Atomicity in injective banach lattices

• 1.    Introduction

• 2.    Boolean Valued Representation

The aim of this note is to examine a Boolean valued interpretation of the concept of atomic Banach lattice and to give a complete description of the corresponding class of injective Banach lattices. Some representation and isometric classification results for general injective Banach lattices were announced in [1, 2].

Section 2 collects some needed Boolean valued representation results following [3]. In Section 3 we demonstrate that a Boolean valued interpretation of atomicity yields some “module atomicity” over a certain f -subalgebra of the center. Section 4 deals with Boolean valued Banach lattices of summable families, which turn out to be “building blocks” for general module atomic injective Banach lattices. Section 5 exposes the main results on representation and classification of injective Banach lattices with atomic Boolean valued representation, i. e. those which are atomic with respect to their natural f -module structure.

The needed information on the theory of Banach lattices can be found in [1, 5]. Recall some definitions and notation. A real Banach lattice X is said to be injective if, for every Banach lattice Y , every closed vector sublattice Y o C Y , and every positive linear operator T o : Yo ^ X there exists a positive linear extension T : Y ^ X of T o with ||T o || = ||T||; see [5, Definition 3.2.3]. Equivalently, X is an injective Banach lattice if, whenever X is lattice isometrically imbedded into a Banach lattice Y , there exists a positive contractive projection from Y onto X ; one more equivalence definition states that each positive operator from X to any Banach lattice admits a norm preserving positive extension to any Banach lattice containing X as a vector sublattice, see [3, Theorem 5.10.6]. This concept was introduced by Lotz [6]; a significant advance towards the structure theory of injectives was made by Cartwright [7] and Haydon [8].

In what follows X stands for a real Banach lattice. We denote by P(X) the Boolean algebra of all band projections in X . A crucial role in the theory of injective Banach lattices is played by the concept of M -pro jection. A band projection π in a Banach lattice X is called an M-projection if || x || = max{|nx|, ||n ^± x|} for all x G X, where n ^± := I x — n. The collection M(X) of all M-projections in X is a subalgebra of the Boolean algebra P(X).

Throughout the sequel B is a complete Boolean algebra with unit 1 and zero (D, while Λ := Λ(B) is a Dedekind complete unital AM -space such that B is isomorphic to P(Λ). The unit of Л is also denoted by 1. A partition of unity in B is a family (b ^ ) g _e = C B such that V g e = b^ = 1 and b^ Л b_n = 1) whenever £ = n We let := denote the assignment by definition, while N, Q, and R symbolize the naturals, the rationals, and the reals.

Boolean valued analysis is an useful tool in studying of injective Banach lattices [9]. We need some Boolean valued representation results as presented in [3] and [25].

Applying the Transfer and Maximum Principles to the ZFC-theorem “There exists a field of reals” we find an element R G V ^{( B )} for which [ R is a field of reals]] = 1. We call R the reals within V ^{( B )} . The following remarkable result due to Gordon [28] tells us that the interpretation of the reals in V ^{( B )} is a universally complete vector lattice with the Boolean algebra of band projections isomorphic to B.

Theorem 2.1. Let R be the reals within V ^{( B )} . Then R4 ( with the descended operations and order ) is a universally complete vector lattice with a weak order unit 1 := 1 ^Л . Moreover, there exists a Boolean isomorphism x of B onto P(R^) such that the equivalences

X ^(b)x = x ^(b) y ^^ ^{b 4 [ x} = У 1, x(b)x 4 x(b)y ^^ b 4 [ x 4 У 1

(G)

hold for all x,y G R4 and b G B .

• <1 See [3, Theorem 2.2.4] and [25, Theorem 10.3.4]. >

Definition 2.2. A complete Boolean algebra of M -projections in X is an arbitrary order complete and order closed subalgebra B C M(X). A Banach lattice X is said to be B-cyclic whenever it is a B-cyclic Banach space with respect to a complete Boolean algebra B of M-projections. If X has the Fatou and Levi properties (see [3, 5.7.2]), then M(X) itself is an order closed subalgebra of the complete Boolean algebra P(X).

