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Владикавказский математический журнал 2015, Том 17, Выпуск 3, С. 44^52
УДК 517.98
ATOMICITY IN INJECTIVE BANACH LATTICES1 A. G. Kusraev
To Бе-тёп Kutateladze on occasion of his 70th birthday
This note is aimed 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.
Mathematics Subject Classification (2010): 46B42, 47B65.
Key words: Injective Banach lattice, atomic Banach lattice, Boolean valued representation, classification.
1. Introduction
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 Y0 С Y, and every positive linear operator T0 : Yo — X there exists a positive linear extension T : Y — X of T0 with ||To|| = ||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].
© 2015 Kusraev A. G.
1 The study was supported by the Russian Foundation for Basic Research, projects № 12-01-00623-a, № 14-01-91339_ННИО-а, and № 15-51-53119^ГФЕН^а.
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-projection. A band projection n in a Banach lattice X is called an M-projection if ||x|| = max{||nx||, Hn^x||} for all x € X, where := IX — 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 A:= A(B) is a Dedekind complete unital AM-space such that B is isomorphic to P(A). The unit of A is also denoted by 1. A partition of unity in B is a family (bg)geS c B such that Vges bg = 1 and bg A bn = © whenever £ = n We let := denote the assignment by definition, while IN, (Q, and H symbolize the naturals, the rationals, and the reals.
2. Boolean Valued Representation
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 € 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 Ri (with the descended operations and order) is a universally complete vector lattice with a weak order unit 1 := 1A. Moreover, there exists a Boolean isomorphism x of B onto P(Ri) such that the equivalences
x(b)x = x(b)y ^^ b < Ix = y 1, (G)
x(b)x ^ x(b)y ^^ b ^ [ x < y ]
hold for all x, y € Ri and b € B.
< 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 A = R4 be the bounded part of the universally complete vector lattice Ri; i. e., A is the order-dense ideal in Ri generated by the weak order unit 1 := 1A € Ri. Take a Banach space % within V(B) and put XJj := {x € : \x\ € A}. Equip JTJJ. with some mixed norm by putting ||a;|| := |||a;|||oo for all x € X, where the order unit norm || ■ is defined as ||A||TO := inf{0 < a € R : |A| ^ al} (A € A). In this situation, (X|| ■ ||) 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-cyc-lic 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 € V(B) from Theorem 2.1 is said to be the Boolean-valued representation of X.
Theorem 2.6. Let X be a Banach lattice with the complete Boolean algebra B = M(X) of M-projections, A be a Dedekind complete unital AM-space such that P(A) 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 $ : X — A with the Levi property such that X = L:($) and ||x|| = ||$(|x|)|U for all x € X.
(4) There is a A-valued additive norm on X such that (X, | |) is a Banach-Kantorovich lattice and Ibll = IIIxlII for all x € X.
II II III I II oo
< 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 XA within V(B). Then [ X is a Banach lattice ] = 1 and Xis lattice B-isometric to C#(Q,X) equipped with the norm ||^>|| = sup{||^>(q)|| : q € dom(^) c Q} (^ € 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 € X+ with y + z ^ x there exists a projection n € B such that ny = 0 and = 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 ati(X) := {x € at(X) : ||x|| = 1}, while B-ati(X) consists of all x € B-at(X) with ||7ra;|| = 1 for all ir € B. It is easy to see that B-ati(X) = {x € B-at(X) : \x\ = 1}.
Proposition 3.2. Let X be a B-cyclic Banach lattice identified with the bounded descent X^ of a Banach lattice X, its Boolean valued representation X € V(B). Then the following assertions hold:
(1) B-at(X) = at(X
(2) B-ati(X) = ati(X
(3) X is B-atomic if and only if [X is atomic] = 1.
