Further, we study some properties of the family of antisymmetric ideals

(1)

DIMITRIE KRAVVARITIS AND GAVRIIL P ˘ALTINEANU Received 16 October 2002

LetEbe 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), whenAis a subset of the real part of the center ofE, andFis a vector subspace ofE.

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 isTheorem 4.2which states that for everyx∈Ewe have x∈Fif and only ifπI(x)∈πI(F) for any minimal (A,F)-antisymmetric idealI, whereπI

denotes the canonical mappingE→E/I.

This theorem is a Bishop’s type approximation theorem and generalizes a similar result forC(X).

Finally, we show that if the pair (A,F) fulfils some supplementary conditions, thenF is dense inE, and also show how Nachbin’s density theorem for weighted spaces follows from this theorem.

2. Preliminaries

In the sequel,Edenotes a real, locally convex, locally solid vector lattice of (AM)-type.

For every closed idealIofE, we will denote byπIthe canonical mappingE→E/Iand by πIit’s adjoint. The centerZ(E) ofEis the algebra of all order-bounded endomorphisms onE, that is, thoseU∈L(E,E) for which there existsλU>0 such that|U(x)| ≤λU|x|, for allx∈E. The real part of the center is ReZ(E)=Z(E)+−Z(E)+.

(2)

Definition 2.1. For every closed idealI ofEand everyU∈ReZ(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 operatorπI(U) is well defined. For everyA⊂Z(E), we denote πI(A)=

πI(U);U∈A. (2.2)

Remark 2.2. IfA⊂ReZ(E), thenπI(A)⊂ReZ(E/I).

Indeed, ifU∈A, then, for everyx∈E, we have πI(U)πI(x)=πI

U(x)=πIU(x)

≤πI λU|x|

=λUπI

|x|

=λUπI(x), (2.3) henceπI(U)∈Z(E/I).

Definition 2.3. LetI andJ be two closed ideals ofEsuch thatI⊂J. Then the following two mappings can be defined:πIJ:E/I→E/Jgiven by

πIJ

πI(x)=πJ(x), x∈E, (2.4)

andMIJ: ReZ(E/I)→ReZ(E/J) given by MIJ(U)πJ(x)=πIJ

UπI(x), U∈ReZ(E/I). (2.5) As a consequence of the inequality,

MIJ(U)πJ(x)=πIJUπI(x)

=πIJUπI(x)≤πIJλUπI(x)

=λUπIJπI(x)=λUπJ

|x|

=λUπJ(x),

(2.6)

for everyx∈E, the range ofMIJis included in ReZ(E/J).

3. Antisymmetric ideals

LetAbe a subset of ReZ(E) containing 0 and letFbe a vector subspace ofE.

Definition 3.1. A closed idealIofEis said to be antisymmetric with respect to the pair (A,F) if, for everyU∈πI(A) with the propertyU[πI(F)]⊂πI(F), it follows that there exists a real numberαsuch thatU=α1E/I, where1E/Iis the identity operator onE/I.

Of course,Eitself is an antisymmetric ideal with respect to the pair (A,F) for every A⊂ReZ(E) and every vector subspaceFofE.

Further, we denote byᏭA,F(E) the family of all (A,F)-antisymmetric ideals ofE.

Now we consider the particular caseE=C(X,R), whereX is a compact Hausdorﬀ space. It is well known that there is a one-to-one correspondence between the class of the closed ideals ofC(X,R) and the class of the closed subsets ofX. Namely, for every closed

(3)

subsetSofX, the setIS= {f ∈C(X,R); f|S=0}is a closed ideal ofC(X,R) and every closed ideal ofC(X,R) has this form.

Definition 3.2. LetAbe a subset ofC(X,R) with 0∈Aand letF be a closed subset of C(X,R). A closed subsetSofXis said to be antisymmetric with respect to the pair (A,F) if every f ∈Awith the property f ·g|S∈F|Sfor everyg∈Fis constant onS.

Remark 3.3. A closed subsetSofXis (A,F)-antisymmetric if and only if the corresponding idealISis (A,F)-antisymmetric in the sense ofDefinition 3.1.

Indeed, it is suﬃcient to observe thatπIS(a)=a|Sfor every subsetSofX.