Definition 2.3. Let Л = R 4 be the bounded part of the universally complete vector lattice R^; i. e., Л is the order-dense ideal in R4 generated by the weak order unit 1 := 1 ^Л G R 4. Take a Banach space X within V ^{( B )} and put X4 := {x G X4 : |x| G Л}. Equip X4 with some mixed norm by putting ||x|| := |||x|||^ for all x G X, where the order unit norm || • ||^ is defined as ||A| ^ := inf{0 < a G R : |A| 4 al} (A G Л). In this situation, (X4, || • ||) is a Banach space called the bounded descent of X. The terms B-isomorphism and B-isometry mean that isomorphism or isometry under consideration commutes with the projections from B, see [3, 5.8.9].

Theorem 2.4. A bounded descent of a Banach lattice from the model V ^{( B )} is a B -cyclic Banach lattice. Conversely, if X is a B -cyclic Banach lattice, then in the model V ^{( B )} there exists up to the isometric isomorphism a unique Banach lattice X whose bounded descent is isometrically B -isomorphic to X . Moreover, B = M(X) if and only if [ there is no M-projection in X other than 0 and X ] = 1 .

• < See [3, Theorem 5.9.1]. >

Definition 2.5. The element X G V (B) from Theorem 2.1 is said to be the Booleanvalued representation of X .

Theorem 2.6. Let X be a Banach lattice with the complete Boolean algebra B = M(X) of M -projections, Λ be a Dedekind complete unital AM -space such that P(Λ) is isomorphic to B . Then the following assertions are equivalent:

• (1)    X is injective.

• (2)    X is lattice B -isometric to the bounded descent of some AL-space from V ^{( B )} .

• (3)    There exists a strictly positive Maharam operator Ф : X ^ Л with the Levi property such that X = L ¹ (Ф) and ||xH = ^(|x|)| ^ for all x G X .

• (4)    There is a Л -valued additive norm on X such that (X, |- |) is a Banach-Kantorovich lattice and |x| = || |x| || ^ for all x G X .

• <1 See [3, Theorem 5.12.5]. >

Theorem 2.7. Suppose that X is a Banach lattice and X is the completion of the metric space X ^л within V ^{( B )} . Then [ X is a Banach lattice ] = 1 and X 4 is lattice B -isometric to C # (Q,X) equipped with the norm |^| = sup{|^(q)| : q G dom(^) C Q} (^ G C#(Q,X )) .

• < The proof is a due modification of [25, 11.3.8]. >

• 3.    Boolean Valued Atomicity

In this section we present Boolean valued interpretation of atomicity.

Definition 3.1. A positive element x of a B-cyclic Banach lattice X is said to be B- indecomposable or a B-atom if for any pair of disjoint elements y,z G X ₊ with y + z ^ x there exists a projection n G B such that ny = 0 and n ^ z = 0, while X is called B-atomic if the only element of X disjoint from every B-atom is the zero element.

Denote by at(X ) and B-at(X) the sets of atoms in X and B-atoms in X, respectively. Let at _i (X) := {x G at(X) : |x| = 1}, while B-at i (X) consists of all x G B-at(X) with |nx| = 1 for all n G B. It is easy to see that B-at _i (X) = {x G B-at(X) : |x| = 1}.

Proposition 3.2. Let X be a B -cyclic Banach lattice identified with the bounded descent X4 of a Banach lattice X, its Boolean valued representation X G V ^{( B )} . Then the following assertions hold:

• (1)    B - at(X) = at(X)4 .

• (2)    B - at i (X) = at i (X)4 .

• (3)    X is B -atomic if and only if [X is atomic ] = 1 .