< (1) Observe that x € at(X) if and only if x € X+ and for any two positive disjoint elements x1,x2 € X with x1 + x2 ^ x we have x1 = 0 or x2 = 0. Now, given x € at(X with y + z ^ x for some disjoint y,z € X+, we put b := [y = 0] and n := x(b). Since [y = 0 — z = 0J = 1, we have [y = 0] ^ [z = 0] and thus b* = [y = 0] ^ [z = 0]. By (G) we have ny = 0 and = x(b*)z = 0. Thus, at(Xc B-at(X) and for the converse inclusion the argument is similar.
(2) Taking into account the representation B-ati(X) = {x € B-at(X) : \x\ = 1} the claim follows easily from the following chain of equivalences:
x € ati(X^ [x € ati(X)] = 1 ^ [x € at(X)] = [||x||X = 1] = 1
^^ xeM-at(X) A \x\ = 1 ^^ xeB-ati(X).
(3) Let for a while X, X, and _L stand for disjoint complements in X, X = Xand Xi, respectively. The third claim is immediate from the first one, since the disjoint complement and the descent commute: (Ax)i = (Ai)see [3, 1.5.3]. Indeed,
(Ax)4 = (Ax)i n X = (Ai)A n X = (Ai n X)A n X = (A4)x,
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))x = {0}. >
Corollary 3.3. Let B, X, and X be the same as in Proposition 3.2 and A = A(B). Then the following assertions hold:
(1) x € X+ is a B-atom if and only if for each 0 ^ y ^ x there exists A € A+ 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 € A+ and u = x + y.
< Interpreting in the model V(B) the well-known claims corresponding to that particular case when B = {0, IX} (see [13, Theorem 26.4.]) and using Proposition 3.2 yields the required properties. >
Definition 3.4. Given a cardinal 7, say that a B-cyclic Banach lattice X is purely (B, 7)-atomic if X = for some subset D0 c B-ati(X) of cardinality 7 and for every nonzero projection n € B and every subset D c B-at1(nX) with nX = we have card(D) ^ 7. Evidently, X is purely ({0,IX},7)-atomic if and only if X is atomic and the cardinality of at1(X) is 7 or, equivalently, X is atomic and the cardinality of the set of atoms in B(X) equals 7. In this case we say also that X is 7-atomic.
Proposition 3.5. A B-cyclic Banach lattice X is purely (B, 7)-atomic for some cardinal 7 if and only if [ 7A is a cardinal and X is 7A-atomic ] = 1.
< Sufficiency. Assume that 7A is a cardinal and X is 7A-atomic within V(B). The latter means that X is atomic and card(at1(X)) = ya within V(B). If A := at1(X) then there exists (/> € V(B) such that [(/> : 7A —>• A is a bijection] = 1. Note that (j)\ embeds 7 into A4-by [3, 1.5.8] and A4- = B-ati(X) by Proposition 3.1. It follows that the set := </^(7) of cardinality 7 is contained in B-at1(X) and X = Dsince A = D| and X = Ax±. Take b € B and a set D' of cardinality ft which is contained in B-at1(X) and generates bX, i.e. bX = (D')±±. Then D't is of cardinality card(^A) and X = (D't)x± within the relative universe V([°>bD. By [3, 1.3.7] [ya = card(7A) ^ card(^A) ^ ] = 1 and so 7 ^
Necessity. Assume now that X is purely (B,7)-atomic and X = Dfor some D c B-at1(X) of cardinality 7. Then within V(B) we have A:= D| c at1(X), X = Ax± and and the cardinalities of A and ya coincide, i. e. card(A) = card(7A). By [3, 1.9.11] the cardinal card(7A) has the representation card(7A) = mixa^7baaA, where (ba)a^7 is a partition of unity in B. It follows that ba ^ [Ax± = X and A is of cardinality aA] = 1. If ba = (D then (ba A A)x± = ba A X and ba A A is of cardinality card(7A) = aA ^ ya in the relative universe V^(Concerning ba A A and ba A X and their properties see [3, 1.3.7].) It is easy that ba A A = (baD)t and so (baD)±± = bX. By hypothesis X is purely (B, 7)-atomic, consequently, a ^ card(baD) ^ 7, so that a = 7, since a ^ 7 if and only if aA ^ ya. Thus, card(7A) = ya whenever ba = O and ya is a cardinal within V(B). >
Definition 3.6. Let 7 is a cardinal. A complete Boolean algebra B (as well as its Stone representation space) is said to be Y-stable whenever V(B) |= ya = card(7A), i.e. [ya is a cardinal ] = 1. An element b € B is called 7-stable if the relative Boolean algebra [©, b] is Y-stable, see [25, Definition 12.3.7]. Finally, say that a partition of unity (nY)7er in B with r a set of cardinals is stable if is Y-stable for all 7 € r.
