CC BY
7
Владикавказский математический журнал 2015, Том 17, Выпуск 1, С. 60-64
УДК 517.98
A CHARACTERIZATION OF ORDER BOUNDED DISJOINTNESS PRESERVING BILINEAR OPERATORS
A. G. Kusraev, S. S. Kutateladze
The paper is aimed to characterize order bounded disjointness preserving bilinear operators in terms of
their null-spaces. To this end the Boolean valued analysis approach is employed.
Mathematics Subject Classification (2000): 46A40, 47A65.
Key words: Boolean valued representation, vector lattice, disjointness preserving operator.
It was observed and employed in [1, 2, 3] that a linear operator T from a vector lattice X to a Dedekind complete vector lattice Y is, in a sense, determined up to an orthomorphism from the family of the kernels of the strata nT of T with n ranging over all band projections on Y. Similar reasoning was involved in [4] to characterize order bounded disjointness preserving bilinear operators. Unfortunately, Theorem 3.4 in [4] is erroneous and this note aims to give correct statement and proof of this result. Unexplained terms can be found on the theory of vector lattices and order bounded operators, in [5, 6], on Boolean valued analysis machinery, in [7, 8].
In what follows X, Y, and Z are Archimedean vector lattices, Zu is a universal completion of Z, and B : X x Y ^ Z is a bilinear operator. We denote the Boolean algebra of band projections in X by P(X). Recall that a linear operator T : X ^ Y is said to be disjointness preserving if x ± y implies Tx ± Ty for all x, y e X. A bilinear operator B : X x Y ^ Z is called disjointness preserving (a lattice bimorphism) if the linear operators B(x, ■) : y ^ B(x,y) (y e Y) and B(-,y) : x ^ B(x,y) (x e X) are disjointness preserving for all x e X and y e Y (lattice homomorphisms for all x e X+ and y e Y+). Denote Xn := f||ker(nB(-,y)) : y e Y} and Yn := f| {ker(nB(x, ■)) : x e X}. Clearly, Xn and Yn are vector subspaces of X and Y, respectively. Now we state the main result of the note.
Theorem. Assume that X, Y, and Z are vector lattices with Z having the projection property. For an order bounded bilinear operator B : X x Y ^ Z the following assertions are equivalent:
(1) B is disjointness preserving.
(2) There are a band projection q e P(Z) and lattice homomorphisms S : X ^ Zu and T : Y ^ Zu such that B(x, y) = qS(x)T(y) - qxS(x)T(y) for all (x, y) e X x Y.
(3) For every n e P(Z) the subspaces Xn and Yn are order ideals respectively in X and Y, and the kernel of every stratum nB of B with n e P(Z) is representable as
ker(nB) = У {Xa x Yr : a,r e P(Z); a V т = n}.
© 2015 Kusraev A. G., Kutateladze S. S.
The proof presented below follows along general lines of [1-4]: Using the canonical embedding and ascent to the Boolean valued universe V(B), we reduce the matter to characterizing disjointness preserving bilinear functional on the product of two vector lattices over dense subfield of the reals R. The resulting scalar problem is solved by the following simple fact.
Lemma 1. Let X and Y be vector lattices. For an order bounded bilinear functional P : X x Y ^ R the following assertions are equivalent:
(i) P is disjointness preserving.
(ii) ker(P) = (X0 x Y) U (X x Y0) for some order ideals X0 C X and Y0 C Y.
(iii) There exist lattice homomorphisms g : X ^ R and h : Y ^ R such that either P(x, y) = g(x)h(y) or P(x, y) = —g(x)h(y) for all x e X and y e Y.
< Assume that ker(P) = (X0 x Y) U (X x Y0) and take y e Y. If y e Y0 then p(■, y) = 0, otherwise ker(P(-,y)) = X0 and P(-,y) is disjointness preserving, since an order bounded linear functional is disjointness preserving if and only if its null-space is an order ideal. Similarly, P(x, ■) is disjointness preserving for all x e X and thus (ii) (i). The implication (i) (iii) was established in [9, Theorem 3.2] and (iii) (i) is trivial with X0 = ker(g) and Y0 = ker(h). >
Let B be a complete Boolean algebra and V(B) the corresponding Boolean valued model with Boolean truth values ] for set-theoretic formulas There exists an element R e V(B) which plays the role of a field of reals within V(B). The descending functor sends every internal algebraic structure A into its descent Aj which is an algebraic structure in conventional sense. Gordon's theorem (see [5, 8.1.2] and [10, Theoren 2.4.2]) tells us that the algebraic structure Rj (with the descended operations and order relation) is an universally complete vector lattice. Moreover, there is a Boolean isomorphism x of B onto P(Rj) such that b ^ [ x = y ] if and only if x(b)x = x(b)y. We identify B with P(Rj) and take x to be /B.