• <    (1) Observe that x G at(X) if and only if x G X + and for any two positive disjoint elements x _i ,x ₂ G X with x _i + x ₂ <  x we have x _i = 0 or x ₂ = 0. Now, given x G at(X)4 with y + z ^ x for some disjoint y,z G X ₊ , we put b := [y = 0] and n := x(b)- Since [y = 0 ^ z = 0] = 1, we have [y = 0] <  [z = 0] and thus b* = [y = 0] <  [z = 0]. By (G) we have ny = 0 and n ^ z = x(b * )z = 0. Thus, at(X)4 C B-at(X) and for the converse inclusion the argument is similar.

• (2)    Taking into account the representation B-at i (X) = {x G B-at(X) : |x| = 1} the claim follows easily from the following chain of equivalences:

x G at _i (X)4 ^^ [x G at _i (X)] = 1 ^^ [x G at(X)] = [|x|X = 1] = 1

^^ x G B-at(X) Л |x| = 1 ^^ x G B-at _i (X).

• (3)    Let for a while X, X, and X stand for disjoint complements in X, X = X4, and X;, respectively. The third claim is immediate from the first one, since the disjoint complement and the descent commute: (A ^- ); = (A;)^x, see [3, 1.5.3]. Indeed,

(A-)4 = (a-); n x = (a;) - n x = (a; n x) - n x = (A4) -, hence putting A:= at(X) and making use of (1) we deduce that at(X)- = {0} within V(B) if and only if (B-at(X))- = {0}. >

Corollary 3.3. Let B , X , and X be the same as in Proposition 3.2 and Λ = Λ(B) . Then the following assertions hold:

• (1)    x € X ₊ is a B -atom if and only if for each 0 C y C x there exists A € Л ₊ with y = Ax .

• (2)    If x and y are B -atoms in X ₊ then there exist a pair of disjoint projections n, p € B such that nx X ny , px = Au and py = ^u for some ^, A € Л ₊ and u = x + y.

• <1    Interpreting in the model V ^{( B )} the well-known claims corresponding to that particular case when B = {0, I x } (see [13, Theorem 26.4.]) and using Proposition 3.2 yields the required properties. >

Definition 3.4. Given a cardinal γ, say that a B-cyclic Banach lattice X is purely (B, γ)- atomic if X = D ^-- for some subset D o C B-at i (X) of cardinality y and for every nonzero projection n € B and every subset D C B-at i (nX) with nX = D ^-- we have card(D) >  7. Evidently, X is purely ({0, Ix },Y)-atomic if and only if X is atomic and the cardinality of at i (X) is y or, equivalently, X is atomic and the cardinality of the set of atoms in B(X) equals γ. In this case we say also that X is γ - atomic.

Proposition 3.5. A B -cyclic Banach lattice X is purely (B, γ) -atomic for some cardinal γ if and only if [ y ^л is a cardinal and X is y ^л -atomic ] = 1 .

• <    Sufficiency. Assume that Y ^л is a cardinal and X is Y ^л -atomic within V ^{( B )} . The latter means that X is atomic and card(at 1 (X)) = y ^л within V ^{( B )} . If A := at i (X) then there exists ф € V ^{( B )} such that [ф : y ^л ^ A is a bijection]] = 1. Note that ф; embeds y into A; by [3, 1.5.8] and A; = B-at 1 (X) by Proposition 3.1. It follows that the set D := ф;(Y) of cardinality y is contained in B-at i (X) and X = D ^-- , since A = D t and X = A ^-- . Take b € B and a set D ‘ of cardinality в which is contained in B-at i (X) and generates bX, i.e. bX = (D ‘ ) ^-- . Then D ‘ t is of cardinality card(в ^л ) and X = (D ‘ t) ^-- within the relative universe V ^([ . By [3, 1.3.7] [7 ^л = card(Y ^л ) C card(в ^л ) С в^л] = 1 and so y С в.

