SOME DENSITY THEOREMS FOR COMPLEX LOCALLY CONVEX LATTICES

FOR COMPLEX LOCALLY CONVEX LATTICES

DIMITRIE KRAVVARITIS and GAVRIIL P ˘ALTINEANU

In this paper we continue the study began in [3] of real locally convex lattices.

Thus, some Bishop type approximation theorems for complex locally convex lat- tices of (M)-type are given. We also show how various approximation theorems, known for weighted spaces, follows from this results.

AMS 2000 Subject Classification: Primary 41A65; Secondary 46A40.

Key words: complex lattice, complex locally convex lattice of (M)-type, antisym- metric ideal, antisymmetric set, weighted space.

1. INTRODUCTION

LetE be a complex locally convex lattice of (M)-type,Aa subset of the center of E, and F a (complex) vector subspace ofE.

First, we introduce the notion of antisymmetric ideal with respect to the pair (A, F). This notion is a generalization, for locally convex lattices, of the notion of antisymmetric set from the theory of function algebras.

Further, we study the main properties of the familyA_F,A of all (A, F)- antisymmetric ideals and state some density theorems of the type

x∈F iffπ_I(x)∈π_I(F), ∀I ∈ A_F,A, I-minimal, where π_I is the canonical mappingE →E/I.

An abstract version of the Stone-Weierstrass theorem for complex locally convex lattices is also given.

2. PRELIMINARIES

According to [9] a complex vector lattice is a pair (E, m), whereE is a complex vector space and m a modulus mapping, i.e. m : E → E with the properties:

i)m(m(x)) =m(x),∀x∈E;

ii) m(αx) =|α|m(x),∀α∈C,∀x∈E;

iii)m(m(m(x) +m(y))−m(x+y)) =m(x) +m(y)−m(x+y),∀x, y∈E;

REV. ROUMAINE MATH. PURES APPL.,53(2008),2–3, 167–179

iv)E= Sp(m(E)) (the (complex) subspace spanned bym(E)).

Now, we recall some results and definitions concerning complex vector lattices. The reader is referred to [4], [8] and [9].

Proposition2.1.If E is a complex lattice, then m(E) is a convex cone and E_R =m(E)−m(E)is a real vector lattice with respect to the order relation

x6yiffy−x∈m(E).

Moreover, x>0 iff x=m(x) and for each x∈E_R we have m(x) =|x|= (−x)∨x.

Proposition2.2. Every x∈E can be written in a unique way as a sum x=x1+ ix2, xi∈E_R, i= 1,2.

We shall denote Rex=x₁ and Imx=x₂. So, for every x ∈E we have x= Rex+ i Imx, and x^∗ = Rex−i Imx.

Definition 2.1. Let E be a complex vector lattice (c.v.l.). A complex vector sublattice ofE, is a complex vector subspaceF ofEwith the following property: x ∈F implies Rex∈F and m(x) ∈F. A subset S of E is said to be solid if m(y) 6m(x),∀x ∈S,∀y ∈E impliesy ∈S. An ideal I ofE is a complex vector subspace I of E which is a solid set.

Proposition2.3.Let Ebe a c.v.l.,I ⊂Ean ideal andπ_I:E →E/I the canonical mapping. The modulus on the quotient vector space E/I is defined by

mc_I(π_I(x)) =π_I(m(x)), ∀x∈E.

The pair (E/I,mc_I) is a complex vector lattice.

Definition 2.2. A complex vector lattice E is called archimedean ifER

is archimedean (i.e. x60 wheneverαx6y for somey∈E₊ and allα >0).

Definition 2.3. Let E be a c.v.l., and let r ∈ E₊. A sequence {x_n} is called r-uniformly convergent to x if for every ε > 0 there exists a natural number n_ε such thatm(x_n−x)6εrfor all n>n_ε.

A sequence {x_n} is called r-uniformly fundamental if for every ε > 0, there exists a natural number n_ε such that m(x_n+p−x_n)6εr for all n>n_ε and all p∈N^∗.

A complex vector lattice E is called relatively uniformly complete if for everyr ∈E₊ and everyr-fundamental sequence {x_n}there existsx∈E such that {x_n} ber-uniformly convergent to x.