Let [X x Y, Rj] e V and [XA x YA, R] e V(B) stand for the sets respectively of all maps from X x Y to Rj and from XA x XA to R (within V(B)). The correspondences / h-> /t, the modified ascent, is a bijection between [X x Y, Rj] and [XA x YA, R ] Given f e [X, Rj], the internal map e [XA,R] is uniquely determined by the relation [/l(irA) = f(x)j = 1 (x £ X). Observe also that vr ^ [/t(icA) = nf(x)} {x G X, vr G P(R|)). This fact specifies for bilinear operators as follows.
Lemma 2. Let B : X x Y ^ Y be a bilinear operator and ¡3 := B^ its modified ascent. Then P : XA x YA ^ R is a RA-bilinear functional within V(B). Moreover, B is order bounded and disjointness preserving if and only [ p is order bounded and disjointness preserving ] = 1.
< The proof goes along similar lines to the proof of Theorem 3.3.3 in [10]. >
Lemma 3. Let B and P be as in Lemma 2. Then [ker(B)A = ker(P)] = 1.
< Using the above mentioned determining property of modified ascent and interpreting the formal definition z e ker(P) o (3 x e XA)(3 y e YA)(z = (x,y) A P(x,y) = 0), the proof is reduced to a straightforward calculation:
[z e ker(P)] = V [z = (xA, yA) A P(xA, yA) = 0]
x€X, y€V
= V [z = (x, y)A A (x,y)A e ker(B)A 1
(x,y)eXxY
62
Kusraev A. G., Kutateladze S. S.
< [z G ker(B)A] = V [z = (x, y)A A (x.y) G ker(B)]
(x,y)eX xY
= V [(z = (xA,yA) A P(xA,yA) = 0]
xex, yeY
^ [z G ker(P)]. >
Lemma 4. Define X and Y within V(B) by X := f| {ker(P(■, Y)) : y G YA| and Y : = P|{ker(P(x, ■)) : x G XA|. Given arbitrary n G P(Z), x G X, and y G Y, the equivalences hold:
n ^ [xA G X] ^ x G Xn, n ^ [yA G Y\ ^ y G Yn.
< For n G P(Z) and x G X we need only to calculate Boolean truth values taking into account that [B(x, y) = P(xA, vA)] = 1 for all x G X and y G Y:
[xA G X] = [(Vv G YA)P(xA, v) = 0] = A [P(xA, vA) = 0] = / [B(x, v) = 0]
veY veY
It follows that n ^ [xA G X] if and only if n ^ [B(x, v) = 0] for all v G Y. By Gorgon's theorem the latter means that nB(x, v) = 0 for all v G Y, that is x G Xn. >
Lemma 5. Let B and P be as in Lemma 2. For arbitrary n G P(Z), x G X, and y G Y, we have n ^ [(xA, yA) G (X x Y) U (X x Y)] if and only if there exist a, t G P(Z) such that a V t = n, x G Xa, and y G YT.
< Denote p:= [(xA, yA) G (X x Y) U (X x Y)] and observe that
p = [(xA G X) V yA G Y] = [xA G X] V [yA G Y]
Clearly, n ^ p if and only if a V t = n for some a ^ [xA G X] and t ^ [yA G Y], so that the required property follows from Lemma 4. >
Proof of the main result. The implication (1) (2) was proved in [9, Corollary 3.3], while (2) (3) is straightforward. Indeed, observe first that if (2) is fulfilled then |B(x,y)| = |B|(|x|, |y|) = |S|(|x|)|T|(|y|), so that we can assume S and T to be lattice homomorphisms, as in this event ker(B) = ker(|B|). Take n G P(Z) and denote a := n — n[Sx] and t := n — n[Ty], where [y] is a band projection onto {y|x±. Observe next that nB(x, y) = 0 if and only if n[Sx] and n[Ty] are disjoint or, what is the same, if a V t = n. Moreover, the map py : x ^ aS(x)T(y) is disjointness preserving for all y G Y and hence Xa = HyeY ker(py) is an order ideal in X. Similarly, YT is an order ideal in Y. Thus, (x, y) G ker(nB) if and only if x G Xa and y G YT for some a, t G P(Z) with a V t = n.
