A DENSITY THEOREM FOR LOCALLY CONVEX LATTICES

In this paper, we introduce the concept of antisymmetric ideal with respect to a pair (A,F), when A is a subset of the real part of the center of E, and F is a vector subspace of E. This notion is a generalization, for locally convex lattices, of the notion of antisymmetric set from the theory of function algebras. Further, we study some properties of the family of antisymmetric ideals. For example, we show that every element of this family contains a unique minimal element belonging to this family. The main result of this paper is Theorem 4.2 which states that for every x ∈ E we have x ∈ F if and only if πI(x)∈ πI(F) for any minimal (A,F)-antisymmetric ideal I , where πI denotes the canonical mapping E→ E/I . This theorem is a Bishop’s type approximation theorem and generalizes a similar result for C(X). Finally, we show that if the pair (A,F) fulfils some supplementary conditions, then F is dense in E, and also show how Nachbin’s density theorem for weighted spaces follows from this theorem.


Introduction
In this paper, we introduce the concept of antisymmetric ideal with respect to a pair (A,F), when A is a subset of the real part of the center of E, and F is a vector subspace of E. This notion is a generalization, for locally convex lattices, of the notion of antisymmetric set from the theory of function algebras.
Further, we study some properties of the family of antisymmetric ideals.For example, we show that every element of this family contains a unique minimal element belonging to this family.
The main result of this paper is Theorem 4.2 which states that for every x ∈ E we have x ∈ F if and only if π I (x) ∈ π I (F) for any minimal (A,F)-antisymmetric ideal I, where π I denotes the canonical mapping E → E/I.This theorem is a Bishop's type approximation theorem and generalizes a similar result for C(X).
Finally, we show that if the pair (A,F) fulfils some supplementary conditions, then F is dense in E, and also show how Nachbin's density theorem for weighted spaces follows from this theorem.

Preliminaries
In the sequel, E denotes a real, locally convex, locally solid vector lattice of (AM)-type.For every closed ideal I of E, we will denote by π I the canonical mapping E → E/I and by π I it's adjoint.The center Z(E) of E is the algebra of all order-bounded endomorphisms on E, that is, those U ∈ L(E,E) for which there exists It is easily seen that the operator π I (U) is well defined.For every A ⊂ Z(E), we denote Indeed, if U ∈ A, then, for every x ∈ E, we have Definition 2.3.Let I and J be two closed ideals of E such that I ⊂ J. Then the following two mappings can be defined: π IJ : E/I → E/J given by and M IJ : ReZ(E/I) → Re Z(E/J) given by As a consequence of the inequality, for every x ∈ E, the range of M IJ is included in ReZ(E/J).

Antisymmetric ideals
Let A be a subset of ReZ(E) containing 0 and let F be a vector subspace of E.
Definition 3.1.A closed ideal I of E is said to be antisymmetric with respect to the pair (A,F) if, for every U ∈ π I (A) with the property U[π I (F)] ⊂ π I (F), it follows that there exists a real number α such that U = α1 E/I , where 1 E/I is the identity operator on E/I.
Of course, E itself is an antisymmetric ideal with respect to the pair (A,F) for every A ⊂ Re Z(E) and every vector subspace F of E.
Further, we denote by Ꮽ A,F (E) the family of all (A,F)-antisymmetric ideals of E. Now we consider the particular case E = C(X,R), where X is a compact Hausdorff space.It is well known that there is a one-to-one correspondence between the class of the closed ideals of C(X,R) and the class of the closed subsets of X. Namely, for every closed subset S of X, the set R) and every closed ideal of C(X,R) has this form.Definition 3.2.Let A be a subset of C(X,R) with 0 ∈ A and let F be a closed subset of C(X,R).A closed subset S of X is said to be antisymmetric with respect to the pair (A,F) if every f ∈ A with the property f • g|S ∈ F|S for every g ∈ F is constant on S.
Remark 3.3.A closed subset S of X is (A,F)-antisymmetric if and only if the corresponding ideal I S is (A,F)-antisymmetric in the sense of Definition 3.1.
Indeed, it is sufficient to observe that π IS (a) = a|S for every subset S of X. (3.1) On the other hand, we have (3.4) Since J = E, it follows that a α = a (constant) for any α.Therefore, for any α, and this involves Proof.Let I ∈ Ꮽ A,F (E) be such that I = E and let I = ∩{J ∈ Ꮽ A,F (E); J ⊂ I}.According to Lemma 3.4, I ∈ Ꮽ A,F (E).It is now obvious that I ⊂ I and I is minimal.
Further, we denote by Ꮽ A,F (E) the family of all minimal closed ideals of E, antisymmetric with respect to the pair (A,F).