Necessity. Assume now that X is purely (B, Y)-atomic and X = D ^-- for some D C B-at i (X) of cardinality 7. Then within V ^{( B )} we have A:= D t C at i (X), X = A ^-- and and the cardinalities of ∆ and γ ^∧ coincide, i. e. card(∆) = card(γ ^∧ ). By [3, 1.9.11] the cardinal card(Y ^л ) has the representation card(Y ^л ) = mix _a c _Y b _a a ^л , where (b _a ) _a c _Y is a partition of unity in B. It follows that b _a C [A ^-- = X and A is of cardinality а^л ]] = 1. If b _a = (D then (b _a Л A) ^-- = b _a Л X and b _a Л A is of cardinality card(Y ^л ) = а ^л С Y ^л in the relative universe V^{[ a ]} . (Concerning b _a Л A and b _a Л X and their properties see [3, 1.3.7].) It is easy that b _a Л A = (b _a D)t and so (b _a D) ^-- = bX. By hypothesis X is purely (B, Y)-atomic, consequently, a >  card(b _a D) >  y , so that a = y , since a C y if and only if а ^л C Y ^л . Thus, card(Y ^л ) = Y ^л whenever b _a = (D and Y ^л is a cardinal within V ^{( B )} . >

Definition 3.6. Let γ is a cardinal. A complete Boolean algebra B (as well as its Stone representation space) is said to be y -stable whenever V ^{( B )} = Y ^л = card(Y ^л ), i.e. [Y ^Л is a cardinal ] = 1. An element b € B is called y -stable if the relative Boolean algebra [(L), b] is 7-stable, see [25, Definition 12.3.7]. Finally, say that a partition of unity (n Y^_ег in B with Г a set of cardinals is stable if n Y is Y-stable for all y € Г.

Theorem 3.7. Let X be a B -atomic B -cyclic Banach lattice. There exist a set of cardinals Г and a partition of unity (n Y ^_er such that B y := [d-),^] is Y -stable and n Y X is purely ( B _y , y ) -atomic for all y € Г .

• <1    If a B-cyclic Banach lattice X is B-atomic then its Boolean valued representation X is atomic within V ^{( B )} according to Proposition 3.1. Denote yo := card(at i (X)). By [3, 1.9.11] yo is a mixture of some set of relatively standard cardinals. More precisely, there are nonempty set of cardinals Г and a partition of unity (b Y ^er in B such that x = mix Y ^ r b Y Y ^A and V ⁽ B ^Y ) |= y ^a = card(Y ^A ) with B _Y := [C), b _Y ] for all y € Г. It follows that b _Y Л X is atomic Banach lattice and y ^a = card(at i (b Y Л X)) within V ⁽B ^Y ) . It remains to apply Proposition 3.5. >

• 4.    The Banach Lattices 1 ¹ (Г,Л) and C#(Q,l1(r))

We now consider some special injective Banach lattices that are building blocks for the class of all B-atomic injective Banach lattices. Recall that Λ = Λ(B).

Given a non-empty set Г, denote by 1 ¹ (Г ^а ) € V ^{( B )} the internal Banach lattice of all summable families x := (x _Y ) _Y e r A in R with the norm ||x| _L := ^2_{Y e} r A |x _Y |.

Let 1 ¹ (Г, Л) stand for the vector space of all order summable families in Л, i.e.

1 1 (Г, Л):= { x :Г ^ Л: |x| i := о-£^ | x ( y )| € л } .

The order on 1 ¹ (Г, Л) is defined by letting x ^ y if and only if x ( y ) ^ y (Y) for all y € Г. Evidently, 1 ¹ (Г, Л) is an order ideal of the Dedekind complete vector lattice Л ^г , hence so is 1 ¹ (Г,Л). Moreover, 1 ¹ (Г,Л) equipped with the norm | x | := || |x IJI^ ( x € 1 ¹ (Г,Л)) is a B-cyclic Banach lattice, since B = B(Λ).

