Every lateral band is the kernel of an orthogonally additive operator

• 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 □ 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 ⊔ y ∈ I for every disjoint x, y ∈ I ;

• (2)    if x € I then y € I for all y € F x .

It is clear that every order ideal of E is a lateral ideal of E. The set F x 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.

• 2.    Main Result

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 (x д ) д е Л in a vector lattice E is called order fundamental if the net (x д — x д ‘ ) (д,д ‘ )еЛхЛ order converges to zero. An order fundamental net (x д ) дел in E is called laterally fundamental if x д □ x д ‘ for all A, A ’ € Л with A C A ’ . A subset D of the vector lattice E is called laterally closed , if every laterally fundamental net (х д ) деЛ 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.

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 lateral ly disjoint and write x ^ y if { z € F x П F y } = 0. We say that two subsets H and D of E are laterally disjoint and use the notation H † D if x † y for every x ∈ H and y ∈ D .

Let E be a vector lattice and G be a lateral ideal in E . Put

• (1)    G ^t := { x € E : x f y for all y € G } ;

• (2)    G ^tt := (G ^t ) ^t .

Lemma 2.1. G ^t is a lateral band in E.

• < 1 Take x € G ^t and y € F x . Since F y C F x we have that y € G ^t . Fix x,y € G ^t with x ± y. We note that

F xuy = { u U v : u € F x , v € F y } .

Suppose that x U y / G ^t . Then there exists 0 = w € F _xUy П G and consequently w = u U v for some u ∈ F _x and v ∈ F _y . It follows that u, v ∈ G and we come to the contradiction. Thus G ^t is a lateral ideal in E. Now we show that G ^t is laterally closed. Suppose that (x д ) д_Gл is a laterally convergent net in G ^t that converges to x € E. We claim that x € G ^t . Indeed, assume that x / G ^t . Then there exists 0 = y € F x П G and since all fragments of y belong to G we have that (x д ) д_eл 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 x † y.

The following example shows that the converse assertion, in general, is not true.

Example 2.1. Let (Q, X,v) be a finite measure spaces, E = L o (v). We denote the characteristic function of a set D € X by 1d . Suppose that x = 1 q and y = kl ^ , for some 0 < k < l. Then | x | Л | y | = x Л y = y >  0. On the other hand we have that

F x = { 1d : D € X } and F x = { kl D : D € X } .

Hence, F x П F y = 0 and therefore x ^ y.

The next lemma clarifies the relation between the lateral and the traditional disjointness.

Lemma 2.3 [2, Proposition 5.7] . Let E be a vector lattice, x, y ∈ E , x † y and x, y ∈ F _v 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 ⊂ G ^†† and G = G^ if G is a lateral band.

• < 1 Take x E G. Then F x П F y = 0 for all y E G^ and therefore x 6 G^. Suppose that G is a lateral band and assume that the inclusion G ⊂ G ^†† is strict. Then there is x ∈ G ^†† such that there exists 0 = y E Fx with F y П G = 0. It follows that y E G t 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 E, the set G(x) = F x П G contains a maximal element x G 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 E there exists the unique decomposition x = Xg U x _G t , with Xg E G and x _G † ∈ G ^† .

• <    Fix x E E. By Lemma 2.5 there is the unique decomposition x = Xg U (x — Xg). We claim that (x — Xg ) E G t . Indeed, assuming the contrary we can find y E F _x-x G such that y ∈ G . Then x G ⊑ x G ⊔ y and it contradicts to the maximality of x G . ⊲

Remark 2.1. We note that orthogonally additive projections p G and p _G † onto lateral bands G and G ^† are defined by the setting p G x := Xg, p G t x := x _G t , x E E and the identity operator Id E on E has the representation Id E = p G U p G t . We observe that different properties of projections onto lateral bands were investigated in [1]. In particular, an orthogonally additive projection p G 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 pro jections p G and p _G † onto G and G ^† coincide with linear order pro jections π G and π _G ⊥ onto projection bands G and G ^⊥ respectively.

• <    It is enough to prove that G ^ = G ^ . The relation G ^ C G ^ is obvious. Assume that there exists x E E with x E G t and x / G ^ . Then | x | Л y = u >  0 for some 0 < y E G. Let n _u is a band projection onto the projection band { u } ^^ . Then v := n u x E G and F x П F v = F v . Thus x / G ^ and G ¹ = G ^± . ▻

We provide an example of a decomposition of a vector lattice into the disjoint sum of “nonlinear” bands.

Example 2.2. Let (Q, X,v) be a finite measure spaces, E = L o (v), x E E and G = F x . Take y ∈ E and define ν -measurable sets:

D G := { t E Q : y(t) = x(t) } , D G _t := { t E Q : y(t) = x(t) } .

Then for every and y E E there is the disjoint decomposition y = y G U y G t , where y G := y1 D y ^and y G t ^:= y 1Dy, .

G ^†

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 p G t : F ^ F is a well defined orthogonally additive operator. By p G t | e we denote the restriction of p G t on E . Clearly, ker p G t | e = G. Put

Tx := | p G t | e x | , x E E.

Since p G t | e preserves disjointness we have that T : E ^ F is a well defined positive, disjointness preserving orthogonally additive operator and ker T = G . ▻