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Владикавказский математический журнал 2021, Том 23, Выпуск 4, С. 115-118
УДК 517.98
DOI 10.46698/e4075-8887-4097-s
EVERY LATERAL BAND IS THE KERNEL OF AN ORTHOGONALLY ADDITIVE OPERATOR*
M. A. Pliev12
1 Southern Mathematical Institute VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia; 2 North Caucasus Center for Mathematical Research VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia E-mail: plimarat@yandex.ru
Abstract. In this paper we continue a study of relationships between the lateral partial order С in a vector lattice (the relation x С y means that x is a fragment of y) and the theory of orthogonally additive operators on vector lattices. It was shown in [1] that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice E and a lateral band G of E, there exists a vector lattice F and a positive, disjointness preserving orthogonally additive operator T: E ^ F such that ker T = G. As a consequence, we partially resolve the following open problem suggested in [1]: Are there a vector lattice E and a lateral ideal in E which is not equal to the kernel of any positive orthogonally additive operator T: E ^ F for any vector lattice F?
Key words: orthogonally operator, lateral ideal, lateral band, lateral disjointness, orthogonally additive projection, vector lattice.
Mathematical Subject Classification (2010): 47H30, 47H99.
For citation: Pliev, M. A. Every Lateral Band is the Kernel of an Orthogonally Additive Operator, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 115-118. DOI: 10.46698/e4075-8887-4097-s.
1. Introduction and Preliminaries
The lateral order introduced in [1] shows its importance for the theory of orthogonally additive operators (OAOs) [1-3]. In this paper we continue investigation of order properties of OAOs connected with the lateral order. For all unexplained notions concerning vector lattices and orthogonally additive operators we refer the reader to [1, 4, 5].
We shall write x = y U z if x = y + z and y ± z. We say that y is a fragment (a component) of x € E, and use the notation y C x, if y ± (x — y).
Definition 1.1. Let E be a vector lattice. A subset I of E is said to be a lateral ideal if the following hold:
(1) x U y € I for every disjoint x, y € I;
(2) if x € I then y € I for all y € Fx.
It is clear that every order ideal of E is a lateral ideal of E. The set Fx of all fragments of an element x € E provides the example of a lateral ideal that is not a linear subspace of E.
The importance of lateral ideals for orthogonally additive operators is demonstrated in the following statement.
Proposition 1.1 [1, Proposition 6.4]. Let E, F be vector lattices and T: E ^ F a positive orthogonally additive operator. Then ker T := {x € E : Tx = 0} is a lateral ideal in E.
#The research is supported by the Ministry of Science and Higher Education of the Russian Federation, agreement № 075-02-2021-1844. © 2021 Pliev, M. A.
The following problem was stated in [1].
Problem 1.2 [1, Problem 6.7]: Are there a vector lattice E and a lateral ideal in E which is not equal to the kernel of any positive orthogonally additive operator T: E ^ F for any vector lattice F?
Below we resolve this problem for lateral bands. That is the special subclass of lateral ideals.
Definition 1.2. We recall that a net (xa)asA in a vector lattice E is called order fundamental if the net (xa — xA')(a,a')eAxA order converges to zero. An order fundamental net (xa)asA in E is called laterally fundamental if xa E xa' for all A, A' € A with A ^ A'. A subset D of the vector lattice E is called laterally closed, if every laterally fundamental net (xa)AgA in D order converges to some x € D. A laterally closed lateral ideal B in E is called a lateral band.
Example 1.1. Every band B of a Dedekind complete vector lattice E is a lateral band of E.
2. Main Result
The next theorem is the main result of these notes.
Theorem 2.1. Let E be a vector lattice and G a lateral band in E. Then there exist a vector lattice F and a positive, disjointness preserving orthogonally additive operator T: E ^ F with ker T = G.
We need some auxiliary constructions to prove Theorem 2.1.
Definition 2.1. Let E be a vector lattice and x, y € E. We say that x and y are laterally disjoint and write xfy if {z € Fx n Fy} = 0. We say that two subsets H and D of E are laterally disjoint and use the notation HfD if xfy for every x € H and y € D.
Let E be a vector lattice and G be a lateral ideal in E. Put
(1) Gt := {x € E : xfy for all y € G};
(2) Gtt := (Gt)t.