Proposition2.4. Every archimedean and relatively uniformly complete c.v.l., has the Riesz decomposition property (i.e. for each x∈E , y1, y2 ∈E+

satisfying m(x) 6 y₁+y₂, there exist x₁, x₂ ∈E such that x =x₁+x₂ and m(x1)6y1, m(x2)6y2).

See [4], Theorem 3.3.

Proposition2.5. Let E be a c.v.l. Then for every x∈E we have m(x) = sup{Reax; a∈C, |a|= 1}.

Definition 2.4. LetE₁ and E₂ be two c.v.l. A (complex) linear operator T : E₁ → E₂ is called order bounded, if T maps order bounded sets into order bounded sets (i.e. for each x∈(E1)+ there existsu ∈(E2)+ such that m₁(y)6x impliesm₂[T(y)]6u).

Proposition 2.6. Let E be an archimedean, relatively uniformly com- plete c.v.l. and let F be an order complete c.v.l. Then every order bounded complex linear operator T :E → F has a linear modulus |T| that is defined for each x∈E+ by

|T|x= sup{m₂(T(y)); m₁(y)6x}= sup{Re(λT)(x); λ∈C, |λ|= 1}

(|T|is additive on E+ and can be extended as a positive linear operator from E to F).

Definition 2.5. A complex locally convex lattice of (M)-type (or of (AM)- type) is an archimedian relatively uniformy complete c.v.l. endowed with a linear topology such that there exists a neighborhood basis of the origin con- sisting of sublattices which are solid and convex.

3. ANTISYMMETRIC IDEALS

In the sequel, E denotes a complex, locally convex lattice of (M)-type.

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 λ_U > 0 such that m(U(x))6λ_Um(x) for allx∈E. The real part of the center is

ReZ(E) =Z(E)+−Z(E)+.

Definition 3.1. For every closed ideal I of E and every U ∈ Z(E), we define UI :E/I →E/I by

(3.1) U_I(π_I(x)) =π_I(U(x)), x∈E.

It is easily seen that the operator UI is well defined and that UI ∈ Z(E/I).

Definition 3.2. Let I and J be two closed ideals of E such that I ⊂J. Define the mappings

πIJ :E/I →E/J, πIJ[πI(x)] =πJ(x), x∈E,

and

M_IJ :Z(E/I)→Z(E/J), M_IJ(T)(π_J(x) =π_IJ(T(π_I(x))), T ∈Z(E/I).

Lemma3.1. The mappings π_IJ and M_IJ have the properties below:

1)π_IJ is a lattice homomorphism;

2)MIJ(T)∈Z(E/J) for every T ∈Z(E/I);

3)M_IJ(T)∈Z(E/J)₊ for every T ∈Z(E/I)₊. Proof. 1) For every x∈E we have

mb_J(π_IJ(π_I(x))) =mb_J(π_J(x)) =π_J(m(x)) =π_IJ(π_I(m(x))) =π_IJ(mb_I(π_I(x))), hence

(3.2) mbJ(πIJ(πI(x))) =πIJ(mbI(πI(x))).

2) For everyT ∈Z(E/I) we have

mb_J(M_IJ(T))(π_J(x)) =mb_J(π_IJ(T(π_I(x)))) =π_IJ(mb_I(T(π_I(x))) 6π_IJ(λ_Tmb_I(π_I(x))) =λ_Tπ_IJ(mb_I(π_I(x)))

=λ_Tπ_IJ(π_I(m(x))) =λ_Tπ_J(m(x)) =λ_Tmb_J(π_J(x)), hence M_IJ(T)∈Z(E/J).

3) If π_J(x)>0 then π_J(x) = mc_J[π_J(x)] =π_J[m(x)]. Further, for every T ∈Z(E/I)+ we have

mb_J[M_IJ(T)(π_J(x))] =mb_J[M_IJ(T)(π_J(m(x)))] =mb_J[π_IJ(T(π_I(m(x))))]

=πIJ[mbI(T(πI(m(x))))] =πIJ[T(πI(m(x)))]

=MIJ(T)(πJ(m(x))) =MIJ(T)(πJ(x)), hence M_IJ(T)>0.

