A Separable Universal Homogeneous Banach Lattice
aa r X i v : . [ m a t h . F A ] A ug A SEPARABLE UNIVERSAL HOMOGENEOUS BANACHLATTICE
M.A. TURSI
Abstract.
We prove the existence of a separable approximately ultra-homogeneous Banach lattice BL that is isometrically universal for sepa-rable Banach lattices. This is done by showing that the class of Banachlattices has the Amalgamation Property, and thus finitely generated Ba-nach lattices form a metric Fra¨ıss´e class. Some additional results aboutthe structural properties of BL are also proven. Introduction
This paper explores homogeneity in the class of separable Banach lattices.We prove the existence of isometrically universal lattices for appropriateclasses with varying levels of homogeneity and give some initial results aboutthe structure of such spaces.Some comparable results are already known in Banach space theory. TheGurarij space G is an isometrically universal separable Banach space withthe following homogeneity property: for any finite dimensional spaces A ⊆ B , any isometric embedding f : A → G , and any C >
1, there exists a map g : B → G extending f such that C k x k ≤ k g ( x ) k ≤ C k x k for all x ∈ B .Such separable Banach spaces are isometrically unique (see [16], as well asa simplified proof by Kubi´s and Solecki in [12]). An alternate construc-tion by Ben Yaacov characterizing G as a metric Fra¨ıss´e limit is found in[5]. As a Fraisse limit, G has another kind of homogeneity that strength-ens isomorphic embeddings with small distortion to isometric embeddingswhile sacrificing full commutativity. In other words, given finite dimensionalspaces A ⊆ B , an isometric embedding f : A → G , and ε >
0, there existsan isometric embedding g : B → G such that k f − g | A k < ε . Since G isa Fra¨ıss´e limit, it is also isometrically unique among separable spaces withthis property.We prove the lattice analogue of the above stated result. Using Fra¨ıss´emachinery, we show that there is a unique isometrically universal separa-ble Banach lattice BL with the following homogeneity property: for anylattices A ⊆ B generated by finitely many elements and lattice isometric Date : August 18, 2020.
M.A. TURSI embeddings f : A → BL , for all ( a , ..., a n ) ⊆ A generating A , and for all ε >
0, there exists a lattice isometric embedding g : B → BL such thatfor each a i , k f ( a i ) − g ( a i ) k < ε (Theorem 4.1). The key to this result isthe fact that Banach lattices have the Amalgamation Property (Theorems3.10 and 3.12). Observe that if A and B are finite dimensional, we canstrengthen almost commutativity of the diagram restricted to generators toalmost commutativity in norm. BL can also be constructed as an inductivelimit of ℓ m ∞ ( ℓ n ) lattices, paralleling the construction of the Gurarij space asa limit of ℓ n ∞ spaces (Theorem 4.3). BL does not have the homogeneity property that originally characterized G , however, because in certain cases one cannot extend a lattice isometricembedding in a way that preserves both lattice structure and full commuta-tivity in the separable setting. In addition, even though G can be ”almost”homogeneous in either of the forms mentioned above, it cannot fully homo-geneous in the sense of requiring both isometric embeddings and full commu-tativity of the diagram. Since it is unique, no separable spaces can have thisstronger property. There exist non-separable Banach spaces, however, thatare fully homogeneous, not just for the class of finite dimensional spaces,but also for separable spaces. Such spaces, referred to as spaces of universaldisposition, are constructed by Avil´es, S´anchez, Castillo, and Moreno in [2].A different construction (which assumes the CH) using Fra¨ıss´e sequences isgiven in [11], where uniqueness is also established. Very recently, Avil´es andTradecete also constructed a (necessarily non-separable) lattice of universaldisposition for separable lattices [3].Homogeneity in sublcasses of Banach lattices has been recently explored atlength by Ferenczi, Lopez-Abad, Mbombo, and Todorcevic [9]. This papertreats on various levels of homogeneity in L p Banach spaces, but it alsoexplores lattice homogeneity. Specifically, for 1 ≤ p < ∞ , the separablespaces L p (0 ,
1) are Fra¨ıss´e limits for ℓ np spaces with lattice embeddings ascorresponding maps. The authors also construct an approximately ultra-homogeneous M -space for the class of finite dimensional M spaces.Outside of the Banach lattice setting, homogeneous structures have beenfound for various classes. Using injective objects, Lupini proved the ex-istence of homogeneous structures for the classes of function systems, p -multinormed spaces, and M q -spaces [15]. Certain C ∗ -algebras can also beconstructed as Fra¨ıss´e limits of appropriate classes with relaxed conditions,including all UHF algebras, the hyperfinite II -factor [7], the Jiang-Su alge-bra [17], and more recently, a projectively universal AF-algebra constructedin [10]. SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 3 Preliminaries
We introduce definitions and notation that will be used in this paper. Forgeneral information about Banach lattices, we refer the reader to [18, 14].Throughout the paper, all Banach lattices are assumed to be real.Given a Banach lattice X , we let B ( X ) and S ( X ) respectively refer to theunit ball and unit sphere of X . For A ⊆ X , let A + denote the elements of A in the positive cone, and let BL ( A ) be the Banach lattice generated by A . We say that x ∈ A is an order extreme point of A if for all y, z ∈ A , if x ≤ ty − (1 − t ) z for 0 < t <
1, then x = y = z . A set A ⊆ X is called solid if for all a ∈ X and b ∈ A , if | a | := a ∨ ( − a ) ≤ | b | , then a ∈ A . It turns outthat for any solid set A ∈ X , a point a ∈ A is an extreme point if and onlyif | a | is order extreme [19, Theorem 19.2]. We also let EP ( A ) be the set ofextreme points of A , let OEP ( A ) be the set of order extreme points of A ,and let SCH ( A ) ( CSCH ( A )) be the (closed) solid convex hull of A .The following is largely taken from [5]: let L be a collection of symbols.These can be either predicate symbols or function symbols . Each predicateor function symbol has an associated number called its arity . We then call A with associated metric space A an L -structure if(1) For every predicate symbol R with arity n , there is a continuousinterpretation R A : A n → R . We can also consider the distance tobe a binary symbol (found in every structure).(2) For every function symbol f with arity n , we have a continuousinterpretation f A : A n → A . Note that if a function symbol c has0-arity, then it is a constant symbol , and c A ∈ A .These are different from the typical definitions of L -structures in continuouslogic as found in [4], the latter which require uniform continuity for functionsand predicates but do not require that X and Y be bounded. The theoryof Banach lattices can be formulated in the language L = (+ , R , ∧ , ∨ ). Inparticular, its function and predicate symbols have corresponding moduliof uniform continuity which are independent of their interpretation in aparticular lattice. Given x = ( x , ..., x n ), y = ( y , ..., y n ) ⊆ X , where X is ametric space, we let d ( x, y ) = max i ≤ n d ( x i , y i ) . As in [4, Chapter 2], we define the modulus of uniform continuity. A func-tion ∆ f : R + → (0 ,
1] is a modulus of uniform continuity for a L -functionor predicate symbol f of arity n if for all L -structures M and x, y ∈ M n , d ( x, y ) < ∆ f ( ε ) implies that d ( f M ( x ) , f M ( y )) < ε . For example, the func-tion symbol ∧ has modulus of continuity ∆( ε ) = ε . That is, given alattice X and ( x , y ) and ( x , y ) ∈ X , if d (( x , y ) , ( x , y )) < ε , then M.A. TURSI k x ∧ y − x ∧ y k < ε . The definition of moduli of continuity in the appen-dix in [4, Chapter 2] assumes that moduli have domains restricted to (0 , R + by letting ∆ f ( r ) = ∆ f (1) for all r >
1. Propositions 2.4 and 2.5 in [4] show that compositions of uniformlycontinuous real functions and L -function and L -predicate symbols also havecorresponding moduli of uniform continuity, since they are also uniformlycontinuous.We say that A is a substructure of B if A is a closed subset of B which is alsoclosed under all combinations of the function symbol operations. For Banachlattices, X is a substructure of Y if it is a sublattice of Y . Let f : A → B bea map between two L structures. If f preserves norms, function operations,and predicate symbols in L , then f is considered a embedding .Let φ : X → Y be a map between two Banach lattices. We say that φ isa lattice homomorphism if it is a bounded linear map that also preservesthe lattice operations (i.e., φ ( x ∧ y ) = φ ( x ) ∧ φ ( y )). To check whethera linear map is also a lattice homomorphism, by [1, Theorem 1.34], it isenough to check that it is positive ( x ≥ ⇒ φ ( x ) ≥
0) and preservesdisjointness. That is, if x, y ∈ X and x ⊥ y (i.e., | x | ∧ | y | = 0), then φ ( x ) ⊥ φ ( y ). For C ≥
1, we say that φ is a C -embedding , if for all x ∈ X , C k x k ≤ k φ ( x ) k ≤ C k x k . If C = 1, then φ is an embedding between Banachlattice structures, so we simply call it an embedding. In subsequent sections,since this paper mainly deals with Banach lattices, we refer to lattice em-beddings simply as embeddings. If a C -embedding is also surjective, then itis called an C - isometry , and if C = 1, it is simply an isometry. Observe thatfor any C ≥
1, if φ : X → Y is a C -embedding, then φ ( X ) is a sublatticeof Y , φ is a C -isometry from X onto φ ( X ), and φ − is a C -isometry from φ ( X ) onto X .Let A ⊆ B . We then let < A > be the substructure generated by A . Thiscan be understood as the smallest set A ⊆ B with A ⊆ A and A a substruc-ture of B . We say that A is finitely generated if there exist ( a , ...a n ) ⊆ A such that A = < ( a , ..., a n ) > . Suppose that K is a class of finitely gener-ated L -structures. If A ∈ K , we say that A is a K - structure if every finitelygenerated substructure of A is also in K .For A = < a > and B = < b > with | a | = | b | , we define d K ( a, b ) = inf φ : A → Cφ : B → C d ( φ ( a ) , φ ( b )) , where φ and φ are both embeddings into some ambient K -structure C . Ifwe clearly understand generating tuples a and b for lattices A and B to be SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 5 in some larger ambient space without necessary reference to explicit embed-dings, we just write d ( a, b ) instead of d ( φ ( a ) , φ ( b )).Let K be a class of finitely generated structures. We then say K is Fra¨ıss´e if: • K has the
Hereditary Property (HP): every member of K is a K -structure. • K has the Joint Embedding Property (JEP): any two K -structuresembed into a third. (Note that if K has the JEP, then d K is definedfor all pairs of tuples in K of the same length.) • K has the Near Amalgamation Property (NAP): for any structures A = < a > , B and B in K with embeddings f i : A → B i , and forall ε >
