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Abstract and Applied Analysis A density theorem for locally convex lattices
A density theorem for locally convex lattices
Kravvaritis, Dimitrie, Păltineanu, Gavriilكم أعجبك هذا الكتاب؟
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المجلد:
2004
عام:
2004
اللغة:
english
مجلة:
Abstract and Applied Analysis
DOI:
10.1155/s1085337504303088
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A DENSITY THEOREM FOR LOCALLY CONVEX LATTICES DIMITRIE KRAVVARITIS AND GAVRIIL PĂLTINEANU Received 16 October 2002 Let E be a real, locally convex, locally solid vector lattice of (AM)type. First, we prove an approximation theorem of Bishop’s type for a vector subspace of such a lattice. Second, using this theorem, we obtain a generalization of Nachbin’s density theorem for weighted spaces. 1. 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 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. 2. 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 πI the canonical mapping E → E/I and by πI it’s adjoint. The center Z(E) of E is the algebra of all orderbounded endomorphisms on E, that is, those U ∈ L(E,E) for which there exists λU > 0 such that U(x) ≤ λU x, for all x ∈ E. The real part of the center is ReZ(E) = Z(E)+ − Z(E)+ . Copyright © 2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:5 (2004) 387–393 2000 Mathematics Subject Classification: 41A65, 46A40 URL: http://dx; .doi.org/10.1155/S1085337504303088 388 A density theorem for locally convex lattices Definition 2.1. For every closed ideal I of E and every U ∈ Re Z(E), πI (U) : E/I → E/I is defined by πI (U) πI (x) = πI U(x) , x ∈ E. (2.1) It is easily seen that the operator πI (U) is well defined. For every A ⊂ Z(E), we denote πI (A) = πI (U); U ∈ A . (2.2) Remark 2.2. If A ⊂ ReZ(E), then πI (A) ⊂ ReZ(E/I). Indeed, if U ∈ A, then, for every x ∈ E, we have πI (U) πI (x) = πI U(x) = πI U(x) ≤ πI λU x = λU πI x = λU πI (x), (2.3) hence πI (U) ∈ Z(E/I). 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: πIJ : E/I → E/J given by πIJ πI (x) = πJ (x), x ∈ E, (2.4) and MIJ : ReZ(E/I) → ReZ(E/J) given by MIJ (U) πJ (x) = πIJ U πI (x) , U ∈ Re Z(E/I). (2.5) As a consequence of the inequality, MIJ (U) πJ (x) = πIJ U πI (x) = πIJ U πI (x) ≤ πIJ λU πI (x) = λU πIJ πI (x) = λU πJ x = λU πJ (x), (2.6) for every x ∈ E, the range of MIJ is included in ReZ(E/J). 3. 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 ∈ πI (A) with the property U[πI (F)] ⊂ πI (F), it follows that there exists a real number α such that U = α1E/I , where 1E/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 ⊂ ReZ(E) and every vector subspace F of E. Further, we denote by ᏭA,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 onetoone 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 D. Kravvaritis and G. Păltineanu 389 subset S of X, the set IS = { f ∈ C(X, R); f S = 0} is a closed ideal of C(X, 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 IS is (A,F)antisymmetric in the sense of Definition 3.1. Indeed, it is suﬃcient to observe that πIS (a) = aS for every subset S of X. Lemma 3.4. Let (Iα ) be a family of elements of ᏭA,F (E) such that J = α Iα = E. Then I = ∩α Iα ∈ ᏭA,F (E). (3.1) Proof. If U ∈ πI (A) has the property U[πI (F)] ⊂ πI (F), then MIIα (U) πIα (F) = πIIα U πI (F) ⊂ πIIα πI (F) = πIα (F). (3.2) Let V ∈ A be such that U = πI (V ). For every x ∈ E, we have = πIIα πI (V ) πI (x) = πIIα πI V (x) = πIα V (x) = πIα (V ) πIα (x) . MIIα (U) πIα (x) = πIIα U πI (x) (3.3) Thus, MIIα (U) = πIα (V ) ∈ πIα (A) ⊂ ReZ(E/Iα ) and MIIα (U)(πIα (F)) ⊂ πIα (F). Since Iα ∈ ᏭA,F (E), it follows that an aα ∈ R exists such that