Definition 3.3. Let A be a subset of Z(E) containing 0, and let F be a complex vector subspace ofE. A closed idealIofEis said to be antisymmetric with respect to the pair (A, F) (or (A, F)-antisymmetric) if for every U ∈A with the properties

i)UI ∈ReZ(E/I);

ii) U_I[π_I(F)]⊂π_I(F),

there exists α∈Csuch thatU_I =α1_E/I, where1_E/I is the identity operator on E/I. Of course, E itself is antisymmetric with respect to any pair (A, F).

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

Now, we consider the particular caseE=C(X,C), whereXis a compact Hausdorff space. Then the center of C(X,C) is C(X,C) itself, namely, if U :C(X,C) → C(X,C) belongs to the center of C(X,C), then there exists

a function aU ∈ C(X,C) such that U(f)(x) = aU(x)f(x), ∀f ∈ C(X,C),

∀x∈X.

The mapping U → aU : Z[C(X,C)] → C(X,C) is an isomorphism of Banach algebras, hence Z[C(X,C)]∼=C(X,C).

On the other hand, it is known that for every closed subset S of X, the set IS={f ∈C(X,C); f|S = 0} is a closed ideal ofC(X,C), and every closed ideal ofC(X,C) has this form.

Definition 3.4. LetAbe a subset ofC(X,C) containing 0, and letF be a complex vector subspace of C(X,C). A closed subset S of X is said to be (A, F)-antisymmetric, if everya∈A, such that

i)a(x)∈R,∀x∈S;

ii) af|S ∈F|S,∀f ∈F, is constant on S.

Remark 3.1. A closed subsetSofXis (A, F)-antisymmetric if and only if the corresponding idealIS={f ∈C(X,C); f|S= 0}is (A, F)-antisymmetric in the sense of Definition 3.3. Indeed, it is sufficient to observe that π_IS(g) = g|S , for everyg∈C(X,C).

Remark 3.2. IfF =Ais a subalgebra ofC(X,C), then a closed subsetS ofX is antisymmetric with respect toAif everya∈Athat is real valued onS is constant on S. Thus, we retrieve the original concept of antisymmetric set with respect to a subalgebra of C(X,C) introduced by E. Bishop. (See [1].)

Remark 3.3. In the context of weighted spaces, the concept of antisym- metric set was introduced by J.B. Prolla in 1971.

Remark 3.4. If E is a real locally convex lattice of (M)-type, then Defi- nition 3.3 coincides with Definition 3.1 of [3].

Theorem3.1. Let E be a complex locally convex lattice of (M)-type, A a subset of the center Z(E) containing 0, and F a complex vector subspace of E. If (Iα) is a family of (A, F)-antisymmetric ideals of E such that J =P

α

I_α6=E, then I =T

α

I_α is an (A, F)-antisymmetric ideal.

Proof. Let U ∈A such that i) U_I ∈ReZ(E/I);

ii) UI[πI(F)]⊂πI(F).

According to Lemma 3.1, M_IIα(U_I)∈ReZ(E/I_α). Further, we have M_IIα(U_I)(π_Iα(F)) =π_IIα[U_I(π_I(F))]⊂π_IIα[π_I(F)] =π_Iα(F).

Since I_α is (A, F)-antisymmetric, we deduce that there isa_α ∈Csuch that (3.3) M_IIα(U_I) =a_α ·1_E/Iα.

On the other hand, we have

M_IIα(U_I)(π_Iα(x)) =π_IIα[U_I(π_I(x))] =π_IIα[π_I(U(x))]

=π_Iα[U(x)] =U_Iα[π_Iα(x)], ∀x∈E, hence

(3.4) M_IIα(U_I) =U_Iα.

It follows from (3.3) and (3.4) that

(3.5) U_Iα =a_α ·1_E/Iα.

Now, it follows from the commutative diagram ReZ(E/I) ^M−→^IIα ReZ(E/I_α)

MIJ& ._MIαJ ReZ(E/J)

that

(3.6) M_IJ(U_I) =M_IαJ(M_IIα(U_I)) =M_IαJ(U_Iα) =a_αM_IαJ(1_E/Iα).