Lemma 2.1.
Gt
is a lateral band in E.
< Take x € Gt and y € Fx. Since Fy C Fx we have that y € Gt. Fix x, y € Gt with x ± y. We note that
FxUy = {u U v : u € Fx, v € Fy}•
Suppose that x U y € Gt. Then there exists 0 = w € FxUy n G and consequently w = u U v for some u € Fx and v € Fy. It follows that u,v € G and we come to the contradiction. Thus Gt
is a lateral ideal in E. Now we show that Gt is laterally closed. Suppose that (xa)asA is a laterally convergent net in Gt that converges to x € E. We claim that x € Gt. Indeed, assume that x € Gt. Then there exists 0 = y € Fx n G and since all fragments of y belong to G we have that (xA)AeA does not laterally converges to x. >
Lemma 2.2 [2, Proposition 5.6]. Let E be a vector lattice, x, y € E and x ± y. Then xfy.
The following example shows that the converse assertion, in general, is not true.
Example 2.1. Let (Q, £,v) be a finite measure spaces, E = L0(v). We denote the characteristic function of a set D € £ by 1D. Suppose that x = 1q and y = Mq, for some 0 < k < 1. Then |x| A |y| = x A y = y > 0. On the other hand we have that
Fx = {1d : D € £} and Fx = {k1D : D € £}.
Hence, Fx n Fy = 0 and therefore xfy.
The next lemma clarifies the relation between the lateral and the traditional disjointness.
Every Lateral Band is the Kernel of an Orthogonally Additive Operator
117
Lemma 2.3 [2, Proposition 5.7]. Let E be a vector lattice, x,y € E, xfy and x,y € Fv for some v € E. Then x ± y.
Lemma 2.4. Let E be a vector lattice and G be a lateral ideal in E. Then G c G"" and G = G"" if G is a lateral band.
< Take x € G. Then Fx n Fy = 0 for all y € G" and therefore x € Gtt. Suppose that G is a lateral band and assume that the inclusion G c G"" is strict. Then there is x € G"" such that there exists 0 = y € Fx with Fy n G = 0. It follows that y € G" and we come to the contradiction. >
Lemma 2.5 [6, Lemma 3]. Let E be a Dedekind complete vector lattice and G be a lateral band in E. Then, for every x € E, the set G(x) = Fx n G contains a maximal element xG with respect to the lateral order.
Lemma 2.6. Let E be a Dedekind complete vector lattice and G be a lateral band in E. Then for every x € E there exists the unique decomposition x = xG U xGt, with xG € G and xGt € G".
< Fix x € E. By Lemma 2.5 there is the unique decomposition x = xG U (x — xG). We claim that (x — xG) € G". Indeed, assuming the contrary we can find y € Fx-xG such that y € G. Then xG C xG U y and it contradicts to the maximality of xG. >
Remark 2.1. We note that orthogonally additive projections pG and pGt onto lateral bands G and G" are defined by the setting pGx := xG, pGtx := xGt, x € E and the identity operator on E has the representation = pG UpGt. We observe that different properties of projections onto lateral bands were investigated in [1]. In particular, an orthogonally additive projection pG onto a lateral band G in E preserves disjointness [1, Theorem 6.9].
Lemma 2.7. Suppose that E is a Dedekind complete vector lattice and G is a band in E. Then the disjoint complement G^ coincides with
G"
and orthogonally additive projections pG and pGt onto G and G" coincide with linear order projections nG and nG± onto projection bands G and Gx respectively.
< It is enough to prove that G" = Gx. The relation Gx c G" is obvious. Assume that there exists x € E with x € G" and x / Gx. Then |x| Ay = u > 0 for some 0 < y € G. Let
is a band projection onto the projection band {u}x±. Then v := nux € G and Fx n Fv = Fv. Thus x / G" and G" = Gx. >
We provide an example of a decomposition of a vector lattice into the disjoint sum of "nonlinear" bands.
Example 2.2. Let (Q, £,v) be a finite measure spaces, E = L0(v), x € E and G = Fx. Take y € E and define v-measurable sets:
DG := {t € Q : y(t) = x(t)}, DGt := {t € Q : y(t) = x(t)}. Then for every and y € E there is the disjoint decomposition y = yG U yGt, where yG := y1Dy
and yGt := ylDa t.