Lemma3.4. Let(Iα)be a family of elements ofᏭA,F(E)such thatJ=

αIα=E. Then

I= ∩αIα∈ᏭA,F(E). (3.1)

Proof. IfU∈πI(A) has the propertyU[πI(F)]⊂πI(F), then MIIα(U)πIα(F)=πIIα

UπI(F)⊂πIIα

πI(F)=πIα(F). (3.2) LetV∈Abe such thatU=πI(V). For everyx∈E, we have

MIIα(U)πIα(x)=πIIα

UπI(x)=πIIα

πI(V)πI(x)

=πIIα

πI

V(x)=πIα

V(x)=πIα(V)πIα(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 anaα∈Rexists such thatMIIα(U)=aα·1E/Iα.

On the other hand, we have

MIJ(U)=MIαJ

MIIα(U)=aα·1E/Iα. (3.4) SinceJ=E, it follows thataα=a(constant) for anyα. Therefore,

MIIα(U)=a·1E/Iα=a·MIIα

1E/I

, (3.5)

hence,

MIIα

U−a·1E/I

=0, (3.6)

for anyα, and this involvesU=a·1E/I.

Corollary3.5. EveryI∈ᏭA,F(E)contains a unique minimal idealI∈ᏭA,F(E).

Proof. LetI∈ᏭA,F(E) be such thatI=Eand letI= ∩{J∈ᏭA,F(E); J⊂I}. According toLemma 3.4,I∈ᏭA,F(E). It is now obvious thatI⊂IandIis minimal.

Further, we denote byᏭA,F(E) the family of all minimal closed ideals ofE, antisymmetric with respect to the pair (A,F).

(4)

4. Bishop’s type approximation theorem

Lemma4.1. LetAbe a subset ofReZ(E)with0∈A, letF be a vector subspace ofE, and letV be a convex and solid neighborhood of the origin ofE, which is also a sublattice. If

f ∈Ext{V⁰∩F⁰}andI= {x∈E;|f|(|x|)=0}, thenI∈ᏭA,F(E).

Proof. LetU∈πI(A) be such thatU[πI(F)]⊂πI(F). We can suppose that 0≤U≤1E/I. Since f ∈I⁰, there exists g ∈(E/I) such that f =π1g. Obviously, g ∈ {[πI(V)]⁰

∩[πI(F)]⁰}. We denoteg1=Ug,g2=(1E/I−U)g, andai=inf{λ >0 : gi∈λ[πI(V)]⁰}=

sup{|gi(y)|: y∈πI(V)}, fori=1, 2.

Sinceg=g1+g2∈(a1+a2)[πI(V)]⁰, it follows that f ∈(a1+a2)V⁰, hencea1+a2≥ 1.

On the other hand, for anyy1,y2∈πI(V), we have g1

y1+g2

y2=gUy1+g1E/I−Uy2

≤ |g|

Uy1∨y2+1E/I−Uy1∨y2

= |g|y1∨y2.

(4.1)

SinceπI(V) is a sublattice andg∈[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, hencea1+a2=1.

Now, we observe that if|g|(|y|)=0, then y=0. Indeed, letx∈Ebe such that y= πI(x).

We have 0= |g|(|πI(x)|)= |π_Ig|(|x|)= |f|(|x|).

If follows thatx∈I, hencey=πI(x)=0.

This remark involves that ifg1=Ug=0, thenU=0 and, analogously,g2=(1E/I− u)g=0 impliesU=1E/I.

Therefore, we can suppose thatgi=0 fori=1, 2, and henceai>0,i=1, 2. Further, we have

g=a1

g1

a1

+a2

g2

a2

, gi

ai ∈

πI(V)⁰∩

πI(F)⁰, 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−a11E/I)(g)=0.

The last equality yieldsU=a11E/I.

The main result concerning antisymmetric ideals is the following Bishop’s type approximation theorem.

Theorem4.2. LetEbe a real, locally convex, locally solid vector lattice of (AM)-type,A⊂ ReZ(E)with0∈A, and letFbe a vector subspace ofE. Then, for anyx∈E,

x∈F⇐⇒πI(x)∈πI(F) (4.3)

for everyI∈ᏭA,F(E).

(5)

Proof. The necessity is clear. We suppose thatπI(x)∈πI(F) for anyI∈ᏭA,F(E) and that x /∈F. Then, there exists f ∈Esuch thatf(x)=0 and f(y)=0 for anyy∈F.