Theorem 3.7. Let X be a B-atomic B-cyclic Banach lattice. There exist a set of cardinals r and a partition of unity (nY)7er such that BY := [©] is 7-stable and X is purely (BY, 7) -atomic for all 7 € r.
< 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 70 := card(ati(X)). By [3, 1.9.11] Y0 is a mixture of some set of relatively standard cardinals. More precisely, there are nonempty set of cardinals r and a partition of unity (bY)7er in B such that x = mix7er bY7A and V(by) |= 7a = card(7A) with B7 := [©, b7] for all 7 € r. It follows that b7 A X is atomic Banach lattice and 7A = card(ati(bY A X)) within V(By). It remains to apply Proposition 3.5. >
4. The Banach Lattices 1i(r, A) and C#(Q,1i(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 A = A(B).
Given a non-empty set r, denote by 1i(rA) € V(B) the internal Banach lattice of all summable families x := (xY)7erA in R with the norm ||x|i := ^7grA |xY|.
Let 1i(r, A) stand for the vector space of all order summable families in A, i.e.
ll(T, A) := {x : T A : ^ := o-J^ |x(7)| G a}.
The order on 1i(r, A) is defined by letting x ^ y if and only if x(y) ^ y(Y) for all 7 € r. Evidently, 1i(r, A) is an order ideal of the Dedekind complete vector lattice Ar, hence so is ¿1(r,A). Moreover, ^(rjA) equipped with the norm ||x|| := Hlx^Hoo (x e ¿1(r,A)) is a B-cyclic Banach lattice, since B = B(A).
Proposition 4.1. 1i(rA) is a Boolean valued representation of 1i(r, A) and thus 1i(r, A) and 1i(rA)^ are lattice B-isometric.
< Straightforward verification shows that 1i(r, A) is a Banach f-module over A, see [3, Definitions 2.11.1 and 5.7.1]. The modified ascent mapping x is a bijection from
onto (Rr see [3, 1.5.9]. It follows from [3, 2.4.7] that | ^ is the bounded descent of || • ||i and hence x € ^(PjA) if and only if [xf € ¿1(rA)] = 1. Moreover, in this event Hx^ = ||xf||i] = 1 so that the modified descent induces an isometric bijection between 1i(r, A) and (1iTA)^. Making use of the definition of modified descent it can be easily checked that this bijection is A-linear and order preserving. >
Proposition 4.2. The Banach lattice 1i(r, A) is B-atomic and injective with M(X) isomorphic to B. Moreover, 1i(r, A) is purely (B, Y)-atomic if and only if [ya = card(rA)J = 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 1i(rA) is card(rA)-atomic within V(B). >
Proposition 4.3. The norm completion of RA-normed space 1i(r)A within V(B) is a Banach lattice which is lattice isometric to the internal Banach lattice 1i(rA).