Proposition 4.1. 1 ¹ (Г ^а ) is a Boolean valued representation of 1 ¹ (Г, Л) and thus 1¹(Г, Л) and 1 ¹ (Г ^а ) 4 are lattice B -isometric.

• <    Straightforward verification shows that 1 ¹ (Г, Л) is a Banach f-module over Л, see [3, Definitions 2.11.1 and 5.7.1]. The modified ascent mapping x h- x ^ is a bijection from (R^) ^r onto (R ^{r A} )^, see [3, 1.5.9]. It follows from [3, 2.4.7] that | • |_l is the bounded descent of || • ||i and hence x € 1 ¹ (Г,Л) if and only if [ x ^ € 1 ¹ (Г^а)]| = 1. Moreover, in this event [ |x| 1 = | x ^| 1 ] = 1 so that the modified descent induces an isometric bijection between l ¹ (Г, Л) and (1 1 Г ^А )4. Making use of the definition of modified descent it can be easily checked that this bijection is Л-linear and order preserving. >

Proposition 4.2. The Banach lattice 1 ¹ (Г, Л) is B -atomic and injective with M(X) isomorphic to B . Moreover, l ¹ (Г, Л) is purely (B, y ) -atomic if and only if [ y ^a = card(Г ^A )]] = 1 .

• <    By Theorem 2.6 (2) and Propositions 3.2 and 4.1 X is injective with M(X) ~ B and B-atomic. The second part follows from Propositions 3.5 and 4.1, since l ¹ (Г ^а ) is card(Г ^A )-atomic within V ^{( B )} . >

Proposition 4.3. The norm completion of R ^A -normed space 1 ¹ (Г) ^а within V ^{( B )} is a Banach lattice which is lattice isometric to the internal Banach lattice 1 ¹ (Г ^а ) .

• <    Denote by L _l the completion of 1 ¹ (Г) ^а inside V ^{( B )} . Let A be the set of all norm-one atoms in 1 ¹ (Г) which is of course bijective with Г. Then A ^A and Г ^А are also bijective and A ^A can be considered as the set of all norm-one atoms in 1 ¹ (Г ^а ). Denote by Q-lin(A) the set of all linear combinations of the members of A with rational coefficients. Then by [12, 8.4.10] we have (Q-lin(A)) ^A = Q ^A -lin(A ^A ). Clearly, Q ^A -lin(A ^A ) is a dense sublattice in 1 ¹ (Г ^а ),

while (Q-lin(A)) ^Л is a dense sublattice in 1 ¹ (Г) ^Л and thus in L i , since Q-lin(A) is dense in 1 ¹ (Г). Moreover, the norms induced in (Q-lin(A)) ^Л by 1 ¹ (Г ^Л ) and l ¹ (Г) ^Л coincide. Indeed, if x E (Q-lin(A)^ is of the form ^2k _{G n} r (k) u(k) whith n E N, r : n ^ Q, and u : n ^ A, then г ^Л : п^Л ^ Q ^Л , и^Л : п^Л ^ А ^Л and x ^л = ^ к _{е п} л г ^Л (к)и ^Л (к); therefore,

»Х» 1 1 (Р) Л = 1МГ = (E „g„ Н^Г = E k ₆ „ « |r * fl-Ф Л ) .

It follows that L i and 1 ¹ (Г ^Л ) are lattice isometric. >

Corollary 4.4. Let Q be the Stone representation space of B = P(Λ) . Then the injective Banach lattices 1 ¹ (Г, Л) and C # (Q,l ¹ (Г)) are lattice B -isometric.

• < 1 This is immediate from Theorem 2.7 and Proposition 4.3. >

Corollary 4.5. Given an arbitrary infinite cardinals Y i and Y 2 , we may find a Boolean algebra B such that the injective Banach lattices l ¹ (Y i , Л) and l ¹ (Y 2 , Л) are lattice B -isometric provided that Λ = Λ(B) . If Q is the Stone representation space of B then the injective Banach lattices C#(Q, 1 ¹ ( y ₁ ) and C# ( Q,1 ¹ ( y ₂ )) are also lattice B -isometric.