Gt
Now we are ready to present a positive orthogonally additive operator that is vanished on a lateral band G.
< Proof of Theorem 2.1. Suppose that F is the Dedekind completion of E. Then by Lemma 2.6 pGt: F — F is a well defined orthogonally additive operator. By pGt |E we denote the restriction of pGt on E. Clearly, ker pGt |E = G. Put
Tx := |pGt|Ex|, x € E.
Since pGt |e preserves disjointness we have that T: E F is a well defined positive, disjointness preserving orthogonally additive operator and ker T = G. >
References
1. Mykhaylyuk, V., Pliev, M. and Popov, M. The Lateral Order on Riesz Spaces and Orthogonally Additive Operators, Positivity, 2021, vol. 25, no. 2, pp. 291-327. DOI: 10.1007/s11117-020-00761-x.
2. Erkursun-Ozcan, N. and Pliev, M. On Orthogonally Additive Operators in C-Complete Vector Lattices, Banach Journal of Mathematical Analysis, 2022, vol. 16, article no. 6. DOI: 10.1007/s43037-021-00158-2.
3. Popov, M. Horizontal Egorov Property of Riesz Spaces, Proceedings of the American Mathematical Society, 2021, vol. 149, no. 1, pp. 323-332. DOI: 10.1090/proc/15235.
4. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, New York, Acad. Press, 1985.
5. Kusraev, A. G. Dominated Operators, Kluwer Academic Publishers, 2000.
6. Pliev, M. and Popov, M. On Extension of Abstract Urysohn Operators, Siberian Mathematical Journal, 2016, vol. 57, no. 3, pp. 552-557. DOI: 10.1134/S0037446616030198.
Received November 2, 2021 Marat A. Pliev
Southern Mathematical Institute VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia, Leading Researcher;
North-Caucasus Center for Mathematical Research VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia, Head of the Research Department E-mail: plimarat@yandex.ru https://orcid.org/0000-0001-8835-8805.
Владикавказский математический журнал 2021, Том 23, Выпуск 4, С. 115-118
КАЖДАЯ ЛАТЕРАЛЬНАЯ ПОЛОСА ЯВЛЯЕТСЯ ЯДРОМ ПОЛОЖИТЕЛЬНОГО ОРТОГОНАЛЬНО АДДИТИВНОГО ОПЕРАТОРА
Плиев М. А.1'2
1 Южный математический институт — филиал ВНЦ РАН, Россия, 362027, Владикавказ, Маркуса, 22;
2 Северо-Кавказский центр математических исследований ВНЦ РАН, Россия, 362027, Владикавказ, Маркуса, 22 E-mail: plimarat@yandex.ru
Аннотация. В данной статье мы продолжим изучение приложений латерального порядка С в векторных решетках (запись x С у означает, что x — это осколок у) к теории ортогонально аддитивных операторов. В работе [1] было установлено, что понятия латерального идеала и латеральной полосы играют такую же важную роль в теории ортогонально аддитивных операторов, как и понятия порядкового идеала и полосы — в теории линейных операторов в векторных решетках. В заметке установлено, что для произвольной векторной решетки E и латеральной полосы G в E найдется векторная решетка F и положительный ортогонально аддитивный оператор T: E ^ F, сохраняющий дизъюнктность, такой, что kerT = G. Данный результат частично решает следующую открытую проблему, указанную в работе [1]. Верно ли, что для любой векторной решетки E и латерального идеала G в E существуют векторная решетка F и положительный ортогонально аддитивный оператор T: E ^ F такие, что ker T = G?
Ключевые слова: ортогонально аддитивный оператор, латеральный идеал, латеральная полоса, латеральная дизъюнктность, ортогонально аддитивный проектор, векторная решетка.
Mathematical Subject Classification (2010): 47H30, 47H99.
Образец цитирования: Pliev M. A. Every Lateral Band is the Kernel of an Orthogonally Additive Operator // Владикавк. мат. журн.—2021.—Т. 23, № 4.—C. 115-118 (in English). DOI: 10.46698/e4075-8887-4097-s.