Next, we have

(3.7) M_IαJ(1_E/Iα)(π_J(x)) =π_IαJ(1_E/Iα)(π_Iα(x)) =

=π_IαJ(π_Iα(x)) =π_J(x) =1_E/J(π_J(x)).

From (3.6) and (3.7) we deduce that

(3.8) M_IJ(U_I) =a_α·1_E/J, ∀α.

Since 1_E/J is not constant on E/J (from the hypothesis E 6= J, hence E/J 6={0}), we have a_α =a (constant). Therefore, from (3.4) and (3.5) we deduce that

(3.9) U_Iα =a·1_E/Iα =M_IIα(U_I).

Now, from (3.9) and from the fact that MIIα(1_E/I) = 1_E/Iα, we de- duce that

(3.10) M_IIα(U_I−a·1_E/I) = 0, ∀α.

SinceM_IIα(U_I−a·1_E/I)(π_Iα(x)) =π_Iα(U(x)−ax), ∀x∈I, ∀α we have U(x)−ax∈Iα, ∀α, hence

(3.11) U(x)−ax∈I, ∀x∈E.

From (3.11) we deduce that U_I =a·1_E/I, thus I is an (A, F)-antisymmetric ideal.

Corollary 3.1. Every (A, F)-antisymmetric ideal contains a unique minimal (A, F)-antisymmetric ideal.

Proof. Let I ∈ A_A,F(E) be such that I 6= E and let Ie= T

{J|J ∈ A_A,F(E), J ⊂I}.According to Theorem 3.1, Ie∈ A_A,F(E). Obviously,Ie⊂I and Ieis minimal.

4. A GENERALIZATION OF DE BRANGE’S LEMMA We start with two remarks.

Remark 4.1. Let E be a complex vector lattice and let U :E → E be a positive linear operator. Then m(U(x)) 6 U(m(x)), ∀x ∈ E. Indeed, if x∈ER thenU(x)∈ER and

m(U(x)) =U(x)∨ −U(x)6U(m(x)).

For an arbitrary x∈E we have m(U(x)) = sup

a∈C

|a|=1

ReaU(x) = sup

θ∈[0,2π]

(cosθReU(x)−sinθImU(x))

= sup

θ∈[0,2π]

(cosθU(Rex)−sinθU(Imx)) = sup

a∈C

|a|=1

U(Re(ax)),

hence m(U(x))6U(m(x)).

Remark 4.2. If g : E → C is an order bounded linear functional and U :E →E is a positive linear operator, then|g(U(x))|6|g|(U(m(x))).This is a consequence of Remark 4.1 and of the formula

|g|(U(m(x))) = sup{|g(z)|; m(z)6U(m(x))}.

Lemma 4.1. Let E be a complex locally convex lattice of (M)-type, A a subset of Z(E) containing 0, F a vector subspace of E and V a con- vex and solid neighborhood of the origin of E, which also is a sublattice. If f ∈ Ext

V⁰∩F⁰ and I = {x∈E; |f|(m(x)) = 0}, then I is an (A, F)- antisymmetric ideal.

Proof. Let U ∈Asuch that i)UI ∈ReZ(E/I);

ii) U_I[π_I(F)]⊂π_I(F).

According to Definition 3.3, we must prove that there exists a∈Csuch that U_I=a·1_E/I.AsU_I ∈ReZ(E/I), we can suppose that

(4.1) 06U_I 61_E/I.

Indeed, let S, T ∈ Z(E/I)+ such that UI =S−T. SinceS ∈ Z(E/I)+, one can find λ_S>0 with the propertym[S(πb _I(x))]6λ_Sm(πb _I(x)), ∀x∈E,hence (4.2) S(πI(x))6λSπI(x), ∀π_I(x)>0.

If we replace S by _λ¹

SS, we get 0 6 S 6 1_E/I. On the other hand, it is clear that S[π_I(F)] ⊂ π_I(F). Now, if we prove that S = α·1_E/I and T = β ·1_E/I, then it will follow that UI = a·1_E/I, a ∈ C; so that our supposition (4.1) is completely justified. Further, let U_I ∈ ReZ(E/I)+ with the properties 0 6U_I 61_E/I andU_I[π_I(F)]⊂π_I(F). Asf ∈I⁰, there exists g∈(E/I)⁰ such thatf =π_I⁰g. Obviously,g∈Ext

π_I(V)⁰∩π_I(F)⁰ . Denote g1 =U_I⁰g, g2= (1_E/I−UI)⁰g and

ai= inf

λ >0|gi∈λ[πI(V)]⁰ = sup{|g_i(πI(x))|; x∈V}, i= 1,2.