• <    The claim follows from Proposition 4.3 and Corollary 4.4 making use of the cardinal collapsing phenomena: There exists a complete Boolean algebra B such that the ordinals Y Л and y ₂ have the same cardinality within V ^{( B )} , see [3, 1.13.9]. >

Definition 4.6. A B-cyclic Banach lattice X is called B- separable, if there is a sequence (x _n ) С X such that the norm closed B-cyclic subspace, generated by the set {bx n : n E N, b € B}, coincides with X. In more detail, X is called B-separable whenever for every x E X and 0 <  e E R there exist an element x _e E X and a partition of unity (n _n ) _{n G} N in B such that ||x — x _e | C e and n n x = n n x n for all n E N. It can be easily seen that X is B-separable if and only if its Boolean valued representation is separable within V ^{( B )} . Denote by ш the countable cardinal and put l ¹ := 1 ¹ (ш).

Corollary 4.7. For every infinite cardinal γ , there exists a Stonean space Q such that the injective Banach lattice C#(Q, 1 ¹ ( y )) is B -separable, with B standing for the Boolean algebra of the characteristic functions of clopen subsets of Q .

• <    Apply Corollary 4.5 with Y i := Y and Y 2 := ш, where ш is the countable cardinal. It follows that C#(Q, 1 ¹ ( y ) and C # (Q,l ¹ (ш)) are lattice B-isometric. Moreover, [1 ¹ (ш ^Л ) is separable]] = 1 by transfer principle. Taking into account Proposition 4.1 it remains to observe that [X is separable]] = 1 if and only if X4 is B-separable. >

• 5.    The Main Results

Now we are able to state and prove the main representation and classification results for B-atomic injective Banach spaces.

Definition 5.1. Let X be an injective Banach lattice. Say that X is centrally atomic if X is B-atomic with B = M(X). According to corollary 3.3 this amounts to saying that there is no nonzero element in X disjoint from all Л-atom, while a Л-atom is any element x E X + such that the principal ideal generated by x is equal to Лx := {Ax : A E Л}. Given a family of Banach lattices (X _Y , || • | _Y ) _{Y G} r , denote by ( ^ ®_G r b _Y X)the I^ю-sum, the Banach lattice of all families x := (x( y )) _{Y G r} with x ( y ) E X _y for all y E Г and | x | := sup {| x ( y ) | _y : Y E Г} <  to .

Lemma 5.2. For a centrally atomic injective Banach lattice X there exist a set of cardinals Г and a stable partition of unity (n Y ) y _G f in M(X) such that n Y X is purely ( y, B y ) -atomic with B y := [C),n Y ] for all y E Г and injective and the representation holds:

X - B ( E г b Y X) •

\ YGr l^

• < 1 This is immediate from Proposition 3.7. ▻

Lemma 5.3. Suppose that the injective Banach lattices C# ( Q,1 ¹ ( y )) and C # (Q,l ¹ (5)) are lattice B -isometric, where Q is the Stone space of B , while γ and δ are infinite cardinals. If B is γ -stable and δ -stable then γ = δ .

• <    If C#(Q, l ¹ (Г)) and C#(Q, 1 1 (A)) are lattice B-isometric then V ^{( B )} = “ 1 ¹ ( y ^a ) and l ¹ (5 ^A ) are lattice isometric” and thus V ^{( B )} = card(Y ^A ) = card(5 ^A ). It remains to observe that B is Y-stable (5-stable) if and only if V ^{( B )} = card(Y ^A ) = Y ^A (respectively card(5 ^A ) = 5 ^A ). ▻

Theorem 5.4. Let X be a centrally atomic injective Banach lattice. Then there is a set of cardinals Г and a stable partition of unity (n Y ^e r in B = M(X) such that the following lattice B -isometry holds:

X B (EYGr l^7 ЛY))l^, where ЛY = nYЛ (7 € Г). If a partition of unity (pg)gGA in B satisfies the same conditions as (nY)Yer, then Г = A, and nY = pY for all 7 € Г.