0, there exists a C ∈ K and embeddings g i : B i → C suchthat d ( g ◦ f ( a ) , g ◦ f ( a )) < ε. If g ◦ f = g ◦ f , then we just say that K has the AmalgamationProperty (AP). Clearly the AP implies the NAP. • K has the Polish Property (PP): if K has the JEP, HP and NAP,then d K is a pseudo-metric over K . If d K is separable and completein K n (the K -structures generated by n many elements): • K has the Continuity Property (CP): every symbol in L is continuouson K : that is, for function symbols, the map ( a, b ) ( a, b, f ( a ) ( a ))is continuous, and for predicate symbols P , the map a P a ( a ) iscontinuous.By [5, Theorem 3.21], if K is Fra¨ıss´e, there exists a separable space M ,known as the Fra¨ıss´e limit , that is universal for K and approximately ultra-homogeneous on K . That is, for all finitely generated structures A = ⊆ M , embeddings f : A → M , and ε >
0, there exists an automorphism φ : M → M such that d ( f ( a ) , φ ( a )) < ε . Conversely, if a space M is ap-proximately ultra-homogeneous, its finitely generated substructures form aFra¨ıss´e class, and M is its limit. Such a space is also isometrically universalfor all separable K structures (including those which are not finitely gener-ated).Instead of the PP and CP, a class K may have the following weakenedconditions: • The
Weak Polish Property (WPP): the metric d K is separable (butnot necessarily complete) • The
Cauchy Continuity Property (CCP): the map ( a, b ) ( a, b, f ( a ) ( a ))sends d K - Cauchy sequences to Cauchy sequences, and for predicatesymbols P , the map a P a ( a ) sends Cauchy sequences to Cauchysequences, M.A. TURSI If K has the HP, JEP, and NAP in addition to the two conditions above,then K is an incomplete Fra¨ıss´e class . A relevant example is that of finitedimensional ℓ p spaces ([5, Section 4.2] gives a brief discussion). These havea (unique) Fra¨ıss´e limit of their completion, which is the class of separable L p spaces, and the limit is L p (0 , C be a class of Banach lattices. A lattice X isof approximately universal disposition for a class C with lattices defined byfinitely many elements if for all A ∈ C and for all embeddings f : A → X , g : A → B , with A ∈ C defined by a and B ∈ C , and for all ε >
0, thereexists an (1 + ε )-embedding h : B → X such that k h ◦ g ( a ) − f ( a ) k < ε . Ap-proximate universal disposition relaxes the condition in approximate ultra-homogeneity of the existence of an embedding down to a (1 + ε )-embeddingfor arbitrarily small ε .Definition by finitely many elements in lattices can occur in more than oneway. One can speak, for example, of finite generation in the context of thelogic of metric structures. On the other hand, one might refer to finite di-mensional lattices. For spaces of approximately universal disposition, if theclass in question is finite dimensional lattices, we can let the finitely manyatoms define the lattice’s basis rather than generators doing so (in fact anyfinite dimensional lattice can be generated by two elements: see Theorem4.4), and we can strengthen the requirement that k h ◦ g ( a ) − f ( a ) k < ε to anorm requirement that k h ◦ g − f k < ε .Throughout the paper we rely on the notion of finite branchability. Let E be a Banach lattice. Let ( A n ) n be a sequence of finite non-empty sets, andlet T = ∪ ∞ k =0 Q kn =1 A n be the tree generated by them. Suppose also that( x σ ) σ ∈ T ⊆ E + . We then say that ( x σ ) is a finitely branching tree in E + iffor all σ with | σ | = k , ( x ( σ ⌢ b ) ) | σ | = k is disjoint, and x σ = X b ∈ A k +1 x ( σ ⌢ b ) . Note that given the property outlined in the definition, span( { ( x σ ) σ ∈ T } ) isa vector lattice in E . If span( { ( x σ ) σ ∈ T } ) is dense in E for some finitelybranching tree ( x σ ), we call E finitely branchable. Finitely branchable lat-tices allow us to reduce problems involving finitely generated, but infinitedimensional lattices to that of finite dimensional lattices, since they are in-ductive limits of finite dimensional lattices. It is also easy to show the otherdirection: If a lattice is the inductive limit of finite dimensional lattices, thenit is finitely branchable. Finally, observe that finitely branchable lattices are
SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 7 separable.Throughout, we will be working with two named classes of Banach lattices:let K be the class of finitely generated lattices, and K ′ be the class of sub-lattices of ℓ m ∞ ( ℓ n ) spaces, with m, n ∈ N . Let K n ⊆ K be the class of latticesgenerated by n elements, and likewise for K ′ n ⊆ K ′ . Here we do not requirethat the generating elements be distinct or minimal. We also will makeuse of the isometrically universal separable lattice U := C (∆ , L [0 , U is finitely branchable and in particularis the inductive limit of an increasing union of lattices in K ′ , which will beuseful later on.We conclude this section with an outline of the rest of this paper. Section 3explores the AP in Banach lattices and is split into two subsections. In thefirst, we show show that any finite dimensional lattice can be approximatedby a lattice in K ′ with arbitrarily small distortion (Lemma 3.2). We use thisresult to prove an approximate amalgamation property for finite dimensionallattices (Theorem 3.5). In particular, it is shown that K ′ has the AP. In thesecond subsection, we use the results in the first subsection to show thatthe class of Banach lattices has the AP (Theorems 3.10 and 3.12). Thekey to expanding the results on K ′ is the use of finitely branchable lattices.We then end the section with some additional results on amalgamation over C -embeddings.In Section 4, we prove the existence of a separable approximately ultra-homogeneous lattice BL by showing that K is a metric Fra¨ıss´e class andexplore some of its structural properties (Theorem 4.1). The subclass K ′ isnot just the first step to amalgamation; it is itself an incomplete Fra¨ıss´e classthat is dense in the class of finitely generated separable lattices accordingto the Fra¨ıss´e metric (Lemma 4.2). We use this fact to show that BL isfinitely branchable (Theorem 4.3). Finitely branchable lattices are them-selves finitely generated (Theorem 4.4), so unlike the Gurarij space, BL isfinitely generated, and in particular can be generated by two elements.In Section 5, we show that any separable lattice of approximately universaldisposition for finitely generated lattices is isometric to BL (Theorem 5.2).We also construct lattices of approximately universal disposition for finitedimensional lattices and show that any such lattice which is also finitelybranchable is isometric to BL (Theorem 5.4). Finally, we show a self-similarity property of BL : any non-trivial projection band in BL is iso-metric to BL (Theorem 5.5). M.A. TURSI Banach lattices and the Amalgamation Property
The bulk of this section is dedicated to proving that the class of Banachlattices has the AP. As this paper was nearing its completion, Avil´es andTradecete independently proved that Banach lattices have the Amalgama-tion Property by generating pushouts using free Banach lattices (see [3,Theorem 4.4]). We give an alternative approach. We first show that K ′ it-self has the AP, and then expand this result to K and to lattices in general.3.1. The Amalgamation Property in K ′ . We start with the followinglemma:
Lemma 3.1.
Let X be a finite dimensional lattice. Then the following areequivalent:(1) OEP ( B ( X )) is finite.(2) EP ( B ( X )) is finite.(3) EP ( B ( X ∗ )) is finite.Proof. (1) is equivalent to (2) by finite dimensionality and Theorem 19.2 in[19]. To show that (2) implies (3), suppose B ( X ) has finitely many extremepoints. Then by Theorem 16 in [8], it is the intersection of finitely manyclosed half-spaces. Let f , ..., f m ∈ S ( X ∗ ) such that B ( X ) = { x ∈ X : f i ( x ) ≤ ≤ i ≤ m } . Then k x k = max f i ( x ) for all x ∈ X , so B ( X ∗ ) = CH { f , ..., f m } . Otherwise, if g ∈ S ( X ∗ ) \ CH { f , ..., f m } , by theHahn-Banach separation theorem there exists some x ∈ S ( X ) such thatsup i f i ( x ) < g ( x ) . By Milman’s theorem, all the extreme points of B ( X ∗ ) are contained in { f , ..., f m } , so B ( X ∗ ) has finitely many extreme points. By reflexivity offinite dimensional lattices, (3) implies (2) as well. (cid:3) Lemma 3.2.
Let X be a finite dimensional Banach lattice. Then for all C > , there exists a C -embedding from X into an ℓ m ∞ ( ℓ M ) space for some m , with M = dim X . If, furthermore, X has finitely many order extremepoints, then X embeds isometrically into ℓ m ∞ ( ℓ M ) space for some m .Proof. Suppose { x ∗ , ..., x ∗ m } is an ε -net on S ( X ∗ ) + , where C < − ε . Thenfor all x ∈ S ( X ), we have C < − ε ≤ sup i x ∗ i ( | x | ) ≤
1. Now X ∗ is alsofinite and is thus generated by its atoms, which are the evaluation function-als e ∗ i for the atoms e i ∈ X , with 1 ≤ i ≤ M . That is, if x = P j c j e j ,then e ∗ i ( x ) = c i . These functionals form a basis in X ∗ , so we can assume x ∗ i = P j a ( i, j ) e ∗ j , with a ( i, j ) ≥
0. Based on this, consider the lattice ℓ m ∞ ( ℓ M ), and let u ( i, j ) ∈ ℓ m ∞ ( ℓ M ) correspond to the j ’th atom in the i ’thcopy of ℓ M . Then let φ ( e j ) = P i a ( i, j ) u ( i, j ). SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 9 φ is a lattice homomorphism, since it is a positive linear map that mapsatoms to disjoint elements. It also is a C -embedding. Indeed, let x = P c j e j ∈ S ( X ) + . Then φ ( x ) = X j c j φ ( e j ) = X j X i c j a ( i, j ) u ( i, j ) , so k φ ( x ) k = sup i X j | c j | a ( i, j ) = sup i X j a ( i, j ) e ∗ j ( x ) = sup i x ∗ i ( x ) . Thus C k x k ≤ k φ ( x ) k ≤ k x k .If B ( X ) has finitely many order extreme points, then by Lemma 3.1, sodoes the dual unit ball B ( X ∗ ). Let { x ∗ , ..., x ∗ m } = OEP ( B ( X ∗ )). Then k x k = sup ≤ i ≤ m x ∗ i ( | x | ). Construct φ in the same way as above, and observethat k φ ( x ) k = sup i x ∗ i ( x ) = k x k . (cid:3) Lattices in K ′ play a key role in subsequent results on homogeneous latticesand their structure. We present some of the notation that will be used insubsequent proofs: Suppose F ∈ K ′ , and let ( e , ..., e m ) be the atoms of F .Let f : F → G := ℓ N ∞ ( ℓ M ) be a C -embedding, with C ≥
1. Let u ( k, j )be j ’th atom in the k ’th copy of ℓ M . We then have, for each e i ∈ F , that f ( e i ) = P k,j a i ( k, j ) u ( k, j ). Note that f maps atoms to disjoint positiveelements, so we can just let a i ( k, j ) = a ( k, j ), and sum up only over atomsthat support f ( e i ). Specifically, we fix a row k and let F ki = { j ≤ M : f ( e i ) ∧ u ( k, j ) > } . Then f ( e i ) = X k X j ∈ F ki a ( k, j ) u ( k, j ) . Observe that F and f induce an N × m matrix A fF , with A fF ( k, i ) = P j ∈ F ki a ( k, j ). If F ki is empty, then A fF ( k, i ) = 0. It turns out the rowsof A fF capture F ’s structure completely, while small distortions in f implysmall distortions in A fF . We give a lemma to this effect. From now on, ifwe have two C -isometries f j : F → G j with j = 1 , C -isometries with C ≥
1, with G and G both ℓ N ∞ ( ℓ M ) spaces, we just let A = A f F and B = A f F . For 1 ≤ l ≤ N , we also let A ( l ) = ( A ( l, , A ( l, , ..., A ( l, m )) and B ( l ) = ( B ( l, , B ( l, , ..., B ( l, m )). Lemma 3.3.
Let f j : F → G j with j = 1 , be C -isometries with C ≥ ,and suppose G and G be ℓ N ∞ ( ℓ M ) spaces. Then for all rows l , we have A ( l ) ∈ C SCH ( { B ( k ) : 1 ≤ k ≤ N } ) . In particular, if each f j is anembedding, then SCH ( { B ( k ) : 1 ≤ k ≤ N } ) = SCH ( { A ( k ) : 1 ≤ k ≤ N } ) . Proof.
