, F BL ∗ h L i is not projective.Proof. L contains either an increasing or a decreasing sequence. Let us suppose withoutloss of generality that it contains an increasing sequence x < x < x < · · · without upperbounds.First, suppose that it has no upper bound. The map ı : ( N , ≤ ) −→ ( L , (cid:22) ) given by ı ( n ) = x n for every n ∈ N is a lattice homomorphism. Let π : L −→ N be the map givenby π ( x ) = (cid:26) x < x ,n if x ∈ [ x n , x n +1 ) for any n ≥ . Notice that π is also a lattice homomorphism and π ◦ ı = id N . We are going to use theuniversal property of the free Banach lattice over a lattice as stated in Definition 1.1. Let φ L and φ N be the canonical inclusion of L and N into F BL ∗ h L i and F BL ∗ h N i , respectively,and let ˜ ı : F BL ∗ h N i −→ F BL ∗ h L i and ˜ π : F BL ∗ h L i −→ F BL ∗ h N i be the correspondingextensions of φ L ◦ ı and φ N ◦ π according to Definition 1.1. The composition ˜ π ◦ ˜ ı and theidentity mapping F BL ∗ h N i −→ F BL ∗ h N i are both extensions of φ N so by the uniquenessin Definition 1.1, ˜ π ◦ ˜ ı = id F BL ∗ h N i . We can apply Lemma 4.3, so if F BL ∗ h L i was projective,then F BL ∗ h N i would also be projective, in contradiction with Lemma 4.2. ROJECTIVITY OF THE FREE BANACH LATTICE GENERATED BY A LATTICE 11
The rest of the proof must be omitted (cid:3)
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Universidad de Murcia, Departamento de Matem´aticas, Campus de Espinardo 30100 Mur-cia, Spain.
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