LetVbe a solid, convex neighborhood of the origin which is also a sublattice ofE. By the Krein-Milman theorem, we may assume that f ∈Ext{V⁰∩F⁰}. If we denoteJ= {x∈ E;|f|(|x|)=0}, then, according toLemma 4.1, we haveJ∈ᏭA,F(E). On the other hand, byCorollary 3.5, it follows that there existsJ0∈ᏭA,F(E) such thatJ0⊂J. SinceπJ0(x)∈ πJ0(F) and f ∈J0⁰∩F⁰, we have f(x)=0, and this contradicts the choice of f. Theorem4.3. LetEbe a real, locally convex, locally solid vector lattice of (AM)-type, letA be a subset ofReZ(E)with0∈A, and letFbe a vector subspace ofEwith the properties

(i)AF⊂F,

(ii)Fis not included in any maximal ideal ofE,

(iii)every closed(A,F)-antisymmetric idealIofEwith the propertyπI(A)⊂R·1E/Iis a maximal ideal.

ThenF=E.

Proof. Letx∈EandI∈ᏭA,F(E). Hypothesis (i) involves thatπI(A)[πI(F)]⊂πI(F), and sinceIis (A,F)-antisymmetric, we haveπI(U)=αU·1E/Ifor anyU∈A. Now, from (iii), it results thatIis a maximal ideal and thus that the dimension ofπI(E) is one.

SinceF⊂E, we have eitherπI(F)= {0}orπI(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 anyI∈ᏭA,F(E). According toTheorem 4.2, it follows thatx∈F.

5. The case of weighted spaces

Typical examples of locally convex lattices are the weighted spaces.

LetXbe a locally compact Hausdorﬀspace and letV be a Nachbin family onX, that is, a set of nonnegative upper semicontinuous functions onXdirected in the sense that, givenv1,v2∈Vandλ >0, av∈Aexists such thatvi≤λv,i=1, 2. We denote byCV0(X) the corresponding weighted spaces, that is,

CV0(X)=

f ∈C(X,R); f vvanishes at infinity for anyv∈V. (5.1) The weighted topology onCV0(X) is denoted byωV and it is determined by the semi- norms{pv}v∈V, where

pv(f)=supf(x)v(x) : x∈X, for any f ∈CV0(X). (5.2) The topologyωVis 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 topologyωV and to the ordering f ≤gif and only iff(x)≤g(x),x∈X.

(6)

A result of Goullet de Rugy [1, Lemma 3.8] states that for every closed ideal I of CV0(X) there exists a closed subsetY ofXsuch that

I=f ∈CV0(X) : f|Y=0. (5.4)

Therefore, there exists a one-to-one map from the family of closed ideals ofCV0(X) onto the family of closed subsets ofX.

If X is a compact Hausdorﬀ space and V = {1}, then CV0(X)=C(X,R) and the weighted topologyωVcoincides with the uniform topology ofC(X,R).

Further, we denote byCb(X,R) the algebra of all real bounded continuous functions onX.

As in the case ofC(X), we have the following definition.

Definition 5.1. LetAbe a subset ofCb(X) with 0∈Aand letF be a vector subspace of CV0(X). A closed subsetSofXis 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·h|S∈F|S, for anyh∈F, is constant onS.

It is easily seen that everyx∈Xbelongs to a maximal (A,F)-antisymmetric setSx. At the same time, ifx=y, we have eitherSx=SyorSx∩Sy= ∅.

Theorem 4.2then involves the following theorem.

Theorem5.2. LetAandFbe as inDefinition 5.1. Then, a function f ∈CV0(X)belongs toFif and only if f|Sx∈F|Sxfor anyx∈X.

The following theorem is a generalization of Nachbin’s density theorem for weighted spaces in the real case.

Theorem5.3. LetAbe a subset ofCb(X,R)with0∈Aand letFbe a vector subspace of CV0(X)with the properties

(i)AF⊂F,

(ii)Aseparates the points ofX,

(iii)for everyx∈X, there is an f ∈Fsuch that f(x)=0.

ThenF=CV0(X).

Proof. Since the centre of the latticeE=CV0(X) is the algebraCb(X) of all continuous bounded functions onX(see, e.g., [2]), it follows thatA⊂ReZ(E). On the other hand, from (iii), it follows thatF is not included in any maximal ideal. SinceAF⊂F andA separates the points ofX, it results that every (A,F)-antisymmetric subsetS ofX is a singleton, and thus the corresponding idealISis a maximal ideal. Thus the hypotheses of Theorem 4.3are satisfied and soTheorem 5.3is proved.

(7)

References

[1] A. Goullet de Rugy, Espaces de fonctions pond´erables, Israel J. Math. 12 (1972), 147–160 (French).

[2] G. P˘altineanu 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

E-mail address:[email protected]

Gavriil P˘altineanu: Department of Mathematics, Technical University of Civil Engineering of Bucharest, Lacul Tei Blvd.124, Sector 2, RO-020396, Bucharest 38, Romania

E-mail address:[email protected]