MIIα (U) = aα · 1E/Iα . On the other hand, we have MIJ (U) = MIα J MIIα (U) = aα · 1E/Iα . (3.4) Since J = E, it follows that aα = a (constant) for any α. Therefore, MIIα (U) = a · 1E/Iα = a · MIIα 1E/I , (3.5) hence, MIIα U − a · 1E/I = 0, for any α, and this involves U = a · 1E/I . (3.6) Corollary 3.5. Every I ∈ ᏭA,F (E) contains a unique minimal ideal I ∈ ᏭA,F (E). Proof. Let I ∈ ᏭA,F (E) be such that I = E and let I = ∩{J ∈ ᏭA,F (E); J ⊂ I }. According to Lemma 3.4, I ∈ ᏭA,F (E). It is now obvious that I ⊂ I and I is minimal. Further, we denote by ᏭA,F (E) the family of all minimal closed ideals of E, antisymmetric with respect to the pair (A,F). 390 A density theorem for locally convex lattices 4. Bishop’s type approximation theorem Lemma 4.1. Let A be a subset of ReZ(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 f ∈ Ext{V 0 ∩ F 0 } and I = {x ∈ E;  f (x) = 0}, then I ∈ ᏭA,F (E). Proof. Let U ∈ πI (A) be such that U[πI (F)] ⊂ πI (F). We can suppose that 0 ≤ U ≤ 1E/I . Since f ∈ I 0 , there exists g ∈ (E/I) such that f = π1 g. Obviously, g ∈ {[πI (V )]0 ∩ [πI (F)]0 }. We denote g1 = U g, g2 = (1E/I − U) g, and ai = inf {λ > 0 : gi ∈ λ[πI (V )]0 } = sup{gi (y) : y ∈ πI (V )}, for i = 1,2. Since g = g1 + g2 ∈ (a1 + a2 )[πI (V )]0 , it follows that f ∈ (a1 + a2 )V 0 , hence a1 + a2 ≥ 1. On the other hand, for any y1 , y2 ∈ πI (V ), we have g1 y1 + g2 y2 = g U y1 + g 1E/I − U y2 ≤ g  U y1 ∨ y2 + 1E/I − U y1 ∨ y2 = g  y 1 ∨ y 2 . (4.1) Since πI (V ) is a sublattice and g ∈ [πI (V )]0 , it follows that  y1  ∨  y2  ∈ πI (V ), hence g ( y1  ∨  y2 ) ≤ 1. Therefore, g1 (y1 ) + g2 (y2 ) ≤ 1 for any y1 , y2 ∈ πI (V ) and this yields a1 + a2 ≤ 1, hence a1 + a2 = 1. Now, we observe that if g ( y ) = 0, then y = 0. Indeed, let x ∈ E be such that y = πI (x). We have 0 = g (πI (x)) = πI g (x) =  f (x). If follows that x ∈ I, hence y = πI (x) = 0. This remark involves that if g1 = U g = 0, then U = 0 and, analogously, g2 = (1E/I − u) g = 0 implies U = 1E/I . Therefore, we can suppose that gi = 0 for i = 1,2, and hence ai > 0, i = 1,2. Further, we have g = a1 g1 g2 + a2 , a1 a2 0 0 gi ∈ πI (V ) ∩ πI (F) , ai i = 1,2. (4.2) Since g ∈ Ext{[π1 (V )]0 ∩ [πI (F)]0 }, either g = g1 /a1 or g = g2 /a2 . In the first case, (U − a1 1E/I ) (g) = 0. The last equality yields U = a1 1E/I . 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 ⊂ ReZ(E) with 0 ∈ A, and let F be a vector subspace of E. Then, for any x ∈ E, x ∈ F ⇐⇒ πI (x) ∈ πI (F) for every I ∈ ᏭA,F (E). (4.3) D. Kravvaritis and G. Păltineanu 391 Proof. The necessity is clear. We suppose that πI (x) ∈ πI (F) for any I ∈ ᏭA,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 KreinMilman 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 ∈ ᏭA,F (E). On the other hand, by Corollary 3.5, it follows that there exists J0 ∈ ᏭA,F (E) such that J0 ⊂ J. Since πJ0 (x) ∈ πJ0 (F) and f ∈ J00 ∩ 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 ReZ(E) with 0 ∈ A, and let F be a vector subspace of E with the properties (i) AF ⊂ F, (ii) F is not included in any maximal ideal of E, (iii) every closed (A,F)antisymmetric ideal I of E with the property πI (A) ⊂ R · 1E/I is a maximal ideal. Then F = E. Proof. Let x ∈ E and I ∈ ᏭA,F (E). Hypothesis (i) involves that πI (A)[πI (F)] ⊂ πI (F), and since I is (A,F)antisymmetric, we have πI (U) = αU · 1E/I for any U ∈ A. Now, from (iii), it results that I is a maximal ideal and thus that the dimension of πI (E) is one. Since F ⊂ E, we have either πI (F) = {0} or πI (F) = πI (E). From (ii), it results that πI (F) = {0}. Therefore, we have πI (F) = πI (E) and thus πI (x) ∈ πI (F) for any I ∈ ᏭA,F (E). According to Theorem 4.2, it follows that x ∈ F. 5. 