Prove the remaining implication (3) (1). Suppose that for every n G P(Y) the representation in (3) holds. Take x, u G X and put n := [xA G X] p := [|u|A ^ |x|A]. By Lemma 4 we have x G Xn. Note also that either p = 0 or p = 1. If p = 1 then |u| ^ |x| and by hypotheses u G Xn. Again by Lemma 4 we get p ^ [uA G X] This estimate is obvious whenever p = 0, so that [xA G X] A [|u|A ^ |x|A] ^ [uA G X] = 1 for all x, u G X. Now, a simple calculation shows that X is an order ideal in XA:
[(Vx, u G XA)(|u| ^ |x| A x G X ^ u G X)]
= / ([x G X] A [|u| < |x|] ^ [u G X]) = 1.
Similarly, Y is an order ideal in YA.
It follows from the hypothesis (3) and Lemma 5 that (x,y) e ker(nB) if and only if п ^ [(хл,ул) e (X x Y) U (X x Y)]. Taking into account Lemma 2 and the observation made before it we conclude that п ^ [(хл,ул) e ker(e)]) if and only if п ^ [(хл,ул) e (X x Y) U (X x Y)J and hence [ker(e) = (X x Y) U (X x Y)] = 1. It remains to apply within V(B) the equivalence (i)^^(iii) in Lemma 1. It follows that B is disjointness preserving according to Lemma 2. >
Corollary. Assume that Y has the projection property. An order bounded linear operator T : X ^ Y is disjointness preserving if and only if ker (bT) is an order ideal in X for every projection b e P(Y).
< Apply the above theorem to the bilinear operator B : X x R ^ Y defined as B(x, A) = AT(x) for all x e X and A e R. >
References
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2. Kutateladze S. S. On Grothendieck subspaces // Sib. Math. J—2005 —Vol. 46, No 3.-P. 620-624.
3. Kutateladze S. S. The Farkas lemma revisited // Sib. Math. J—2010—Vol. 51, No l.-P. 78-87.
4. Kusraev A. G., Kutateladze S. S. On order bounded disjointness preserving operators // Sib. Math. J.—2014.—Vol. 55, № 5.-P. 915-928.
5. Kusraev A. G. Dominated Operators.—Dordrecht: Kluwer, 2000.
6. Aliprantis C. D., Burkinshaw O. Positive Operators.—N. Y.: Academic Press, 1985.—xvi+367 p.
7. Bell J. L. Boolean Valued Models and Independence Proofs in Set Theory.—N. Y.: Clarendon Press, 1985.—xx+165 p.
8. Kusraev A. G., Kutateladze S. S. Boolean Valued Analysis.—Novosibirsk: Nauka, 1999; Dordrecht: Kluwer, 1999.
9. Kusraev A. G., Tabuev S. N. On multiplicative representation of disjointness preserving bilinear operators I I Sib. Math. J.-2008.-Vol. 49, № 2.-P. 357-366.
10. Kusraev A. G., Kutateladze S. S. Boolean Valued Analysis: Selected Topics.—Vladikavkaz: SMI VSC RAS, 2014.—iv+400 p.—(Trends in Science: The South of Russia. A Mathematical Monograph. Issue 6).
Received February 16, 2015.
Kusraev Anatoly Georgievich Southern Mathematical Institute Vladikavkaz Science Center of the RAS, Director 22 Markus street, Vladikavkaz, 362027, Russia E-mail: kusraev@smath.ru
Kutateladze Semen Samsonovich Sobolev Institute of Mathematics, senior staff scientist 4 Koptyug Avenue, Novosibirsk, 630090, Russia E-mail: sskutOmember. ams. org
О ХАРАТЕРИЗАЦИИ ПОРЯДКОВО ОГРАНИЧЕННЫХ БИЛИНЕЙНЫХ ОПЕРАТОРОВ, СОХРАНЯЮЩИХ ДИЗЪЮНКТНОСТЬ
Кусраев А. Г., Кутателадзе С. С.
Цель заметки — дать характеризацию сохраняющих дизъюнктиость иорядково ограниченных билинейных операторов в векторных решетках в терминах ядер. В доказательстве основного результата используется булевозначный подход.
Ключевые слова: булевозначное представление, векторная решетка, сохраняющий дизъюнктность оператор.