< Denote by Li the completion of 1i(r)A inside V(B). Let A be the set of all norm-one atoms in 1i(r) which is of course bijective with r. Then AA and rA are also bijective and AA can be considered as the set of all norm-one atoms in 1i(rA). 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 = QA-lin(AA). Clearly, QA-lin(AA) is a dense sublattice in 1i(rA),
while (Q-lin(A))A is a dense sublattice in 11(r)A and thus in L1, since Q-lin(A) is dense in 11(r). Moreover, the norms induced in (Q-lin(A))A by 11(rA) and 11(r)A coincide. Indeed, if x € (Q-lin(A))A is of the form ^fcen r(k) u(k) whith n € N, r : n — Q, and u : n — A, then rA : nA — QA, uA : nA — AA and xA = YjfcenA rA(k)uA(k); therefore,
INI^a = ||x||A = (|r(k)0 A = Efce„A |rA(k)l = INI^a).
It follows that L1 and 11(rA) are lattice isometric. >
Corollary 4.4. Let Q be the Stone representation space of B = P(A). Then the injective Banach lattices 11(r, A) and C#(Q,11(r)) are lattice B-isometric.
< This is immediate from Theorem 2.7 and Proposition 4.3. >
Corollary 4.5. Given an arbitrary infinite cardinals 71 and 72, we may find a Boolean algebra B such that the injective Banach lattices 11(71, A) and 11(72, A) are lattice B-isometric provided that A = A(B). If Q is the Stone representation space of B then the injective Banach lattices C#(Q, 11(71) and C#(Q,11(72)) 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 7A and 7A 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 (xn) c X such that the norm closed B-cyclic subspace, generated by the set {bxn : n € N, b € B}, coincides with X. In more detail, X is called B-separable whenever for every x € X and 0 < e € R there exist an element x£ € X and a partition of unity (nn)neN in B such that ||x — x£|| ^ e and nnx = nnxn for all n € 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 w the countable cardinal and put l1 := 11(w).
Corollary 4.7. For every infinite cardinal 7, there exists a Stonean space Q such that the injective Banach lattice C#(Q, 11(7)) is B-separable, with B standing for the Boolean algebra of the characteristic functions of clopen subsets of Q.
< Apply Corollary 4.5 with 71 := 7 and 72 := w, where w is the countable cardinal. It follows that C#(Q, 11(7) and C#(Q,11(w)) are lattice B-isometric. Moreover, [11(wA) is separable ]] = 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 A-atom, while a A- atom is any element x € X+ such that the principal ideal generated by x is equal to Ax := {Ax : A € A}. Given a family of Banach lattices (XY, || ■ ||Y)7gr, denote by (bYX^ the I™-sum, the Banach lattice of all families x:= (x(7))with x(7) € XY for all 7 € r and ||x|| := sup{||x(7)||Y : 7 € r} < to.
Lemma 5.2. For a centrally atomic injective Banach lattice X there exist a set of cardinals r and a stable partition of unity (nY)7er in M(X) such that X is purely (7, BY)-atomic with BY := [©, ] for all 7 € r and injective and the representation holds:
X -B (£% X L-
< This is immediate from Proposition 3.7. >
Lemma 5.3. Suppose that the injective Banach lattices c#(q,1i(y)) and C#(Q, ^(5)) are lattice B-isometric, where Q is the Stone space of B, while 7 and 5 are infinite cardinals. If B is y-stable and 5-stable then 7 = 5.
< If C#(Q, 1i(r)) and C#(Q, 1i(A)) are lattice B-isometric then V(B) =="1i(ya) and 1i(5A) are lattice isometric" and thus V(B) == card(7A) = card(5A). It remains to observe that B is 7-stable (5-stable) if and only if V(B) == card(7A) = ya (respectively card(5A) = 5A). >
Theorem 5.4. Let X be a centrally atomic injective Banach lattice. Then there is a set of cardinals r and a stable partition of unity (nY)7er in B = M(X) such that the following lattice B-isometry holds:
X -b(£:er 1i (Y, A7)
where AY = A (7 € r). If a partition of unity (p^)^eA in B satisfies the same conditions as (nY)7gr, then r = A, and = pY for all 7 € r.
< The required representation follows from Proposition 4.2 and Lemma 5.2.