Since g=g1+g2 ∈(a1+a2)πI(V)⁰,we get

(4.3) a1+a2 >1.

On the other hand, for everyx₁, x₂∈V we have

|g₁(π_I(x₁))|+|g₂(π_I(x₂))|=|g(U_I(π_I(x₁)))|+

g(1_E/I−U_I)(π_I(x₂)) . If we denote y =mcI(πI(x1))∨mcI(πI(x2)), and take into account that πI(V) is a solid set and a sublattice, we get y ∈π_I(V).

Now, according to Remark 4.2 we have

|g₁(π_I(x₁))|+|g₂(π_I(x₂))|6|g|(U_I(y)) +|g|(1_E/I−U_I)(y)

=|g|(U_I(y) +y−U_I(y)) =|g|(y)61.

Therefore,|g₁(πI(x1))|+|g₂(πI(x2))|61, for anyx1, x2∈V, hencea1+a261.

From (4.3) we deduce that

(4.4) a₁+a₂ = 1.

Let us note that

(4.5)

U_I⁰g

=U_I⁰(|g|).

Indeed, as 0 6 U_I⁰ 6 1_(E/I)⁰, we have U_I⁰(g+) = (U_I⁰g)+ and U_I⁰(g−) = (U_I⁰g)− thus,

U_I⁰g

=U_I⁰(g₊) +U_I⁰(g−) =U_I⁰(|g|).

Further, from (4.5) and Remark (4.1) we have (4.6) |f|(m(U(x)))6|f|(U(m(x))) =

π_I⁰g

(U(m(x)))

=π_I⁰ |g|(U(m(x))) =|g|(UI(πI(m(x)))) = U_I⁰g

(πI(m(x))).

If we now suppose that g₁= 0, then it follows from (4.6) that|f|(m(U(x))) = 0, ∀x ∈ E, hence UI[πI(x)] = πI[U(x)] = 0, ∀x ∈ E. Thus, g1 = 0 implies

UI = 0 = 0·1_E/I and the proof is complete. Using a similar argument, we deduce thatg₂ = 0 implies 1_E/I−U_I = 0, hence U_I =1_E/I.

Therefore, from now on we can suppose that gi 6= 0 for i = 1,2; thus ai > 0 for i = 1,2. On the other hand, it follows from the assumption U_I[π_I(F)] ⊂ π_I(F), that g_i ∈π_I(F)⁰, i= 1,2, hence ^g_aⁱ

i ∈ π_I(F)⁰∩π_I(V)⁰, for i = 1,2. Since g is an extremal element of the set π_I(F)⁰∩π_I(V)⁰, and g = a_1a^g1

1 +a₂^g_a²

2, either g = _a^g1

1 or g = _a^g2

2. On the other hand, the equality g1 = a1g implies (a1 ·1_E/I−UI)⁰g = 0, hence UI = a1 ·1_E/I. By a similar argument, g₂ =a₂g involvesU_I =a₂·1_E/I and this concludes the proof.

5. AN APPROXIMATION THEOREM The main result of the paper is as follows.

Theorem5.1. Let E be a complex locally convex lattice of (M)-type, A a subset of Z(E) containing 0, and F a vector subspace of E. If we denote by Ae_F,A the family of all minimal (A, F)-antisymmetric ideals of E, then for every x∈E we have

x∈F ⇐⇒ πI(x)∈πI(F), ∀I ∈Ae_F,A.