Let r ∈ B ( ℓ M ∞ ) + . Then there exists some row k such that for allrows l , C P r n B ( l, n ) ≤ k P r n e n k ≤ C P r n A ( k, n ). Now that B ( l ) / ∈ C SCH ( { A ( k ) : k ≤ N } ) for some l . Then by [19, Proposition 19.7] thereexists some r ∈ B ( ℓ M ∞ ) + such that C sup y ∈ SCH ( A ( k )) X r n y n < X r n B ( l, n ) ≤ C X r n A ( k, n )for some k , which is a contradiction. (cid:3) In the case that C = 1, recall that the construction in Lemma 3.2 used N rows of ℓ M to correspond to the N order extreme points in the unit ball of X ∗ . Since we can think of the rows in A fF as elements in the dual space F ∗ , where A ( l )( P c i e i ) = P c i A ( l, i ), then by Lemma 3.3, we actually have SCH ( { A ( l ) } ) = B ( F ∗ ). In particular, any F ∈ K ′ has finitely many orderextreme points. Combined with Lemma 3.1, we thus have the followingresult: Corollary 3.4.
The four following properties are equivalent for finite di-mensional lattices X :(1) OEP ( B ( X )) is finite.(2) EP ( B ( X )) is finite.(3) EP ( B ( X ∗ )) is finite.(4) X ∈ K ′ . We now prove the following:
Theorem 3.5.
Suppose for j = 1 , , f j : E → F j are C -embeddings with F and F in K ′ with C ≥ . Then there exist G ∈ K ′ and C -embeddings g j : F j → G such that g ◦ f = g ◦ f . That is, the following diagramcommutes: E FF G f f g g In particular, K ′ has the AP.Proof. We can assume that F j = ℓ N ∞ ( ℓ M j ), where M j = dim F j for j = 1 , e , ..., e n be the atoms in E , and for F and F , we let u ( k, j ) and v ( k, j ),respectively, correspond to the j ’th atom in the k ’th row (that is, the k ’thcopy of ℓ M j ). For row l and atom i , we let F l ,i = { j ≤ M : f ( e i ) ∧ u ( l, j ) > } , and similarly, we let F l ,i = { j ≤ M : f ( e i ) ∧ v ( l, j ) > } SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 11
We now define g and g . Let F ′ j be the the lattice ideal in F j generated by f j ( E ), and let ( F ′ ⊗ F ′ ) ⊕ F ⊕ F be understood as a vector lattice withatoms of the form u ( k, j ) ⊗ v ( l, m ), u ( k, j ), and v ( l, m ). For u ( k, j ) with j ∈ F i ,k , let g ( u ( k, j )) = u ( k, j ) ⊗ f ( e i ). If u ( k, j ) / ∈ F i ,k for any i , let g ( u ( k, j )) = u ( k, j ). For v ( l, m ) ∈ F i ,l , let g ( v ( l, m )) = f ( e i ) ⊗ v ( l, m ),and if v ( l, m ) / ∈ F i ,l for any i , let g ( v ( l, m )) = v ( l, m ). First, we show that g ◦ f = g ◦ f . Indeed, we have g ◦ f (cid:18) X i c i e i (cid:19) = X i c i g (cid:18) X k X j ∈ F i ,k a ( k, j ) u ( k, j ) (cid:19) = X i c i (cid:18) X k X j ∈ F i ,k a ( k, j ) u ( k, j ) (cid:19) ⊗ f ( e i )= X c i ( f ( e i ) ⊗ f ( e i )) , and similarly: g ◦ f (cid:18) X i c i e i (cid:19) = X i c i g (cid:18) X l X m ∈ F i ,l b ( l, m ) v ( l, m ) (cid:19) = X i c i f ( e i ) ⊗ (cid:18) X l X m ∈ F i ,l b ( l, m ) v ( l, m ) (cid:19) = X c i ( f ( e i ) ⊗ f ( e i )) . Let G = BL ( g ( F ) ∪ g ( F )), and let the unit ball of G be SCH (cid:0) g ( B ( F )) ∪ g ( B ( F )) (cid:1) .Note that g and g are both contractive maps. We now show that they arealso C -embeddings. This will be sufficient, because then we can replace g and g with Cg and Cg while still preserving commutativity in the dia-gram. These latter maps are themselves C -embeddings, thus proving thetheorem. Since g i ( B ( F i )) has only finitely many order extreme points, andsince the resulting space is finite dimensional, SCH (cid:0) g ( B ( F )) ∪ g ( B ( F )) (cid:1) is also closed. So we need only to show without loss of generality that if x, y ∈ S ( F ) + , z ∈ S ( F ) + , and rg ( x ) ≤ tg ( y )+(1 − t ) g ( z ) with 0 ≤ t ≤ r < C .Suppose x, y and z are as above. Since for some k ≤ N , (cid:13)(cid:13) x ∧ (cid:0) P j u ( k, j ) (cid:1)(cid:13)(cid:13) =1 and g (cid:0) x ∧ ( P j u ( k, j )) (cid:1) ≤ g ( x ), we can assume that x = P j c j u ( k, j )with P c j = 1. Furthermore, since rg ( x ) ≤ tg ( y ) ∧ rg ( x ) + (1 − t ) g ( z ) ∧ rg ( x ) , we can also assume that tg ( y ) ≤ rg ( x ), so y = P d j u ( k, j ), with P d j ≤ z = P M µ n z n , where z n is an order extreme point in F ;that is, there is a sequence s n = ( s nl ) l of length N such that z n = P l v ( l, s nl ),and furthermore, µ n > P µ n = 1. Then0 ≤ g ( rx − ty ) ≤ g (cid:18) X i X l X n : s nl = m ∈ F i ,l µ n v ( l, m ) (cid:19) . Now both sides of the inequality are supported, and the left hand side fullysupported, by atoms of the form u ( k, j ) ⊗ v ( l, m ) where j ∈ F i ,k and m ∈ F i ,l .Thus, for any u ( k, j ) ∈ F ′⊥ , we must have rc j − td j = 0, since g ( F ′⊥ ) isdisjoint from g ( F ), and similarly g ( F ′⊥ ) is disjoint from g ( F ). Therefore rx − ty = X i X j ∈ F i ,k ( rc j − td j ) u ( k, j )Recall that for each coefficient the left hand side must be less than or equalto the right hand side. Evaluating both sides, we thus have that X i (cid:18) X j ∈ F i ,k ( rc j − td j ) u ( k, j ) (cid:19) ⊗ f ( e i )= X i (cid:20)(cid:18) X j ∈ F i ,k ( rc j − td j ) u ( k, j ) (cid:19) ⊗ (cid:18) X l X m ∈ F l ,i b ( l, m ) v ( l, m ) (cid:19)(cid:21) = X i (cid:20) X j ∈ F i ,k X l X m ∈ F l ,i ( rc j − td j ) b ( l, m ) u ( k, j ) ⊗ v ( l, m ) (cid:21) ≤ (1 − t ) X i (cid:20)(cid:18) f ( e i ) ∧ X j u ( k, j ) (cid:19) ⊗ (cid:18) X l X n : s nl = m ∈ F i ,l µ n v ( l, m ) (cid:19)(cid:21) = X i (1 − t ) (cid:20)(cid:18) X j ∈ F i ,k a ( k, j ) u ( k, j ) (cid:19) ⊗ (cid:18) X l X n : s nl = m ∈ F i ,l µ n v ( l, m ) (cid:19)(cid:21) = X i (cid:20) X j ∈ F i ,k X l X m ∈ F i ,l (1 − t ) a ( k, j ) (cid:18) X n : s nl = m µ n (cid:19) u ( k, j ) ⊗ v ( l, m ) (cid:21) For each i , for all j ∈ F i ,k , for each l , and for all m ∈ F i ,l , the coefficient of u ( k, j ) ⊗ v ( l, m ) on the left hand side is ( rc j − td j ) b ( l, m ), and on the righthand side, we have (1 − t ) a ( k, j ) X n : s nl = m µ n . Thus( rc j − td j ) b ( l, m ) ≤ (1 − t ) a ( k, j ) X n : s nl = m µ n . SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 13
Let A = A f E and B = A f E , as defined prior to Lemma 3.3. Adding acrossall m ∈ F i ,l , we have:( rc j − td j ) X m ∈ F i ,l b ( l, m ) ≤ (1 − t ) a ( k, j ) X m ∈ F i ,l X n : s nl = m µ n so ( rc j − td j ) B ( l, i ) ≤ (1 − t ) a ( k, j ) λ il , where λ il = X m ∈ F i ,l X n : s nl = m µ n . Observe that P i λ il = 1 for all rows l . Add up terms over all j ∈ F i ,k . Thus X j ∈ F i ,k ( rc j − td j ) B ( l, i ) ≤ X j ∈ F i ,k (1 − t ) a ( k, j ) λ il = ⇒ ( rC i − tD i ) B ( l, i ) ≤ (1 − t ) A ( k, i ) λ il , where C i = P j ∈ F i ,k c j and D i = P j ∈ F i ,k d j . Now if C ′ = X u ( k,j ) ∈ F ′⊥ c j and D ′ = X u ( k,j ) ∈ F ′⊥ d j , then P i C i + C ′ = 1 and P i D i + D ′ ≤
1. Since rx − ty ≥
0, it follows that rC i − tD i ≥ j , u ( k, j ) ∈ F ′⊥ implies rc j − t j d j = 0, wehave rC ′ − tD ′ = 0. By Lemma 3.3, there exists a finite sequence ( ν l ) Nl =1 suchthat A ( k ) ≤ C P ν l B ( l ), with P l ν l = 1 and ν l ≥
0. Then in particular,( rC i − tD i ) A ( k, i ) ≤ C (1 − t ) A ( k, i ) X l ν l λ il If A ( k, i ) = 0, then C i and D i are also 0, since F i ,k is empty. Otherwise A ( k, i ) >
0, so for all i , ( rC i − tD i ) ≤ C (1 − t ) X l ν l λ il = ⇒ X i ( rC i − tD i ) + rC ′ − tD ′ ≤ C (1 − t ) X l X i ν l λ il = ⇒ r − t ≤ r − t ( X i D i + D ′ ) ≤ C (1 − t ) X l ν l ( X i λ il ) = ⇒ r − t ≤ C (1 − t ) = ⇒ r ≤ C Thus g (and by similar argument g ) is a C -embedding.Finally, G itself has finitely many order extreme points, so by Lemma 3.2 itcan be embedded into a ℓ m ∞ ( ℓ n ) space, implying that G ∈ K ′ . (cid:3) Corollary 3.6.
Let
E, F , F be finite dimensional lattices, let C ≥ , andsuppose f : E → F and f : E → F are C - embeddings. Then for all ε > , there exist a lattice G ∈ K ′ and ( C + ε ) -embeddings g : F → G and g : F → G such that g ◦ f = g ◦ f .Proof. Pick δ such that (1 + δ ) C < C + ε , and pick N such that there are(1 + δ )-embeddings φ j : F j → F ′ j := ℓ N ∞ ( ℓ dim F j ). Then each φ j ◦ f j : E → F ′ j is a C (1 + δ )-embedding. By Theorem 3.5, there exists G ∈ K ′ and C (1 + δ )-embeddings g ′ j : F ′ j → G for j ∈ { , } such that g ′ ◦ φ ◦ f = g ′ ◦ φ ◦ f .Now let g j = g ′ j ◦ φ j , and observe that each g j is a (1 + δ ) C -embedding,and g ◦ f = g ◦ f . Since (1 + δ ) C < C + ε , we are done. (cid:3) The Amalgamation Property for arbitrary Banach lattices.