Assume now that a partition of unity (p^)^€a in B satisfies the same conditions as (nY)7gr. Fix 5 € A and put := p<s for arbitrary 7 € r. If = 0, then the injective Banach lattices 1i(y, A) and 1i(5, A) are lattice [©, ]-isometric to the same band X. By Lemma 5.3 7 = 5 and thus A c r and p^ ^ for all 5 € A. Similarly, r c A and p^ ^ for all y € r. >
Remark 5.5. Let Q be the Stone representation space of B. Corollary 4.4 enables us to replace 1i (7, A7) by C#(QY,1i(y)) in Theorem 5.4 with a stable partition of unity (QY)7er in he Boolean algebra of clopen subsets of Q. Moreover, if some partition of unity (P<s)<seA satisfies the same conditions, then r = A, and PY = QY for all 7 € r.
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, 1i), 1i = 1i(w).
< In Theorem 5.4 each component 1i (7, A^ is BY-separable and hence its Boolean valued representation is a separable Banach lattice which is lattice isometric to the internal Banach lattice 1i(wA). It follows that 1i(7, A7) is lattice BY-isometric to C#(QY,1i) for all 7 € r by Proposition 4.1 and Corollary 4.4. From this it is obvious that X is B-isometric to C#(Q,1i). >
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 (Xa)aSA. This l^-sum is evidently atomic if and only if X„ 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 r such that the following lattice isometry holds:
x - (s:£r 'i(y)»1.
< In Remark 5.5 each qy is a one-point space by Pr°p°sition 5.8 and hence (q7, 1i(7)) is lattice isometric to 1i(y). >
Definition 5.9. The partition of unity (nY)7er 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)7er and d(Y) = (pY)7er of centrally atomic injective Banach lattices X and Y are congruent if there exists a Boolean isomorphism t from M(X) onto M(Y) such that t(nY) = pY for all 7 € r.
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) = (nY)7er and d(Y) = (pY)7er and let X and Y be their respective Boolean valued representations. We identify X and Y with X4 and Y4, respectively. Denote B := M(X) and D := M(Y) and assume that there exists a Boolean isomorphism t from B onto D such that t(nY) = pY for all 7 € r. Recall that there is a bijective mapping t* : V(B) — V(D) such that a ZFC-formula ^(x1,..., xn) is true within V(B) if and only if ^>(t*x1,..., t*xn) is true within V(D) for all x1,... ,xn € V(B), see [3, 1.3.1, 1.3.2, and 1.3.5 (2)]. It follows that t*(X) is an atomic injective Banach lattice within V(D). Moreover, the mapping x — t*(x) (x € X4) ia a lattice isometry from X4 onto t*(X)4. If a = card(at1(X)) and ft = card(at1(Y)), then t*(a) = mixYer t(nY)7A and ^ = mixYer p7A, so that ^ = t*(a). By [3, 1.3.5(2)] we have t*(a) = card(at1(T*(X))) and card(at1(Y)) = card(at1(T*(X))). It follows that t*(X) and Y are lattice isometric and hence t*(X)4 and Y4 are lattice B-isometric.
Necessity. Suppose that h is a lattice isomorphism from X onto Y. Then the mapping t from B onto D defined by t(n) = h o t o h-1 is a Boolean isomorphism. Moreover, h(B-at1(nX)) = B-at1(T(n)Y). Now it can be easily verified that nX is ([©, n], 7)-atomic if and only if t(n)Y is ([©, t(n)], 7)-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)Yer, with r a set of cardinals, such that QY is 7-stable for all 7 € r and the following lattice B-isometry holds:
X E®er,l1(Y))),
^er
If some family (Ps)s€a of Stonean spaces satisfies the above conditions, then r = A, and PY is homeomorphic with QY for all 7 € r.
< 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#)#:= LB(X#, A). 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 (Qk)keN, and an increasing sequence of naturals (nk) such that the following lattice B-isometry holds:
X EL C#(Qk ^Vk))) joo
If some family (Pk)keN of Stonean spaces satisfies the above conditions, then Qk and Pk are homeomorphic for all k € N.