Proof. The necessity is clear. Conversely, if we suppose that πI(x) ∈ π_I(F), ∀I ∈ Ae_F,A and x /∈ F, then we can find f ∈ E⁰ such that f(x) 6= 0 and f(y) = 0, ∀y ∈F. Let V be a solid, convex neighborhood of the origin of E which is a sublatice of E . Obviously we can suppose thatf ∈V⁰.Since f ∈ F⁰, we deduce that f ∈ F⁰ ∩V⁰ and by the Krein-Milman theorem we may assume that f ∈Ext

F⁰∩V⁰ .If we putJ ={z∈E; |f|(m(z)) = 0}

then, according to Lemma 4.1, that J is an (A, F)-antisymmetric ideal. On the other hand, by Corollary 3.1, there exists a minimal (A, F)-antisymmetric ideal J₀ ⊂J. Since f ∈J₀⁰∩F⁰ and by the hypothesis π_J0(x) ∈π_J0(F), we have f(x) = 0, and this contradicts the choice off.

Remark 5.1. LetE, AandF be as in Definition 3.3, with the additional hypothesis AF ⊂F. Then a closed ideal I ofE is antisymmetric with respect to (A, F) (or A-antisymmetric) if for every U ∈A for whichUI ∈ReZ(E/I) there exists α∈Csuch thatU_I=α·1_E/I.

Let us denote byAe_A the family of all minimalA-antisymmetric ideal of E. From Theorem 5.1 we deduce

Theorem5.2. Let E, Aand F as in Theorem5.1, with the additional hypothesis AF ⊂F. Then for any x∈E we have

x∈F iff πI(x)∈πI(F), ∀I ∈Ae_A.

Remark 5.2. Theorem 5.2 is a generalization of Prolla’s approximation theorem for weighted spaces (see Theorem 6.2 below).

Remark 5.3. IfEis a real locally convex lattice of (M)-type, Theorem 5.1 coincides with Theorem 3.1 of [3].

Remark 5.4. We recall that ifJ is a maximal ideal of a real vector lattice E, then E/J has dimension 1. (See [7], pp. 66.)

On the other hand, if I is an ideal in a complex vector lattice E and J =I∩E_R, thenJ is an ideal in the real latticeERandI =J+ iJ. Moreover, ifI is maximal in E,thenJ is maximal in E_R, hence E/I has dimension 2.

The next result is an abstract version of the Stone-Weierstrass theorem for locally convex lattices.

Theorem5.3. Let E, Aand F as in Theorem5.1, with the additional properties:

i)Ais a (complex) selfadjoint vector subspace of Z(E) (i.e. U = ReU+ i ImU ∈A implies U^∗ = ReU −i ImU ∈A).

ii) AF ⊂F;

iii) F is not included in any maximal ideal of E;

iv) every closed ideal I of E such that {U_I|U ∈A} ⊂ C·1_E/I is a maximal ideal.

Then F =E.

Proof. Let x ∈ E and I ∈ Ae_F,A. Hypothesis i) implies that ReU ∈ A and ImU ∈A for every U ∈A. It follows from ii) that ReU_I =a·1_E/I and ImUI =b·1_E/I, for somea∈R and b∈R, henceUI= (a+ ib)·1_E/I.

According to hypothesis iv),I is a maximal ideal, thus,E/I has dimen- sion 2.

As{0} ⊂π_I(F)⊂π_I(E) =E/I,we have eitherπ_I(F) ={0}orπ_I(F) = π_I(E). It follows from (iii) thatπ_I(F)6={0}. Therefore,π_I(F) =π_I(E), thus π_I(x)∈π_I(F) for everyI ∈Ae_F,A. According to Theorem 5.1, we have x∈F.

6. SOME APPROXIMATION THEOREMS FOR WEIGHTED SPACES

The aim of this section is to give some applications of Theorem 5.1.

LetX be a locally compact Hausdorff space andV a Nachbin family on X, i.e., a set of nonnegative upper semicontinuous functions on X such that for every v₁, v₂∈V and every λ >0 there isv∈V such thatv₁, v₂6λv. We shall denote the corresponding weighted space by

CV0(X,C) ={f ∈C(X,C); f v vanishes at infinity, ∀v∈V}.

The weighted topology on CV0(X,C) is denoted by ωV and is determined by the seminorms {p_v}_v∈V, where

pv(f) = sup{|f(x)|v(x); x∈X} for any f ∈CV0(X,C).

Obviously, CV0(X,C) is a complex locally convex lattice of (M)-type with respect to the weighted topology and to the modulus mapping m(f) = |f|,

∀f ∈CV0(X,C).