The above approach works well with finite dimensional lattices, but expand-ing to finitely generated lattices will lead to some additional complicationssince finitely generated lattices need not be finite dimensional. In fact, theseparable isometrically universal lattice U = C (∆ , L (0 , E is a Banach lattice. Let α be a limit ordinal, and let( E γ ) γ<α be a sequence of increasing sublattices of E such that ∪ γ<α E γ = E .considering ( γ ) γ<α as a net, define E ⊆ Q E γ by E = { ( x γ ) α<γ : lim γ x α = x ∈ E } . Essentially, E is a lattice of α -length sequences converging to elements in E ,with norm k ( x α ) k E = sup α k x α k . Lemma 3.7.
Let E and E be as above, and let E be the ideal in E of nullsequences. Then E is isometric to E / E .Proof. Let x ∈ E , and let ( x γ ) γ → x , and let [( x γ )] denote the equivalenceclass induced by E .We will now show that the map g : E → E / E with g ( x ) [( x γ ) γ ] is anisometry.First, it is well defined: if ( x γ ) γ and ( y γ ) γ converge to x , then ( y γ − x γ ) γ ∈ E ,so [( x γ ) γ ] = [( y γ ) γ ]. By continuity of scalar multiplication and addition, g ( x ) is linear. It also preserves norms. Note that k g ( x ) k E / E = inf {k ( x γ ) γ k :( x γ ) γ → x } , so k g ( x ) k E / E ≥ k x k , since k x γ k → k x k . For ǫ > x γ ) → x , there exists some β < α such that for all γ > β, k x γ − x k < ǫ consider then the α -sequence ( x ′ γ ) with x ′ γ = 0 for all γ < β and x ′ γ = x γ otherwise. Then g ( x ) = [( x ′ γ ) γ ], and so k g ( x ) k E / E ≤ k x k + ε . In addition, SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 15 the map is clearly surjective, since any [( y γ )] = g ( x ) where ( y γ ) → x . Thus g is a linear isometry.Finally, g preserves lattice operations. First of all, g is positive. If x ≥ x γ ) γ → x , then 0 ≤ ( x γ ∨ γ → x ∨ x . Since E is a lattice ideal, g ( x ) = [( x γ ∨ γ ] = [( x γ ) γ ] ∨ [0] ≥
0. In addition, g ( x ) preserves disjointness:if x ∧ y = 0, then if x γ → x and y γ → y , then x γ ∧ y γ → x ∧ y = 0. Then[( x γ )] ∧ [( y γ )] = [( x γ ∧ y γ )] = [0] , so g is a lattice homomorphism. Therefore g is a lattice isometry. (cid:3) Given a separable lattice E , by [13, Proposition 2.2], there exists a finitelybranchable lattice E ′ such that E ⊆ E ′ ⊆ E ∗∗ . Let ( x σ ) T E ′ be the corre-sponding branching tree. Let k n ↑ ∞ where k n ∈ N be a strictly increasingsequence, let E ′ k n = span( x σ : | σ | = k n ), and let E ′ ⊆ Q n E ′ k n be the latticedefined by E ′ = { ( x i ) i : x i → x ∈ E ′ } , with lattice norm k x k E ′ = sup n k x n k . Finally, let E ′ = { x ∈ E ′ : x n → } .By Lemma 3.7, E ′ / E ′ is lattice isometric to E ′ itself. Furthermore, any finitedimensional lattice F ∈ E can be approximated by a sublattice of some E n for some n : Lemma 3.8.
Let E = ∪ n E n where ( E n ) is an increasing sequence of lattices.Suppose F ⊆ E is a finite dimensional sublattice. Then for all ε > thereexist n ∈ N and a (1 + ε ) -isometry g : F → E n such that k g − Id | F k < ε .Proof. Let m = dim F , and let h ( x , ..., x m ) = x − x ∧ ( W i ≥ x i ). Let ( e i ) i be the atoms of F , and let e i = ( e , ..., e i − , e i +1 , ..., e m ). Now h is contin-uous, and for any sequence ( x , ..., x m ) of positive elements, the elements h ( x , x ) , ..., h ( x m , x m ) are mutually disjoint and positive. Thus since ∪ E n is dense in X , for some n there exist corresponding positive ( f , ..., f m ) ⊆ E n such that k h ( e i , e i ) − h ( f i , f i ) k < δ/m . Now since the e i ’s are mutually dis-joint, h ( e i , e i ) = e i . Let g : F → E n be the lattice homomorphism generatedby g ( e i ) = h ( f i , f i ). Then for any P a i e i ∈ S ( F ), we have k X e i − X a i g ( e i ) k ≤ m X i | a i |k e i − g ( e i ) k < δ. It follows that 1 − δ < k g ( P mi a i e i ) k < δ , so g is a δ − δ -isometry. If we let δ − δ < ε , we have both that g is a (1 + ε )-isometry and k Id | F − g k < ε . (cid:3) We now state the following lemma:
Lemma 3.9.
Let E and A be a finitely branchable Banach lattices, andsuppose φ : E → A is an embedding. Let ( x σ ) σ ∈ T E and ( y σ ) σ ∈ T A be linearly dense spanning trees for E and A , respectively. Then for all ε > , there exista strictly increasing sequence ( k n ) n ⊆ N and (1 + ε ) -embedding φ ′ : E → A generated by a sequence of maps φ n : E n → A k n such that:(1) The following diagram commutes: E A
E A φ ′ q E q A φ (2) For each n , φ n : E n → A k n is a (1 + ε/ n ) -embedding.Proof. Let
E ⊆ Q n E n . We will construct k n as follows. Begin with x ∅ ∈ E + ,and suppose that k x ∅ k = 1. Pick k ∈ N and z ∅ ∈ span( { y σ : | σ | = k } )with z ∅ ≥ k z − φ ( x ∅ ) k < ε . We then let φ ′ ( x ∅ ) = z ∅ . For n > E n is finite dimensional and embeds into A , by Lemma 3.8, pick k n insuch a way that such a way that there is a φ n : E n → A k n with distortionlevel at most (1 + ε/ n ).Let φ ′ = ( φ n ) n . Note that φ ′ sends atoms to disjoint elements and is apositive linear map. To show that property 1 is also fulfilled, we mustfirst show that φ takes elements in E to elements in A . Let x ∈ E , with( x i ) → x ′ ∈ E . Now φ ( x i ) ∈ A , and by continuity φ ( x i ) → φ ( x ′ ) ∈ A aswell. Yet k φ ( x i ) − φ i ( x i ) k ≤ ε i , so φ i ( x i ) → φ ( x ′ ) ∈ A . In addition, if x i → x ∈ E , then q A ◦ φ ′ (( x i ) i ) = φ ( x ), which gives us commutativity, thusfulfilling property 1. (cid:3) We are now ready to prove the following:
Theorem 3.10.
Let
E, A , A be separable Banach lattices, and let f : E → A and f : E → A be embeddings. Then there exists a separableBanach lattice G and embeddings g : A → G and g : A → G such that g ◦ f = g ◦ f .Proof. Since each f i : E → A i is a lattice embedding for i = 1 ,
2, by [18, The-orem 1.4.19], each f ∗∗ i : E ∗∗ → A ∗∗ i is a lattice embedding. By Proposition2.2 in [13], there exists a separable finitely branchable lattice E ⊂ E ′ ⊆ E ∗∗ with a finitely branching tree ( x σ ) σ ∈ T E ′ . Similarly, we can take the Banachlattice generated by f ∗∗ i ( E ′ ) and A i , and inject it into a finitely branchable A ′ i with a corresponding finite branching tree ( y σ ) σ ∈ T A ′ i . Thus we can rede-fine f and f to be extended to E ′ .Let ε >
0, and using Lemma 3.9, pick appropriate increasing sequencesof natural numbers k n ↑ ∞ and k n ↑ ∞ generating A ′ and A ′ with ac-companying (1 + ε )-isometries f ′ and f ′ such that the following diagramcommutes: SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 17 E ′ A ′ E ′ A ′ A ′ E A ′ A A f ′ q E ′ f ′ q A ′ f f q A ′ Id f f Id Id
By the assumptions on Lemma 3.9, f ′ j = ( φ jn ) n , where φ jn : E ′ n → A ′ j k jn is a(1+ ε/ n )-isometry. Use Corollary 3.6 to get G n and (1+ ε/ n − )-embeddings ψ n and ψ n such that ψ n ◦ φ n = ψ n ◦ φ n , and let g ′ = ( ψ n ) n and g ′ = ( ψ n ) n .Let G ′ ⊆ Q n G n be the sublattice generated by g ′ ( A ′ ) and g ′ ( A ′ ), and equip G ′ with the sup-norm; that is, if x ∈ G ′ , let k x k G ′ = sup k x n k G n . Now each g ′ j is a (1 + 2 ε )-embedding. Let G ′ be the ideal consisting of elements x ∈ G ′ such that k x n k G n →
0, and let G = G ′ / G ′ . Note that for each j ∈ { , } , wehave g ′ j ( A ′ j ) ⊆ G ′ . Thus g ′ j induces well defined maps g j : A ′ j → G , with g j = q G ◦ g ′ j ◦ q − A ′ j . We therefore have the following commuting diagram: E ′ E ′ A ′ A ′ A ′ A ′ G ′ G f ′ q E ′ f ′ f f q A ′ g ′ q A ′ g ′ g g q G It remains to show that each g j is in fact an embedding. To this end, we notethat if z ∈ G , then k z k = inf {k y k : q G ( y ) = z } . Let x ∈ A ′ . Pick y ∈ A ′ with k y k < δ such that y i → x . This can be done by picking n such thatfor all n ≥ N k x − y n k < δ , ε/ n − < δ , and furthermore, we can assumethat for all n < N , y n = 0. It then follows that δ ) ≤ k g ′ ( y ) k G ′ ≤ (1+ δ ) ,so k q G g ′ ( y ) k ≤ (1 + δ ) , Thus k g ( x ) k G ′ ≤ (1 + δ ) . In addition, for any z ∈ G ′ , since for all δ ′ > k z n k G n ≤ δ ′ for all large enough n , itfollows that k z − g ′ ( y ) k > δ ) − δ ′ . Thus k g ( x ) k G ≥ δ ) . δ can be chosen to be arbitrarily small, so k g ( x ) k G = 1.Finally, we show that g j preserves disjointness and is a positive map. Let x ∈ A ′ j + , and chose a sequence y = ( y i ) i ∈ A ′ j + with y i → x . Then g ′ j ( y ) ≥
0, so q G g ′ j ( y ) = g j ( x ) ≥
0. To show preservation of disjointness, let x, x ′ ≥ y = ( y i ) i ∈ q − A ′ j ( x ) and similarly let y ′ = ( y ′ i ) i ∈ q − A ′ j ( x ′ ). Then y ∧ y ′ ∈ A ′ j ; since ( y i ) i → x and ( y ′ i ) i → x ′ , wehave y ∧ y ′ = ( y i ∧ y ′ i ) i → x ∧ x ′ = 0 , so g ′ j ( y ) ∧ g ′ j ( y ′ ) = g ′ j ( y ∧ y ′ ) ∈ G ′ , which means that g j ( x ) ∧ g j ( x ′ ) = q G g ′ j ( y ) ∧ q G g ′ j ( y ′ ) = q G g ′ j ( y ∧ y ′ ) = 0. Thus g j is an embedding.To show separability, we simply restrict g and g to A and A , and replace G with the lattice generated by g ( A ) S g ( A ). Thus if A and A areboth separable, then so is G . (cid:3) Remark 3.11.