• <    The required representation follows from Proposition 4.2 and Lemma 5.2.

Assume now that a partition of unity (pg ) g _G A in B satisfies the same conditions as (n Y ) ygf - Fix 5 € A and put o_Yg := n _Y pg for arbitrary 7 € Г. If a_Yg = 0, then the injective Banach lattices l¹(Y,o_Yg Л) and l¹(5,a_Yg Л) are lattice [(L ),a g _Y ]-isometric to the same band a g _Y X. By Lemma 5.3 7 = 5 and thus A С Г and pg ^ n Y for all 5 € A. Similarly, Г C A and pg ^ n Y for all y € Г. ▻

Remark 5.5. Let Q be the Stone representation space of B. Corollary 4.4 enables us to replace l¹ (7, Л^ by C#(Q_Y , 1 ¹ ( y )) in Theorem 5.4 with a stable partition of unity (Q _Y ) _{Y e} r in he Boolean algebra of clopen subsets of Q. Moreover, if some partition of unity (P g ) g _G A satisfies the same conditions, then Г = A, and P y = Qy for all 7 € Г.

Corollary 5.6. Let X be an injective Banach lattice and Q the Stone representation space of B = M(X) . If X is B -separable, then X is lattice B -isometric to C # (Q, l ¹ ) , l ¹ = l ¹ (w) .

• <    In Theorem 5.4 each component l ¹ (7, Л^ is B _Y -separable and hence its Boolean valued representation is a separable Banach lattice which is lattice isometric to the internal Banach lattice l ¹ (w ^A ). It follows that l ¹ (7, Л^ is lattice B _Y -isometric to C#(Q_Y , l ¹ ) for all 7 € Г by Proposition 4.1 and Corollary 4.4. From this it is obvious that X is B-isometric to c # (Q,l ¹ ). ▻

Proposition 5.7. A B -cyclic Banach lattice is atomic if and only if it is B -atomic and the Boolean algebra B is atomic.

• <    The complete Boolean algebra B is atomic if and only if B = P (A) for some set A and then X is the l ^∞ -sum of a family of Banach lattices (X _a ) _{a ∈} A . This l ^∞ -sum is evidently atomic if and only if X a is atomic for all a € A. ▻

The following corollary should be compared with [7, Theorem 5.6].

Corollary 5.8. An injective Banach lattice X is atomic if and only if there is a set of cardinals Γ such that the following lattice isometry holds:

X ^ ^ -^L-

• <    In Remark 5.5 each Q _Y is a one-point space by Proposition 5.8 and hence C # (Q_y , l ¹ (7)) is lattice isometric to l ¹ (7). ▻

Definition 5.9. The partition of unity (nY)YGr in B = M(X) satisfying the claim of Theorem 5.4 is called the decomposition series of X and is denoted by d(X). Say that the decomposition series d(X) = (nY)YGr and d(Y) = (pY)YGr of centrally atomic injective Banach lattices X and Y are congruent if there exists a Boolean isomorphism τ from M(X) onto M(Y) such that т (nY) = pY for all 7 € Г.

Theorem 5.10. Centrally atomic injective Banach lattices X and Y are lattice isometric if and only if the Boolean algebras M(X) and M(Y ) are isomorphic and the decomposition series d(X) and d(Y ) are congruent.