Remark 6.1. For every closed idealI of CV0(X,C) there exists a closed subset S_I of X such that

I ={f ∈CV₀(X,C); f|S_I = 0}.

Indeed, if we putJ =I∩CV₀(X,R), thenJ is a closed ideal ofCV0(X,R) and according to a result of Portenier (see [2], Lemma 3.8), there is a closed subset SI of X such that I = {f ∈CV0(X,R); f|S_I = 0}. On the other hand, we have I =J + iJ, hence I ={f ∈CV₀(X,C); f|S_I = 0}.

Remark 6.2. The center of CV0(X,C) is Cb(X,C) – the Banach space of all bounded continuous complex valued function on X. Indeed, it follows from Proposition 2 of [6] that if U belongs to the center of CV₀(X,C). Then there exists a bounded continuous function aU :X→Csuch that

U(f)(x) =aU(x)f(x), ∀f ∈CV0(X,C), ∀x∈X.

The mapping U → a_U : Z(CV₀(X,C)) → C_b(X,C) is an isomorphism of Banach algebras.

Definition 6.1. Let A be a subset of C_b(X,C) containing 0, and let F be a vector subspace of CV0(X,C). A closed subset S of X is said to be (A, F)-antisymmetric if everya∈A, real valued onS, such thataf|S ∈F|S,

∀f ∈F, is constant onS.

Clearly, a closed subsetSofX is (A, F)-antisymmetric if and only if the corresponding ideal I_S ={f ∈CV₀(X,C); f|S = 0} is (A, F)-antisymmetric in the sense of Definition 3.3.

We shall denote by S_F,A the family of all (A, F)-antisymmetric subsets of X. Theorem 3.1 has the following dual version.

Theorem6.1. Let A and F be as in Definition 6.1. If S_α ∈ S_F,A, ∀α and T

α

S_α 6=∅, then S

α

S_α ∈ S_F,A.

Corollary6.1. For every x ∈X there exists a maximal closed subset S_x of X, antisymmetric with respect to the pair (A, F), such that x ∈ S_x. Moreover, if x6=y then Sx and Sy are either equal or disjoint.

See Theorem 3 of [5].

From Theorem 5.1 we deduce

Theorem6.2. Let A be a subset of Cb(X,C) containing 0and let F be a vector subspace of CV₀(X,C). Then, for every f ∈CV₀(X,C), we have

f ∈F if and only if f|S_x ∈F|S_x, ∀x∈X,

where (Sx)x∈X is the family of all maximal (A, F)-antisymmetric subsets of X.

Remark 6.3. Theorem 6.2 coincides with Corollary 1 of [5].

Remark6.4. IfAF ⊂F, then a closed subsetSofXis (A, F)-antisymmetric if any a∈A, real valued onS, is constant onS.

We shall denote bySe_A the family of all maximal antisymmetric subsets of X.

From Theorem 6.1 we deduce

Theorem6.3 (Prolla). Let E, A and F be as in the Theorem 6.1, with the additional property AF ⊂F. Then, for every f ∈CV0(X,C), we have

f ∈F ifff|S ∈F|S , ∀S∈Se_A.

Nachbin’s density theorem below follows from Theorem 5.3.

Theorem6.4. Let A be a self adjoint subalgebra of C_b(X,C) and let F be a vector subspace of CV0(X,C). Assume that

i)AF ⊂F;

ii) Aseparates the points of X;

iii) for every x∈X there is f ∈F such that f(x)6= 0.

Then F =CV₀(X).

Proof. It follows from (iii) thatF is not included in any maximal ideal.

Since AF ⊂ F and A separates the points of X, every (A, F)-antisymmetric subsetSofXis a singleton, thus the corresponding idealI_S is a maximal ideal.

So, the hypotheses of Theorem 5.3 are fulfilled. Consequently, F = CV₀(X).

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(Romanian)

Received 17 July 2007 National Technical University of Athens Department of Mathematics TK 157-80, Zografou Campus

Athens, Greece [email protected]

and Technical University of Civil Engineering Bucharest

Department of Mathematics Bl. Lacul Tei 124, Sector 2 020396, Bucharest 38, Romania

[email protected]