We can also ensure that G is finitely generated, since wecan embed G into U if necessary. Thus K has the AP.We can expand Theorem 3.10 for arbitrary lattices with a similar proof. Theorem 3.12.
Let
E, F , F be Banach lattices, and let f i : E → F i ,with i ∈ { , } be embeddings. Then there exists a lattice G and isometricembeddings g i : F i → G such that g ◦ f = g ◦ f . Furthermore, if F i hasdensity character no more than κ , we can ensure that G does as well.Proof. We prove this by ordinal induction over the density character κ . Forthe base case of κ = ℵ , this was already proven in Theorem 3.10. Sup-pose now that we have shown the same for all lattices of density characterless than κ . Let ( z γ ) γ<κ be a κ -sequence dense in E , and let ( x iα ) α<κ be κ -length sequences dense in F i . Let E β = BL (( z α ) α<β ) and let F βi = BL (( x iα ) α<β ∪ f i ( E β )). Then E β ↑ E , F β ↑ F , and f i ( E β ) ⊆ F βi . Now each f i induces an embedding φ i : E → F i , where φ i (( y β ) β<κ ) = ( f i ( y β )) β<κ .Both E β and the F βi ’s have dense subsets of size strictly less than κ , so byinduction, pick G β and embeddings ψ βi : F βi → G β such that ψ β ◦ f | E β = ψ β ◦ f | E β . Let ψ i = ( ψ βi ) β<α , and let G be the sublattice of Q β G β generatedby the elements of ψ i ( F i ). Let G be the ideal in G of nets convergingin norm to 0, and let G = G / G . Now let g i = q G ◦ ψ i ◦ q − E . Use thesame argument as in Theorem 3.10 to show that each g i is well defined, anembedding, and together with G give the desired amalgamation. Finally, G has the desired density character if we restrict it to the lattice generated by g ( F ) ∪ g ( F ). (cid:3) SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 19
We end this section with some additional results on the interplay betweenthe AP and C -embeddings. In each of these cases, we can perturb latticesor maps that change C -embeddings into embeddings in exchange for fullcommutativity or preservation of the original norm: Theorem 3.13.
Let f : A → X be a C - embedding. Then there exists a C -equivalent renorming |||·||| of X such that f : A → ( X, |||·||| ) is an embedding.Furthermore, • if f is an expansion (that is, if f − is contractive), then we can make |||·||| ≤ k·k . • if f is a contraction, then we can make |||·||| ≥ k · k . • if A and X are both in K ′ , then we can ensure that ( X, |||·||| ) is alsoin K ′ .Proof. We start with a proof for the case when f is an expansion. Let B ′ = CSCH (cid:0) f ( B ( A )) ∪ B ( X ) (cid:1) be the unit ball of |||·||| . Observe that B ′ ⊇ B ( X ) and f ( B ( A )) ⊆ C B ( X ), so C k · k ≤ |||·||| ≤ k · k .We now show that f : A → ( X, |||·||| ) is an embedding. Suppose that thereexist z n ≤ t n f ( x n )+(1 − t n ) y n with (1+ α ) f ( x ) = lim n z n , with x, x n , y n ≥ α ≥ k x n k , k y n k ≤
1, 0 ≤ t n ≤
1, and k x k = 1. By compactness, we cansuppose t n converges to t , and just let z n ≤ tf ( x n ) + (1 − t ) y n . Furthermore,we can assume that k f ((1 + α ) x − tx n ) k → b x . Then for all n , we have f ((1 + α ) x − tx n ) ≤ (1 − t ) y n + δ n with k δ n k →
0. We then have1 + α − t ≤ k f ((1 + α ) x − tx n ) k ≤ (1 − t ) + k δ n k . Thus 1 + α − t ≤ b x ≤ − t , so α = 0.For contractive f , let B ′ be the closed solid convex hull of f ( B ( A )) ∪ C B ( X ).Note here that C B ( X ) ⊆ B ′ ⊆ B ( X ), so k · k ≤ |||·||| ≤ C k · k . Then use thesame type of argument.For the general case, observe that Cf is an expansion which is also a C -embedding. Then by the proof of the first case, there is C -equivalentrenorming |||·||| ≤ k · k of X with Cf : A → ( X, |||·||| ) an embedding. Now takethe new norm of X to be C |||·||| . Then f : A → ( X, C |||·||| ) is an embedding,and C |||·||| is C -equivalent to k · k .Finally, if A, X ∈ K ′ , the unit ball of the renormed lattice ( X, |||·||| ) hasfinitely many order extreme points, so by Corollary 3.4, ( X, |||·||| ) is also in K ′ . (cid:3) Theorem 3.13 can be used to generalize Theorem 3.12 to diagrams involving C -isometries: Corollary 3.14.
Let f i : E → F i with i = 1 , be C i -embeddings for lattices E , F , and F . Then: • There exist a lattice G and C i -embeddings g i : F i → G such that g ◦ f = g ◦ f . • There exist a lattice G , an embedding g : F → G , and a C C -embedding g : F → G such that g ◦ f = g ◦ f . • If E, F , and F are in K ′ , we can ensure G ∈ K ′ as well.Proof. For the first part, let F ′ i = ( F i , |||·||| ) be C i -equivalent renormingssuch that f i is an embedding into F ′ i . By Theorem 3.12 (Theorem 3.5),there exists G and embeddings g i : F ′ i → G such that g ◦ f = g ◦ f . Since F ′ i is C i -equivalent to F i , each g i is a C i -embedding on F i . For the secondpart, use Theorem 3.13 to simply renorm G with a C -equivalent norm |||·||| so that g : F → ( G, |||·||| ) is now an embedding. Then g : F → ( G, |||·||| ) isa C C -embedding. For both parts, G can be in K ′ if E, F , and F are in K ′ . (cid:3) Theorem 3.15.
Suppose f : X → Y is a (1 + ε ) -embedding, and suppose X, Y are in K (or K ′ ). Then there exists a lattice Z ∈ K ( K ′ ) and embeddings g : X → Z and h : Y → Z such that k g − h ◦ f k ≤ ε .Proof. Let j : X → X ⊕ ∞ f ( X ), with j ( x ) = x ⊕ ε f ( x ). Let j : f ( X ) → X ⊕ ∞ f ( X ) with j ( f ( x )) = ε x ⊕ f ( x ). Note then that since ε k f ( x ) k ≤ k x k ≤ (1 + ε ) k f ( x ) k , j and j are both embeddings. Then k j ( x ) − j f ( x ) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) −
11 + ε (cid:19) x ⊕ (cid:18)
11 + ε − (cid:19) f ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = ε ε k x ⊕ − f ( x ) k ≤ ε k x k . If f is surjective, then let g = j and h = j , and we are done. Otherwise, f ( X ) ⊆ Y and j : f ( X ) → X ⊕ ∞ f ( X ) in an embedding, so use Theorem3.12 (or Theorem 3.5) to get a lattice Z in K (respectively K ′ ) and embed-dings h : Y → Z and h : X ⊕ ∞ f ( X ) → Z such that h | f ( X ) = h ◦ j .Then for all x ∈ X , k h j ( x ) − h f ( x ) k = k h j ( x ) − h j f ( x ) k = k j ( x ) − h ( f ( x )) k ≤ ε k x k . Let g = h ◦ j and h = h , and we are done. (cid:3) Corollary 3.16.
Let
E, F , F be lattices in K (or K ′ ), and let f j : E → F j be (1 + ε ) -embeddings. Then there exist H ∈ K ( K ′ ) and embeddings g j : F j → G such that k g ◦ f − g ◦ f k ≤ ε .Proof. By Theorem 3.15 there exist F ′ j ∈ K ( K ′ ) and embeddings f ′ j : E → F ′ j and φ j : F j → F ′ j such that k f ′ j − φ j ◦ f j k ≤ ε . Now use Theorem 3.12 SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 21 (or Theorem 3.5) to get H ∈ K ( K ′ ) and embeddings g ′ j : F ′ j → H with g ′ ◦ f ′ = g ′ ◦ f ′ . Let g j = g ′ j ◦ φ j . Then k g ◦ f − g ◦ f k = k g ′ ◦ φ ◦ f − g ′ ◦ φ ◦ f k≤k g ′ ◦ ( φ ◦ f − f ′ ) k + k g ′ ◦ ( φ ◦ f − f ′ ) k ≤ ε. (cid:3) The approximately ultra-homogeneous separable lattice BL The main result of this section is:
Theorem 4.1.
The class K of finitely generated separable Banach lattices isa Fra¨ıss´e class. Thus there exists a separable approximately ultra-homogeneousBanach lattice BL . The level of homogeneity in BL cannot significantly be strengthened. Forone, BL can only be ”approximately” ultra-homogeneous, since no latticeautomorphism can map non-weak units to weak units. BL is clearly alsoatomless. If there were an atom e in BL , any automorphism would have tomap it to another atom. Thus two embeddings g : R → < e > ⊆ BL and g : R → < x > ⊆ BL such that x both is disjoint from e and not an atomcannot be arbitrarily approximated by an automorphism. BL is isometrically universal for separable Banach lattices, but it is notisometric to U because the latter is not approximately ultra-homogeneousand Fra¨ıss´e limits are unique up to isometry. Indeed, let < e > be a one-dimensional lattice generated by e , let f ( e ) = a := ~ ∆ ⊗ χ [0 , , f ( e ) = b := ~ K ⊗ χ [0 , , where K ⊆ ∆ is a proper clopen subset, and let b ′ = a − b . Let φ be any automorphism over U . Now the sets K b = { k ∈ ∆ : k φ ( b )( k ) k = 1 } and K b ′ = { k ∈ ∆ : k φ ( b ′ )( k ) k = 1 } are non-empty, and furthermore b ( K b ′ ) = 0, and vice versa, since φ ( < b, b ′ > ) is isometric to ℓ ∞ . It followsthat for k ∈ K b ′ , we have k φ ( b )( k ) − a ( k ) k = 1, so k φ ( b ) − a k ≥ Proof of theorem.