⊳ Sufficiency. Let X and Y be centrally atomic injective Banach lattices with d(X) = (n _Y ) _{Y G} r and d(Y) = (p _Y ) _{Y G} r and let X and Y be their respective Boolean valued representations. We identify X and Y with X4 and Y 4, respectively. Denote B := M(X) and D := M(Y ) and assume that there exists a Boolean isomorphism τ from B onto D such that т (n _Y ) = p _Y for all y € Г. Recall that there is a bijective mapping т * : V ^{( B )} ^ V ^{( D )} such that a ZFC-formula y(x i ,..., x _n ) is true within V ^{( B )} if and only if ^(т * x i ,..., т*x_n) is true within V ^{( D )} for all x ₁ ,... ,x _n € V ^{( B )} , see [3, 1.3.1, 1.3.2, and 1.3.5 (2)]. It follows that т * (X) is an atomic injective Banach lattice within V ^{( D )} . Moreover, the mapping x н- т * (x) (x € X4) ia a lattice isometry from X4 onto т * (X)4- If a = card(at ₁ (X)) and в = card(at ₁ (Y)), then т * (a) = mix _{Y G} r т (n _Y ) y ^a and в = mix _{Y G} r PY ^л , so that в = т * (a). By [3, 1.3.5 (2)] we have т * (a) = card(at ₁ (т * (X))) and card(at ₁ (Y)) = card(at ₁ (т * (X))). It follows that т * (X) and Y are lattice isometric and hence т * (X)4 and Y 4 are lattice B-isometric.

Necessity. Suppose that h is a lattice isomorphism from X onto Y . Then the mapping т from B onto D defined by т (n) = h о т о h ^-1 is a Boolean isomorphism. Moreover, h(B-at i (nX)) = B-at 1 (т(n)Y). Now it can be easily verified that nX is ([(L),n], Y)-atomic if and only if т (n)Y is ([(L),т(n)], Y)-atomic. It follows that d(X) and d(Y) are congruent. >

Corollary 5.11. Let X be a centrally atomic injective Banach lattice. Then there is a family of Stonean spaces ( Qy ^er , with Г a set of cardinals, such that Q y is Y -stable for all Y € Г and the following lattice B -isometry holds:

X ^ b (E ®₆ r c #№y .^ ))) l „

If some family (Pg ) ^ _g a of Stonean spaces satisfies the above conditions, then Г = A, and P y is homeomorphic with Q _y for all y € Г .

⊳ This is immediate from Theorem 5.10 and since Corollary 4.4 (see Remark 5.5). ⊲

Definition 5.12. The second B-dual of a B-cyclic Banach space is defined by X ^## := (X ^# ) ^# := L B (X ^# , Λ). A B-cyclic Banach space is said to be B-reflexive if the image of X under the canonical embedding X ^ X ## coincide with X ## , see [3, p. 316].

Theorem 5.13. Let X be a B -reflexive injective Banach lattice with B = M(X) . Then there are a sequence of Stonean spaces (Q _k ) _{k ∈ N} , and an increasing sequence of naturals (n _k ) such that the following lattice B -isometry holds:

X ^ (E^ ^C # k' ■•

If some family (P _k ) _{k ∈ N} of Stonean spaces satisfies the above conditions, then Q _k and P _k are homeomorphic for all k € N .

• < 1 Again identify X with X4, where X is an AL-space in V ^{( B )} . It follows from Theorem [3, Theorem 5.8.12] that X * 4 = X4 # and X ** 4 = X4 ## . Therefore, X is B-reflexive if and only if [X is reflexive ] = 1. Since a reflexive AL-space is finite-dimensional, we have

• 1    = [(3 n € N ^л ) dim(X) = n] = V [dim(X) = п^л ].

n ∈ N

This relation enables us to choose a countable partition of unity (bn) in B such that bn ^ [X is a n ^∧ -dimensional AL -space]. Pick the sequence ( n k ) of indices of nonzero projections in (b n ) and denote by Qk the Stonean space of a Boolean algebra B k := [C У,П_кк ]. Now, by the Transfer Principle we conclude that V ^{( B k} ) = “ bn_k Л X is lattice isometric to 1 1 (пЛ)”. The proof is concluded with the help of Theorem 5.10 taking into consideration that for each finite cardinal y every complete Boolean algebra is y -stable and y ^л is a finite cardinal within V ^{( B )} . >