Bishop's type approximation theorem
Lemma 4.1.Let A be a subset of Re Z(E) with 0 ∈ A, let F be a vector subspace of E, and let V be a convex and solid neighborhood of the origin of E, which is also a sublattice.If On the other hand, for any y 1 , y 2 ∈ π I (V ), we have (4.1) Now, we observe that if |g|(|y|) = 0, then y = 0. Indeed, let x ∈ E be such that y = π I (x).
We have 0 If follows that x ∈ I, hence y = π I (x) = 0.This remark involves that if g 1 = U g = 0, then U = 0 and, analogously, Therefore, we can suppose that g i = 0 for i = 1,2, and hence a i > 0, i = 1,2.Further, we have The last equality yields The main result concerning antisymmetric ideals is the following Bishop's type approximation theorem.
Theorem 4.2.Let E be a real, locally convex, locally solid vector lattice of (AM)-type, A ⊂ Re Z(E) with 0 ∈ A, and let F be a vector subspace of E.Then, for any x ∈ E, for every Proof.The necessity is clear.We suppose that π I (x) ∈ π I (F) for any I ∈ Ꮽ A,F (E) and that x / ∈ F.Then, there exists f ∈ E such that f (x) = 0 and f (y) = 0 for any y ∈ F. Let V be a solid, convex neighborhood of the origin which is also a sublattice of E. By the Krein-Milman theorem, we may assume that f ∈ Ext{V 0 ∩ F 0 }.If we denote J = {x ∈ E; | f |(|x|) = 0}, then, according to Lemma 4.1, we have J ∈ Ꮽ A,F (E).On the other hand, by Corollary 3.5, it follows that there exists J 0 ∈ Ꮽ A,F (E) such that J 0 ⊂ J. Since π J0 (x) ∈ π J0 (F) and f ∈ J 0 0 ∩ F 0 , we have f (x) = 0, and this contradicts the choice of f .Theorem 4.3.Let E be a real, locally convex, locally solid vector lattice of (AM)-type, let A be a subset of Re Z(E) with 0 ∈ A, and let F be a vector subspace of E with the properties Then F = E.
Proof.Let x ∈ E and I ∈ Ꮽ A,F (E).Hypothesis (i) involves that π I (A)[π I (F)] ⊂ π I (F), and since I is (A,F)-antisymmetric, we have π I (U) = α U • 1 E/I for any U ∈ A. Now, from (iii), it results that I is a maximal ideal and thus that the dimension of π I (E) is one.Since F ⊂ E, we have either π I (F) = {0} or π I (F) = π I (E).From (ii), it results that π I (F) = {0}.Therefore, we have π I (F) = π I (E) and thus π I (x) ∈ π I (F) for any I ∈ Ꮽ A,F (E).According to Theorem 4.2, it follows that x ∈ F.

The case of weighted spaces
Typical examples of locally convex lattices are the weighted spaces.
Let X be a locally compact Hausdorff space and let V be a Nachbin family on X, that is, a set of nonnegative upper semicontinuous functions on X directed in the sense that, given v 1 ,v 2 ∈ V and λ > 0, a v ∈ A exists such that v i ≤ λv, i = 1,2.We denote by CV 0 (X) the corresponding weighted spaces, that is, CV 0 (X) = f ∈ C(X,R); f v vanishes at infinity for any v ∈ V . (5.1) The weighted topology on CV 0 (X) is denoted by ω V and it is determined by the seminorms {p v } v∈V , where p v ( f ) = sup f (x) v(x) : x ∈ X , for any f ∈ CV 0 (X). (5. 2) The topology ω V is locally convex and has a basis of open neighborhoods of the origin of the form (5.3) Clearly, CV 0 (X) is a locally convex, locally solid vector lattice of (AM)-type with respect to the topology ω V and to the ordering f ≤ g if and only if f (x) ≤ g(x), x ∈ X.