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 v1 ,v2 ∈ V and λ > 0, a v ∈ A exists such that vi ≤ λv, i = 1,2. We denote by CV0 (X) the corresponding weighted spaces, that is, CV0 (X) = f ∈ C(X, R); f v vanishes at infinity for any v ∈ V . (5.1) The weighted topology on CV0 (X) is denoted by ωV and it is determined by the seminorms { pv }v∈V , where pv ( f ) = sup f (x)v(x) : x ∈ X , for any f ∈ CV0 (X). (5.2) The topology ωV is locally convex and has a basis of open neighborhoods of the origin of the form Dv = f ∈ CV0 (x) : pv ( f ) < 1 . (5.3) Clearly, CV0 (X) is a locally convex, locally solid vector lattice of (AM)type with respect to the topology ωV and to the ordering f ≤ g if and only if f (x) ≤ g(x), x ∈ X. 392 A density theorem for locally convex lattices A result of Goullet de Rugy [1, Lemma 3.8] states that for every closed ideal I of CV0 (X) there exists a closed subset Y of X such that I = f ∈ CV0 (X) : f Y = 0 . (5.4) Therefore, there exists a onetoone map from the family of closed ideals of CV0 (X) onto the family of closed subsets of X. If X is a compact Hausdorﬀ space and V = {1}, then CV0 (X) = C(X,R) and the weighted topology ωV coincides with the uniform topology of C(X, R). Further, we denote by Cb (X, R) the algebra of all real bounded continuous functions on X. As in the case of C(X), we have the following definition. Definition 5.1. Let A be a subset of Cb (X) with 0 ∈ A and let F be a vector subspace of CV0 (X). A closed subset S of X is called antisymmetric with respect to the pair (A,F) if and only if the corresponding ideal IS = f ∈ CV0 (X) : f S = 0 (5.5) is an (A,F)antisymmetric ideal, and this means that every a ∈ A with the property α · hS ∈ F S, for any h ∈ F, is constant on S. It is easily seen that every x ∈ X belongs to a maximal (A,F)antisymmetric set Sx . At the same time, if x = y, we have either Sx = S y or Sx ∩ S y = ∅. Theorem 4.2 then involves the following theorem. Theorem 5.2. Let A and F be as in Definition 5.1. Then, a function f ∈ CV0 (X) belongs to F if and only if f Sx ∈ F Sx for any x ∈ X. The following theorem is a generalization of Nachbin’s density theorem for weighted spaces in the real case. Theorem 5.3. Let A be a subset of Cb (X, R) with 0 ∈ A and let F be a vector subspace of CV0 (X) with the properties (i) AF ⊂ F, (ii) A separates the points of X, (iii) for every x ∈ X, there is an f ∈ F such that f (x) = 0. Then F = CV0 (X). Proof. Since the centre of the lattice E = CV0 (X) is the algebra Cb (X) of all continuous bounded functions on X (see, e.g., [2]), it follows that A ⊂ ReZ(E). On the other hand, from (iii), it follows that F is not included in any maximal ideal. Since AF ⊂ F and A separates the points of X, it results that every (A,F)antisymmetric subset S of X is a singleton, and thus the corresponding ideal IS is a maximal ideal. Thus the hypotheses of Theorem 4.3 are satisfied and so Theorem 5.3 is proved. D. Kravvaritis and G. Păltineanu 393 References [1] [2] A. Goullet de Rugy, Espaces de fonctions pondérables, Israel J. Math. 12 (1972), 147–160 (French). G. Păltineanu and D. T. Vuza, A generalisation of the Bishop approximation theorem for locally convex lattices of (AM)type, Rend. Circ. Mat. Palermo (2) Suppl. II (1998), no. 52, 687–694. Dimitrie Kravvaritis: Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Zografou, Greece Email address: dkrav@math.ntua.gr Gavriil Păltineanu: Department of Mathematics, Technical University of Civil Engineering of Bucharest, Lacul Tei Blvd.124, Sector 2, RO020396, Bucharest 38, Romania Email address: gpalt@hidro.utcb.ro Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014