< Again identify X with Xwhere X is an AL-space in V(B). It follows from Theorem [3, Theorem 5.8.12] that X= X4# and X= XTherefore, X is B-reflexive if and only if [X is reflexive ] = 1. Since a reflexive AL-space is finite-dimensional, we have
1 = [(3n € NA) dim(X) = n] = V [dim(X) = nA].
v ragN
This relation enables us to choose a countable partition of unity (bn) in B such that bn ^ [X is a nA-dimensional AL-space] Pick the sequence (nk) of indices of nonzero projections in (bn) and denote by Qk the Stonean space of a Boolean algebra Bk := [©,bnk]. Now, by the Transfer Principle we conclude that V(Bk) == "bnk A X is lattice isometric to 1i(nA)". The proof is concluded with the help of Theorem 5.10 taking into consideration that for each finite cardinal 7 every complete Boolean algebra is 7-stable and 7A is a finite cardinal within V(B). >
References
1. Kusraev A. G. Boolean valued analysis and injective Banach lattices // Dokl. RAS.—2012.—Vol. 444, № 2.—P. 143-145.
2. Kusraev A. G. Classification of injective Banach lattices // Dokl. RAS—2013—Vol. 453, № 1—P. 12-16.
3. Kusraev A. G. and Kutateladze S. S. Boolean Valued Analysis: Selected Topics.—Vladikavkaz: Vladikavkaz Scientific Center Press, 2014.^iv+400 p.^(Trends in Science. The South of Russia).
4. Aliprantis C. D. and Burkinshaw O. Positive Operators.^London etc.: Acad. Press Inc., 1985.^ xvi+367 p.
5. Meyer-Nieberg P. Banach Lattices.—Berlin etc.: Springer, 1991.—xvi+395 p.
6. Lotz H. P. Extensions and liftings of positive linear mappings on Banach lattices I I Trans. Amer. Math. Soc.—1975.—Vol. 211—P. 85-100.
7. Cartwright D. I. Extension of positive operators between Banach lattices // Memoirs Amer. Math. Soc.—1975.—164 p.
8. Haydon R. Injective Banach lattices // Math. Z—1974—Vol. 156—P. 19-47.
9. Kusraev A. G. A Boolean transfer principle for injective Banach lattices // Sib. Mat. Zh.—2015.— Vol. 56, № 5.—[DOI 10.17377/smzh.2015.56.511].
10. Kusraev A. G. and Kutateladze S. S. Introduction to Boolean Valued Aiuilvsis. M.: Nauka, 2005.^ 526 p.—[in Russian].
11. Gordon E. I. Real numbers in Boolean-valued models of set theory and K-spaces // Dokl. Akad. Nauk SSSR.—1977.—Vol. 237, № I. 1>. 773-775.
12. Kusraev A. G. Dominated Operators.—M.: Nauka, 2003.^ 619 p.—[in Russian].
13. Luxemburg W. A. J. and Zaanen A. C. Riesz Spaces. Vol. 1.—Amsterdam and London: North-Holland, 1971.-514 p.
Received August 31, 2015.
Kusraev Anatoly Georgievich Vladikavkaz Science Center of the RAS, Chaiman 22 Markus Street, Vladikavkaz, 362027, Russia; K. L. Khetagurov North Ossetian State University 44-46 Vatutin Street, Vladikavkaz, 362025, Russia E-mail: [email protected]
АТОМИЧНОСТЬ В ИНЪЕКТИВНЫХ БАНАХОВЫХ РЕШЕТКАХ
Кусраев А. Г.
Цель заметки — рассмотреть булевозначную интерпретацию понятия атомической банаховой решетки и дать полное описание соответствующего класса инъективных банаховых решеток.
Ключевые слова: инъективная банахова решетка, атомическая банахова решетка, булевозначное представление, классификация.