It is clear that K has the HP and the JEP. It also hasthe CP by virtue of the fact that each function symbol in the language ofBanach lattices has a fixed modulus of continuity independent of its inter-pretation. By Theorem 3.10, it has the AP. It remains to show that it hasthe PP. We need to show that the class of finitely generated Banach latticesis both separable and complete under the metric d K . For separability, let( x n ) n be a countable dense subset of U . Then the set { < x i , ..., x i n > } oflattices generated by finitely many elements in ( x n ) n is itself a countabledense subset of K n . To show completeness, we use Theorem 3.10 and the fact that Banach lat-tices are closed under direct limits. Let ( a i ) i be a Cauchy sequence of tuplesgenerating structures in K n . By passing to a subsequence if necessary, wecan assume that d K ( a i , a i +1 ) < i +1 . For a i and a i +1 , let B i be a finitelygenerated lattice containing isometric copies of < a i > and < a i +1 > suchthat d ( a i , a i +1 ) < d K ( a i , a i +1 ) + i +1 . Note then for each i , we have embed-dings < a i +1 > → B i , B i +1 , so use amalgamation to embed B i and B i +1 into some finitely generated space B i where the associated diagram com-mutes. Proceed inductively in a similar manner: each B ki +1 injects into B k +1 i and B k +1 i +1 , so use amalgamation to inject them into some finitely generated B k +2 i . The resulting commutative diagram illustrates the process: < a > B B B . . .< a > B B < a > B ...Let X be the closed inductive limit of the sequence of lattices ( B n ) n . X isitself separable, though it need not be finitely generated. It also contains anisometric copy of each < a i > and for each a i , a j ⊆ X with i ≤ j , we have d ( a i , a j ) < j − X k = i (cid:18) d K ( a k , a k +1 ) + 12 k +1 (cid:19) < j − X k = i − k < − i +1 . Thus ( a i ) i , as a sequence of tuples in X , is Cauchy. Let a = lim i a i . Since X is complete, the sublattice < a > exists, which implies the completion ofthe metric d K . Thus K has the PP, and we are done. (cid:3) We continue with an additional characterization of BL . In particular, BL is finitely branchable and finitely generated. To this end, we concentrate onthe sub-class K ′ .The ℓ m ∞ ( ℓ n ) lattices are in certain ways analogues of ℓ n ∞ spaces. For one,recall the definition of an injective Banach space E : If T : F → E is a linearmap and F is a subpace of G , then there exists a linear map ˆ T : G → E extending T such that k T k = k ˆ T k . There is also a lattice analogue ofinjectivity: We say E is an injective lattice if for all lattices F ⊆ G andany positive linear maps T : F → E , then there exists a positive linear SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 23 map ˆ T : G → E extending T such that k T k = k ˆ T k . The injective finitedimensional Banach spaces are exactly the ℓ n ∞ spaces. By [6, Theorem 5.2],the ℓ ∞ -sums of finite dimensional ℓ spaces make up the collection of finitedimensional injective lattices. Furthermore, the Gurarij space in particularcan be constructed as an inductive limit of ℓ n ∞ Banach spaces. We willnow also show that BL can be constructed as an inductive limit of ℓ m ∞ ( ℓ n )lattices. Lemma 4.2. K ′ is an incomplete Fra¨ıss´e class that is dense in K . Inparticular, the Fra¨ıss´e metric d K ′ isometrically coincides with d K .Proof. K ′ has the HP by its definition. By Theorem 3.5, it has the AP.Clearly it also has the JEP: given two X, Y ∈ K ′ , we also have A ⊕ ∞ B ∈ K ′ .To show density in K , let < a > be a finitely generated lattice, and embed < a > into U . Let ( x σ ) σ ∈ T be the finitely branching tree comprised of ele-ments in U of the form χ N σ ⊗ χ Q k where Q k is a diadic interval of length 2 − n , | σ | = n , and N σ ⊆ ∆ = { , } N is the set consisting of all infinite branchesstarting with σ . Now span(( x σ ) σ ∈ T ) is dense in U . Let S n := { x σ : | σ | = n } ,and observe that span { S n } is itself a ℓ n ∞ ( ℓ n ) space. Given ε >
0, choose n and x ⊆ span( S n ) such that d ( a, x ) < ε . Then the lattice < x > ∈ K ′ issufficiently close to < a > in d K .To show separability of K ′ and the CCP, it is sufficient to show that d K | K ′ = d K ′ . Clearly d K | K ′ ≤ d K ′ , so we need only to show the opposite inequality.Let d K ( a, b ) = δ , and let ε >
0. Choose embeddings φ A and φ B from < a > and < b > into U such that d ( φ A ( a ) , φ B ( b )) < δ + ε . By Lemma 3.8, given ε >
0, there exist n ∈ N and (1 + ε )-embeddings f A : A → D := span( S n )and f B : B → D such that k f A − φ A k < ε and k f B − φ B k < ε . Nownote that d ( f A ( a ) , f B ( b )) < δ + 3 ε . Use Theorem 3.13 to renorm D with a(1 + ε )-equivalent renorming |||·||| so that f B : B → D ′ = ( D, |||·||| ) ∈ K ′ is anembedding. Then f A is a (1 + ε ) -embedding into D ′ . Finally, use Theorem3.15 to get some C ∈ K ′ and embeddings g A : A → C and g D ′ : D ′ → C such that k g D ′ ◦ f A − g A k ≤ ε + ε . Then for each i , we have k g A ( a i ) − g D ′ f B ( b i ) k C ≤ k g A ( a i ) − g D ′ f A ( a i ) k C + k g D ′ f A ( a i ) − g D ′ f B ( b i ) k C ≤ ε + ε + k f A ( a i ) − f B ( b i ) k D ′ ≤ ε + ε + (1 + ε ) d ( f A ( a ) , f B ( b )) ≤ ε + ε + (1 + ε )( δ + 3 ε ) . We can let ε get arbitrarily small, so d K ′ ≤ d K | K ′ , and we are done. (cid:3) It is known that incomplete Fra¨ıss´e classes admit a Fra¨ıss´e limit for the com-pletion of the class, but here we will explicitly show that the constructionof the limit BL need only involve an increasing sequence of lattices in K ′ .In order to prove the following theorem, we use approximate isometries as described in [5], that is, bi-Katetov maps ψ : X × Y → [0 , ∞ ] with X and Y both metric spaces. Recall that ψ is bi-Katetov if for all x, x ∈ X and y, y ∈ Y , | ψ ( x, y ) − d ( x, x ) | ≤ ψ ( x , y ) and | ψ ( x, y ) − d ( y, y ) | ≤ ψ ( x, y ).In this context, approximate isometries provide information about how gen-erating tuples a and b relate in ambient spaces.Approximate isometries can be induced by finite partial embeddings, i.e.,partial functions f : X ⇀ Y , with dom( f ) = X a finite set, which inducelattice embeddings f : < X > → Y . More generally, for any X ⊆ X and a(not necessarily finite) partial embedding f : X → Y , we let ψ f : X × Y → R be defined by ψ f ( x, y ) = inf z ∈ X k x − z k + k y − f ( z ) k . Observe that if x ∈ X ,then ψ f ( x, y ) = k f ( x ) − y k . If X ⊆ X , we also have an approximate isome-try ψ Id X : X × X → R with Id X the inclusion maps from X to X , where ψ Id X ( x, y ) = k x − y k .There is also a “pseudoinverse” operation: if ψ ( x, y ) is an approximate isom-etry, we let ψ ∗ ( y, x ) = ψ ( x, y ). Clearly ψ ∗∗ = ψ . We can also “compose” ap-proximate isometries. If φ : X × Y → R and ψ : Y × Z → R are approximateisometries, then ψφ : X × Z → R with ψφ ( x, z ) = inf y ∈ Y ( φ ( x, y ) + ψ ( y, z ))is also an approximate isometry by [5, Lemma 2.3(i)]. For example, if f : A → C and g : B → C generate embeddings from < A > and < B > to C respectively, then the map ψ ∗ g ψ f : < A > × < B > → R ,where ψ ∗ g ψ f ( x, y ) = inf z ∈ C ( ψ f ( x, z ) + ψ g ( y, z )) . is also an approximate isometry. Note that if A = a , B = b , and d ( f ( a ) , g ( b )) is small, then ψ ∗ g ψ f ( a i , b i ) will also be small, and the converseholds true as well. In fact, for x ∈ A and y ∈ B , we have ψ ∗ g ψ f ( x, y ) = k f ( x ) − g ( y ) k (here A and B need not be finite). Thus we can see ap-proximate isometries as marking conditions for the ”strength” of a jointembedding. An approximate isometry ψ may originally be defined on some X × Y , with X ⊆ X and Y ⊆ Y , but it can be extended to X × Y bythe composition ψ Id Y ψψ ∗ Id X . However, if the ambient spaces are clear fromcontext, we will just write ψ to refer to the extended approximate isometry.Finally, composition and involution as described above work analogously to-gether like the multiplication and inversion group operations. In particular,composition is associative and ( ψφ ) ∗ = φ ∗ ψ ∗ (see [5, Lemma 2.3(ii)]).We say that ψ is refined by , or coarsens φ if φ ( x, y ) ≤ ψ ( x, y ) for all( x, y ) ∈ X × Y . Given lattices X and Y , we let A px ( X, Y ) ⊆ [0 , ∞ ] X × Y ,equipped with the product topology on [0 , ∞ ] X × Y , be the set of all approx-imate isometries generated by finite partial embeddings between elementsin K ′ , composition, coarsening, and any point-wise limit of such maps. For ψ ∈ A px ( X, Y ), we let A px <ψ ( X, Y ) be the interior of the set of refinements
SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 25 of ψ . If A px <ψ ( X, Y ) = ∅ , we say that ψ is a strictly approximate isom-etry and use the notation φ < ψ to mean φ ∈ A px <ψ ( X, Y ). Intuitively,strictly approximate isometries do not impose strong conditions on possiblejoint embeddings except on some finite set (see Lemma [4, Lemma 3.8(ii)]), so they leave much room for refinement. While the set A px ( X, Y ) seemscomplicated, by [5, Lemma 3.8(iv)], it actually is comprised of the closureof coarsening and pointwise limits of approximate isometries in the form of ψ ∗ g ψ f (extended to X × Y ) where f and g are finite partial embeddings.Suppose ψ : X × Y → R is an approximate isometry, and let r >
0. Wesay that ψ is r -total if ψ ∗ ψ ≤ ψ Id X + 2 r . It is not hard to show that if f : X → Y is an embedding, then ψ ∗ f ψ f = ψ Id X , so any such ψ f is r -totalfor all r > A, B ∈ K ′ , < a > ⊆ A and embedding f : < a > → B , there exist C ∈ K ′ and embeddings g : A → C and h : B → C such that ψ ∗ h ψ g ≤ ψ f (with thenecessary extensions on f ). See [4, Definition 3.5(iii)] for the generalizeddefinition of the NAP using approximate isometries. Using the fact that ψ ∗ h ψ g ( x, y ) = k g ( x ) − h ( y ) k and ψ f ( x, y ) = k f ( x ) − y k , one can easily showthat this definition is equivalent to our current working definition of the AP.Note also the inequality; this is due to the fact that f is only defined on < a > while g is defined on all of A , so the extension of ψ f to A × B containsless limiting information than ψ ∗ h ψ g ( x, y ).We are now ready to prove the following: Theorem 4.3. BL can be constructed as the limit of an increasing sequenceof ℓ m ∞ ( ℓ n ) lattices. In particular, it is finitely branchable.Proof. We construct an increasing sequence of finite dimensional lattices A n as in the proof of Lemma 3.17 in [5] with BL isometric to S n A n . Let A ∈ K ′ , and let K n, be a countable dense subset of K ′ n . Since K ′ is densein K , we have K n, dense in K n . We proceed by induction. Suppose A k hasbeen defined for all k ≤ n . Suppose also that A k, ⊆ A k is countable anddense in A k for all k ≤ n , with A k, ⊆ A k +1 , for each k < n .By [5, Lemma 3.8(ii)], for any finite tuples a and b we can ensure the exis-tence of a countable set C ( a, b ) ⊆ K ′ such that every C ∈ C ( a, b ) containsan isometric copy of < a > and < b > , and every strictly approximate isom-etry ψ : b × a → Q can be refined in < b > × < a > by some ψ ∗ f ψ g with f : < a > → C and g : < b > → C for some C ∈ C ( a, b ) (by [5, Lemma 2.8(ii)], such strictly approximate isometries ψ : b × a → Q actually exist). Let C k = [ b ∈ K n, a ⊆ A k, C ( a, b ) . To construct A n +1 , take the first n lattices C k , ..., C nk in each C k for k ≤ n ,and amalgamate them one after another. Here a i,j ⊆ A i, for some i ≤ n ,and < a i,j > and < b i,j > both into C ji ∈ C ( a i,j , b i,j ) ⊆ C i : A n ∗ . . . ∗ ∗ . . . ∗ A n +1 < a , > C < a i,j > C ji . . . < a n,n > C nn < b , > < b i,j > < b n,n > ι ′ f hι Note that in each case, A k ∈ K ′ , so we can if necessary enlarge A k andassume that A k = ℓ m k ∞ ( ℓ n k ) for some m k , n k ∈ N .Observe that for each tuple a ⊆ A k , b ∈ K k, , and each strictly approximateisometry ψ : b × a → Q , we have some m > k such that a = a i,j , b = b i,j for some i, j ≤ m with C ji ∈ C ( a, b ) and embeddings f : < a i,j > → C ji and ι : < b i,j > → C ji such that ψ ∗ f ψ ι < ψ . Additionally, there is an embedding h : C ji → A m with ψ ∗ h ψ ι ′ ≤ ψ f . In particular, we have ψ hι | × = ψ ∗ f ψ ι :Indeed, given x ∈ < a i,j > and y ∈ < b i,j > , ψ hι ( y, x ) = k hι ( y ) − x k = k hι ( y ) − ι ′ ( x ) k = k hι ( y ) − hf ( x ) k = k ι ( y ) − f ( x ) k = ψ ∗ f ψ ι . Thus ψ hι ≤ ψ ∗ f ψ ι < ψ . Furthermore, ψ hι is r -total on b i,j for all r >
0, since hι is an embedding. Thus by [5, Lemma 3.16], S n A n is a Fra¨ıss´e limit for K ′ -structures, where K ′ is the Fra¨ıss´e completion of K ′ . By Theorem 4.2,we have K ′ = K . This implies that S n A n is also a Fra¨ıss´e limit of K , so byuniqueness, S n A n is isometric to BL . (cid:3) A lattice that is finitely branchable can be expressed as an inductive limitof finite dimensional lattices. We also have the following:
Theorem 4.4.
Any finitely branchable lattice can be generated by two ele-ments.
SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 27
Proof.
Let X be finitely branchable, and let ( x σ ) σ ∈ T be a finitely branchingtree densely spanning X . Recall that as a tree, T ⊆ S M Q Mn A n , where each A n is a finite nonempty set. We will find two elements u and v such that X = < u, v > . Let u = x ∅ and let S n = { x σ : | σ | = n } as in the proof ofLemma 4.2. Consider now X = span( S ), and let v = X | σ | =1 a σ x σ , where 0 < a σ and the a σ ’s are mutually distinct. The mutual distinctionenables each x σ to be produced using lattice operations over u and v . Forexample, take a ρ = max a σ , and pick c such that cα ρ > τ = ρ , ca τ <
1. Recall that u = x ∅ = P | σ | =1 x σ , so ( cv − u ) ∨ cα ρ − x ρ . Wethen make the same argument, but for u − x ρ and v − a ρ x ρ , thus generatingeach successive x σ for all σ ∈ S .Suppose that for all k ≤ n , v k has been selected and that for each k wehave a finite sequence of functions ( φ ik ( x, y )) i generated by lattice operations+ , ∧ , r · (where r is real), with corresponding moduli of continuity ∆ ik : R + → (0 , • For each k ≤ n , v k = P | σ | = k a σ x σ , with a σ > • For each k ≤ n , < u, v k > = span( S k ) • For each k ≤ n , ( φ ik ( u, v k )) i is a 2 − k -net in the unit ball of span( S k ). • For each k < n , k v k − v k +1 k < min i,j ≤ k ( (∆ ij (2 − k )) i )2 k Note that v k = X | σ | = k a σ X m ∈ A k +1 x σ ⌢ m , so for each m ∈ A n +1 and σ ∈ Q n A k , pick positive, mutually distinct a σ ⌢ m such that | a σ − a σ ⌢ m | < min i,j ≤ n ( (∆ ij (2 − n )) i )2 n | S n +1 | . Thus if v n +1 := P | σ | = n +1 a σ x σ , we have k v n − v n +1 k < min i,j ≤ n ( (∆ ij (2 − n )) i )2 n .Now < u, v n +1 > = span( S n +1 ). For each σ with | σ | = n + 1, pick 0 < s r or τ < s . Let x = ( v n +1 − su ) + and y = ( v n +1 − ru ) + . Then for some large enough C ,( Cy − x ) + is a multiple of x σ . Finally, let ( φ in +1 ( x, y )) i be a finite collectionof functions generated by lattice operations such that ( φ in +1 ( u, v n +1 )) i is a2 − n − -net in the unit ball of span( S n +1 ). Let v = lim v n . We show that < u, v > = X . Observe that the set { φ in ( u, v n ) : n ∈ N } is dense in X by the above properties. Let ε >
0, and pick n such that2 − n < ε . Then k v n − v n +1 k < ∆ in ( ε )2 n +1 and φ in ( u, v n ) is ε -dense in B (span( S n )).Furthermore, for all m > n , we have k v n − v m k < min i ∆ in ( ε ) P mj = n +1 − j .Thus k v n − v k ≤ ∆ in ( ε ), so k φ in ( u, v n ) − φ in ( u, v ) k < ε for all φ in . This impliesthat the set { φ in ( u, v ) | n ∈ N } is dense in X , so we are done. (cid:3) Theorem 4.3 combined with Theorem 4.4 yields a surprising result:
Corollary 4.5.
The lattice BL is finitely generated. Remark 4.6.
Finite generation implies that BL is not stably homogeneousin the sense defined by Lupini in [15]. That is, given finitely generated A and embeddings f : A → BL and g : A → BL , we cannot guarantee thatfor all ε >
0, there is some automorphism φ on BL such that k φ ◦ f − g k < ε .Thus approximating over a finite number of elements rather than by normsis the best, in some sense, that can be done in terms of homogeneity.Suppose otherwise. Since BL can be generated by two elements x , x , weconsider e ∈ S ( BL ) + and embedding f : < x , x > → BL such that theimage does not have full support and f ( e ) is disjoint from e (we can dothis, for example, by finding a copy of BL ⊕ ∞ R that is in BL , and pick e ∈ S ( R ) + ). If BL were stably homogeneous, there would exist a latticeautomorphism φ on BL such that k f − φ k < . Then k f ( φ − ( e )) − e k = k ( f − φ )( φ − ( e )) k < /
2, but f ( φ − ( e )) is disjoint from e , which means k f ( φ − ( e )) − e k ≥
1, a contradiction.5.
An alternate construction of BL and some of its properties The Fra¨ıss´e limit BL is clearly of approximately universal disposition bothfor finite dimensional lattices and for finitely generated lattices. In thissection, we show that separable lattices of approximately universal disposi-tion for finitely generated lattices are isometric to BL . In addition, finitelybranchable lattices of approximately universal disposition for finite dimen-sional lattices are isometric to BL . A bi-product of the latter is a simplifiedconstruction of BL which allows us to explore some of its structural prop-erties.We first show that approximate universal disposition can be broadened toinclude extensions of ε -isometries: Lemma 5.1.
Suppose X is of approximately universal disposition for finitelygenerated Banach lattices, let < a > = A ⊆ X and B be finitely generated,and let f : A → B be a (1 + ε ′ ) -embedding. Then for all ε > ε ′ and for all δ > , there exists a (1+ δ ) -embedding g : B → X such that k g ◦ f ( a ) − a k < ε . SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 29
Proof.
By Theorem 3.15 there is a lattice Z and embeddings h : A → Z and h : B → Z such that k h ◦ f − h k ≤ ε ′ . We can assume that Z isfinitely generated as well, since we can embed into U if necessary. Decreasing δ as necessary, we can suppose that (1 + δ ) ε ′ + δ < ε . Then there exists a(1 + δ )-embedding g ′ : Z → X such that k g ′ h ( a ) − a k < δ . Then k g ′ h f ( a ) − a k ≤ k g ′ kk h ( a ) − h ( a ) k + k g ′ h ( a ) − a k < (1 + δ ) ε ′ + δ < ε. Let g = g ′ ◦ h , and we are done. (cid:3) We can now show the following:
Theorem 5.2.
Any separable Banach lattice of approximately universal dis-position for finitely generated lattices is isometric to BL .Proof. The proof follows that of Theorem 1.1 in [12]. Suppose X and Y arelattices of approximately universal disposition for finitely generated lattices.We will then construct a lattice isometry. Let ( x n ), ( y n ) be dense in X and Y , with x = 0 and y ≥
0. Given ε >
0, let ε n ↓ ε n < − n − . Throughout, we let x n = ( x , ..., x n ),and let X n = < x n > , with the same notation for y n and Y n . Finally, let f : X → Y be the trivial isometry.We begin our construction: let g : Y → X be a (1 + ε )-isometry. Notethis isometry exists. Now take ˜ X = BL ( X ∪ g ( Y )). This lattice is alsofinitely generated by ˜ x = x ∪ g ( y ), so we have the map g : Y → ˜ X ,and by Lemma 5.1, pick a (1 + ε )-embedding f : ˜ X → Y such that d ( f g ( y ) , y ) < . Now let ˜ Y = BL ( f ( ˜ X ) ∪ Y ). Use Lemma 5.1 againto generate a (1+ ε )- embedding g : ˜ Y → X such that d ( g f ( ˜ x ) , ˜ x ) < .We can proceed inductively by constructing finitely generated subspaces˜ X n = BL ( X n ∪ g n − ( ˜ Y n − )) and ˜ Y n = BL ( Y n ∪ f n ( ˜ X n )) with correspondingtuples ˜ x n = x n ∪ g n − ( y n − ) and ˜ y n = y n ∪ f n ( y n ), as well as (1 + ε n − )-embeddings f n : ˜ X n → ˜ Y n and (1 + ε n − )-embeddings g n : ˜ Y n → ˜ X n +1 suchthat d ( ˜ y n , f n +1 g n (˜ y n )) < n − and d ( ˜ x n , g n f n (˜ x n )) < n − . Note that foreach k ≤ n , we have k f n +1 ( x k ) − f n ( x k ) k = k f n +1 ( x k − g n f n ( x k ) + g n f n ( x k )) − f n +1 ( x k ) k≤k f n +1 ( x k − g n f n ( x k )) k + k ( f n +1 g n − Id ) f n ( x k ) k≤ n − + 22 n − = 12 n − so the sequence f n ( x k ) n ≥ k is Cauchy. The same is true for g n ( y k ) n ≥ k . Let f = lim f n , and let g = lim g n . These exist, since ( x k ) and y k are dense in X and Y . Furthermore, f and g are inverses of each other, and they are eachisometries. (cid:3) We now construct a separable lattice of approximately universal dispositionfor finite dimensional lattices. The approach is a modification of that in [2,Section 5] for the Gurarij space.Let J be the collection of embeddings between finite dimensional lattices in K . Since any such lattice isometrically embeds into U , we can assume that J is a set by limiting it to embeddings between finite dimensional sublatticesof U . Let J be a countable dense subset of J in the following sense: for allembeddings f : A → B with B finite dimensional and for all ε >
0, thereexists u : A ′ → B ′ ∈ J , and (1 + ε )-isometries ι A : A → A ′ and ι B : B → B ′ such that u ◦ ι A = ι B ◦ f . In addition, for a separable lattice X , let L ( X )be the set of all maps v : A ′ → X which are C -embeddings for some C ≥ A ′ ∈ Dom( J ). Let L be a countable subset of L ( X ) which is densein the following sense: for all ε > ε ′ > ε ′ )-embeddings f : A ′ → X with A ′ ∈ Dom( J ), L contains an (1 + ε )-embedding v : A ′ → X such that k v − f k < ε .Let X , = X , and suppose now that X n,k has been constructed. Let L n be a countable subset of L ( X n, ) which is dense in the manner describedfor L and L ( X ). Finally, let Γ n = { ( u, v ) ∈ J × L n : dom( u ) = dom( v ) } ,and let (( u ni , v ni )) i be an enumeration of Γ n . We then construct X n,k +1 by amalgamating as follows. Given ( u nk , v nk ) ∈ Γ n with v nk a C -embedding,we use part 2 of Corollary 3.14 to get an embedding ι : X n,k → X n,k +1 and C -embedding w : cod( u nk ) → X n,k +1 such that the following diagramcommutes: dom( u nk ) cod( u nk ) X n,k X n,k +1 u nk v nk wι Finally, we let X n +1 , = S k ∈ N X n,k , and then let X ω ( X ) = S n ∈ N X n, . Theorem 5.3. X ω ( X ) is of approximately universal disposition for finitedimensional lattices.Proof. Let f : A → B and g : A → X ω ( X ) be embeddings, with A and B finite dimensional. Given ε >
0, by density of embeddings and spaces in J and L n , pick an embedding u : A ′ → B ′ with u ∈ J such that thereare (1 + ε/ i : A → A ′ and i : B → B ′ with u ◦ i = i ◦ f .Since g ◦ i − is also a (1 + ε/ n and a (1 + ε )-embedding v : A ′ → X n, such that v ∈ L n and k v − g ◦ i − k < ε . Then k v ◦ i − g k < (1 + ε ) ε . Using the construction above, we thus have thefollowing commutative diagram: SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 31
A BA ′ B ′ X n, X n +1 , fi i uv ι B ′ ι In the diagram, i , i , v and ι B ′ are each (1 + ε )-embeddings. We now let h : B → X ω ( X ) = ι B ′ ◦ i . This is clearly a (1 + ε ) -embedding. Finally,for all x ∈ B ( A ), we have k g ( x ) − hf ( x ) k = k g ( x ) − ι B ′ i f ( x ) k = k g ( x ) − vi ( x ) k < (1 + ε ) ε. Thus X ω ( X ) is of approximately universal disposition for finite dimensionallattices. (cid:3) One can use a similar argument to construct (non-separable) lattices of uni-versal disposition by amalgamating over all combinations of embeddings ofseparable spaces ω times rather than selecting a countable subset each step(see [3, Theorem 5.3]).We can adapt our construction with additional conditions as a way to discernthe structure of BL . For instance, we can start with X ∈ K ′ in particular,and then inductively construct increasing lattices X = X ⊆ X ⊆ ... ⊆ X n ⊆ X n +1 ⊆ .... with each X n ∈ K ′ . First, we ensure that J and each L n consist only of maps between lattices in K ′ . This is possible assuming X isin K ′ and by Lemma 3.2. Then given X n ∈ K ′ , we can construct X n +1 ∈ K ′ by applying parts 2 and 3 of Corollary 3.14 over the n th pair ( u kn , g kn ) ∈ Γ k for each k < n , followed by the first n pairs in Γ n . dom( u n ) cod( u n ) dom( u n ) cod( u n ) . . . dom( u nn ) cod( u nn ) X n ∗ ∗ ∗ X n +1 u n v n u n v n u nn v nn ... Thus for any pair ( u kn , v kn ) ∈ Γ k with v kn a C -embedding for some C , thereexists some m > k (here m = max( k + 1 , n + 1) ) and a C -embedding ι : cod( u kn ) → X m such that v kn = ι ◦ u kn . The lattice S n X n is then a limitof finite dimensional lattices and is thus finitely branchable, and a smallvariation of the argument in Theorem 5.3 can be used to show that it is alsoof approximately universal disposition for finite dimensional lattices. It turnsout, however, that we have derived an alternate, simplified construction of BL : Theorem 5.4.
Any two finitely branchable lattices of approximately uni-versal disposition for finite dimensional lattices are isometric. In particular,they are isometric to BL and are thus of approximately universal dispositionfor finitely generated lattices.Proof. Suppose X and Y are two finitely branchable separable lattices ofapproximately universal disposition for finite dimensional lattices. As in theproof of Theorem 5.2, we simply construct an isometry f : X → Y with itsinverse g : Y → X .Let ( X n ) and ( Y n ) be sequences of finite dimensional lattices generated bycorresponding spanning trees, (here we let X n = span( { x σ : | σ | = n } )such that X = S X n and Y = S Y n . Let ε n ↓ Q (1 + ε n ) < ε . We then proceed just like in the proof of Theorem 5.2,but with a modification. Let f : X → Y be an isometry (this is possiblebecause X and Y are simply 1-dimensional lattices spanned by x ∅ and y ∅ ,respectively). Now, let g : Y → X be a (1 + ε )-embedding such that k x ∅ − g f ( x ∅ ) k ≤ , By density of S X n , and since Y is finite dimensional,we can in fact ensure that g maps into some X k for some k ∈ N .Rather than generating lattices ˜ X n and ˜ Y n , using Lemmas 5.1 and 3.8,we can pick (1 + ε n − )-embeddings f n : X k n → Y k ′ n and (1 + ε n − )-embeddings g n : Y k ′ n → X k n +1 such that k g n f n − Id X k < n − , and similarly, k f n g n − − Id Y k ≤ n − . Then k f n +1 − f n k = k f n +1 − f n +1 g n f n + f n +1 g n f n − f n k≤k f n +1 kk Id X − g n f n k + k f n +1 g n − Id Y kk f n k≤ n − + 22 n − = 12 n − A similar argument can be made for the g n ’s. Thus for all n and for all x ∈ X k n the sequence ( f m ( x )) m>n is Cauchy. The same is true for any y ∈ Y k ′ n and sequence ( g m ( y )) m>n . Let f = lim f n and g = lim g n , and weare done. (cid:3) The assumption of finite branchability is essential in the above proof. Itis currently unknown, however, if there are lattices which are not finitelybranchable but are of approximately universal disposition for finite dimen-sional lattices.Since BL is finitely branchable, it contains many non-trivial projectionbands. Recall that a ideal sublattice B ⊆ X is a band if for all x ∈ X and sets A ⊆ B , if x = sup A, then x ∈ B . Given a set A ⊆ X , we let A ⊥ = { x ∈ X : x ⊥ a for all a ∈ A } . A ⊥ is itself a band, and if B is a band, SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 33 then B ⊥⊥ = B (see [1, Theorem 1.28]). A band B is a projection band if X = B ⊕ B ⊥ ; that is, every x ∈ X can be uniquely written as x + x with x ∈ B and x ∈ B ⊥ . Note that if B is a projection band, it induces a latticeprojection P : X → B , that is, a contractive lattice homomorphism onto B with P = P and in particular, P | B = Id | B . Let ( x σ ) σ ∈ T be a linearly densespanning tree in BL . Then it is clear that BL = ⊕ | σ | = n span( { x τ : τ ⊇ σ } ),and that each sublattice span( { x τ : τ ⊇ σ } ) is a projection band. We thushave the following: Theorem 5.5.
Every non-trivial projection band in
B ⊆ BL is itself iso-metric to BL .Proof. First note that B itself is finitely branchable. Given a finitely branch-ing tree ( x σ ) σ ∈ T ⊆ BL and lattice projection P : BL → B , we get ( P ( x σ )) σ ∈ T as a spanning tree for B .We now show that B is of approximately universal disposition for finite di-mensional lattices. By Theorem 5.4, this implies that B must in fact beisometric to BL .Let J , Γ n , L n , and X n denote the sets and lattices used in the construc-tion preceding Theorem 5.4. Let A ⊆ B , and f : A → B be an isomericembedding between finite dimensional A and B . Given ε >
0, the goal is toconstruct a (1 + ε )-embedding g : B → B such that k g ◦ f − Id | A k < ε .We begin with the case where f ( A ) fully supports B . Let ( a i ) i and ( b i ) i be finite sequences enumerating the atoms of A and B , and let N be suchthat N f (cid:0) P i a i (cid:1) ≥ P i b i . Finally, let x B be a weak unit in B such that P i a i ≤ x B . Then given δ >
0, by Lemma 3.8 and density of ∪ X n in BL , there exist m ∈ N , a (1 + δ )-embedding v : A ′ → X m with v ∈ L m , u : A ′ → B ′ with u ∈ J , j : A → A ′ , j : B → B ′ , and x ′ B ∈ X m such that(1) For all i , k N a i − N vj ( a i ) k < δ dim A (2) j and j are (1 + δ )- isometries with u ◦ j = j ◦ f (3) k N x B − N x ′ B k < δ and x ′ B ≥ vj (cid:0) P i a i (cid:1) .If necessary, we can replace x B with x B ∨ P ( x ′ B ). All prior conditions willstill be fulfilled, so we can assume x B ≥ P ( x ′ B ). Now ( u, v ) ∈ Γ m , so thereexists some n > m such that B ′ also (1 + δ )-embeds into X n : A BA ′ B ′ X m X nfj j uv ι B ′ ι Furthermore,( ∗ ) N x ′ B ≥ N X i vj ( a i ) = N X i ιvj ( a i ) = N X i ι B ′ uj ( a i )= X i ι B ′ ι (cid:0) N X i f ( a i ) (cid:1) ≥ ι B ′ ι (cid:0) X i b i (cid:1) . .Let g : B → B be defined by g = P ◦ ι B ′ ◦ j . By ( ∗ ), we have that N x B ≥ N P ( x ′ B ) ≥ g (cid:0) P i b i (cid:1) . Furthermore, since P is a band projection, itfollows that for all P i c i ι B ′ j ( b i ) with 0 ≤ c i ≤ N x B ∧ (cid:0) ι B ′ j X i c i b i (cid:1) = N x B ∧ P ι B ′ j (cid:0) X i c i b i (cid:1) = g (cid:0) X i c i b i (cid:1) . Therefore (cid:13)(cid:13)(cid:13)(cid:13) X i c i ι B ′ j ( b i ) − X i c i g ( b i ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) N x ′ B ∧ X i c i ι B ′ j ( b i ) − N x B ∧ X i c i ι B ′ j ( b i ) (cid:13)(cid:13)(cid:13)(cid:13) < δ, so k ι B ′ ◦ j − g k < δ, and since ι B ′ ◦ j is a (1 + δ ) -embedding, g is a (1+ δ ) − δ -embedding. Finally, since the diagram commutes, we have gf ( a i ) = P ι B ′ j f ( a i ) = P vj ( a i ). Since a i ∈ B , P ( a i ) = a i , so for all P i c i a i ∈ S ( A ),by condition 1 we have (cid:13)(cid:13)(cid:13)(cid:13) gf (cid:0) X i c i a i (cid:1) − X i c i a i (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) X i c i (cid:0) P vj ( a i ) − P ( a i ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X i c i k vj ( a i ) − a i k < δ. Now δ can be arbitrarily small, so assume that (1+ δ ) − δ < ε . Since g = P ◦ ι B ′ ◦ j , it sends elements of B into B thanks to composition by P .Thus g satisfies the requirements.Suppose now that B is not fully supported by A . For all δ >
0, we canperturb f with a (1 + δ )-embedding f ′ : A → B such that f ′ ( A ) fullysupports B and k f − f ′ k < δ . By Lemma 3.13, let B ′ be a copy of B witha (1 + δ )-equivalent renorming so that f ′ : A → B ′ is an embedding. SEPARABLE UNIVERSAL HOMOGENEOUS BANACH LATTICE 35
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Dept. of Mathematics, University of Illinois, Urbana IL 61801, USA
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