Duality theory of spaces of vector-valued continuous functions

Duality theory of spaces

of vector-valued continuous functions

Marian Nowak, Aleksandra Rzepka

Abstract. LetX be a completely regular Hausdorff space,E a real normed space, and let Cb(X, E) be the space of all bounded continuous E-valued functions on X. We develop the general duality theory of the space Cb(X, E) endowed with locally solid topologies; in particular with the strict topologies βz(X, E) forz=σ, τ, t. As an ap- plication, we consider criteria for relative weak-star compactness in the spaces of vector measures Mz(X, E^′) for z = σ, τ, t. It is shown that if a subset H of Mz(X, E^′) is relatively σ(Mz(X, E^′), Cb(X, E))-compact, then the set conv(S(H)) is still relatively σ(Mz(X, E^′), Cb(X, E))-compact (S(H) = the solid hull ofHinMz(X, E^′)). A Mackey- Arens type theorem for locally convex-solid topologies onCb(X, E) is obtained.

Keywords: vector-valued continuous functions, strict topologies, locally solid topologies, weak-star compactness, vector measures

Classification: 46E10, 46E15, 46E40, 46G10

1. Introduction and preliminaries

LetXbe a completely regular Hausdorff space and let (E,k·k_E) be a real normed space. LetB_E and S_E stand for the closed unit ball and the unit sphere inE, and letE^′ stand for the topological dual of (E,k · kE). LetC_b(X, E) be the space of all bounded continuous functions f : X → E. We will write C_b(X) instead of Cb(X,R), where R is the field of real numbers. For a function f ∈Cb(X, E) we will writekfk(x) = kf(x)k_E for x∈ X. Thenkfk ∈ C_b(X) and the space C_b(X, E) can be equipped with the norm kfk_∞ = sup_x∈Xkfk(x) = kkfkk_∞, wherekuk_∞= sup_x∈X|u(x)| foru∈C_b(X).

It turns out that the notion of solidness in the Riesz space (= vector lattice) C_b(X) can be lifted in a natural way toC_b(X, E) (see [NR]). Recall that a subset HofC_b(X, E) is said to besolidwheneverkf₁k ≤ kf₂k(i.e.,kf₁(x)k_E ≤ kf₂(x)k_E for allx∈X) andf₁∈C_b(X, E),f₂ ∈H imply f₁∈H. A linear topologyτ on C_b(X, E) is said to belocally solid if it has a local base at 0 consisting of solid sets. A linear topologyτ onC_b(X, E) that is at the same time locally convex and locally solid will be called alocally convex-solid topology.

In [NR] we examine the general properties of locally solid topologies on the spaceC_b(X, E). In particular, we consider the mutual relationship between locally solid topologies on C_b(X, E) and C_b(X). It is well known that the so-called

strict topologiesβz(X, E) onC_b(X, E) (z = t, τ, σ, g, p) are locally convex-solid topologies (see [Kh, Theorem 8.1], [KhO₂, Theorem 6], [KhV₁, Theorem 5]).

For a linear topological space (L, ξ), by (L, ξ)^′ (orL^′_ξ) we will denote its topo- logical dual. We will write C_b(X, E)^′ and C_b(X)^′ instead of (C_b(X, E),k · k_∞)^′ and (C_b(X),k · k_∞)^′ respectively. By σ(L, M) and τ(L, M) we will denote the weak topology and the Mackey topology with respect to a dual pairhL, Mi. For terminology concerning locally solid Riesz spaces we refer to [AB₁], [AB₂].

In the present paper, we develop the duality theory of the space C_b(X, E) endowed with locally solid topologies (in particular, the strict topologiesβz(X, E), wherez=σ, τ, t).

In Section 2 we examine the topological dual of C_b(X, E) endowed with a locally solid topologyτ. We obtain that (C_b(X, E), τ)^′ is an ideal of C_b(X, E)^′. We consider a mutual relationship between topological duals of the spacesC_b(X) andC_b(X, E), which allows us to examine in a unified manner continuous linear functionals onC_b(X, E) by means of continuous linear functionals onC_b(X).

In Section 3 we consider criteria for relative weak-star compactness in spaces of vector measuresMz(X, E^′) forz=σ, τ, t. In particular, we show that if a subset H ofMz(X, E^′) is relativelyσ(Mz(X, E^′), Cb(X, E))-compact, then conv (S(H)) is still relativelyσ(Mz(X, E^′), Cb(X, E))-compact (hereS(H) stand for the solid hull ofH inMz(X, E^′); see Definition 3.1 below).

Section 4 deals with the absolute weak and the absolute Mackey topologies on C_b(X, E). A Mackey-Arens type theorem for locally convex-solid topologies on C_b(X, E) is obtained.

Now we recall some properties of locally solid topologies onC_b(X, E) as set out in [NR]. A seminormρonC_b(X, E) is said to besolid wheneverρ(f₁)≤ρ(f₂) if f₁, f₂∈C_b(X, E) andkf₁k ≤ kf₂k.

Note that a solid seminorm on the vector latticeC_b(X) is usually called a Riesz seminorm (see [AB₁]).

Theorem 1.1 (see [NR, Theorem 2.2]). For a locally convex topology τ on C_b(X, E)the following statements are equivalent:

(i) τ is generated by some family of solid seminorms;

(ii) τ is a locally convex-solid topology.

From Theorem 1.1 it follows that any locally convex-solid topology τ on C_b(X, E) admits a local base at 0 formed by sets which are simultaneously abso- lutely convex and solid.

Recall that the algebraic tensor productC_b(X)⊗Eis the subspace ofC_b(X, E) spanned by the functions of the formu⊗e, (u⊗e)(x) =u(x)e, whereu∈C_b(X) ande∈E.

Now we briefly explain the general relationship between locally convex-solid topologies on C_b(X) and C_b(X, E) (see [NR]). Given a Riesz seminorm p on

C_b(X) let us set

p^∨(f) :=p kfk

for all f ∈C_b(X, E).

It is seen thatp^∨ is a solid seminorm onC_b(X, E). From now on lete₀ ∈S_E be fixed. Given a solid seminormρonC_b(X, E) one can define a Riesz seminormρ^∧ onC_b(X) by:

ρ^∧(u) :=ρ(u⊗e₀) for all u∈C_b(X).

One can easily show:

Lemma 1.2 (see [NR, Lemma 3.1]). (i) If ρ is a solid seminorm onC_b(X, E), then(ρ^∧)^∨(f) =ρ(f)for allf ∈C_b(X, E).

(ii)If pis a Riesz seminorm onC_b(X), then(p^∨)^∧(u) =p(u)for allu∈C_b(X).

Letτ be a locally convex-solid topology onC_b(X, E) and let{ρ_α:α∈ A}be a family of solid seminorms onC_b(X, E) that generatesτ. Byτ^∧ we will denote the locally convex-solid topology onCb(X) generated by the family{ρ^∧_α:α∈ A}.

Next, letξbe a locally convex-solid topology onCb(X) and let{pα:α∈ A}be a family of solid seminorms onCb(X) that generatesξ. Byξ^∨ we will denote the locally convex-solid topology onC_b(X, E) generated by the family{p^∨_α :α∈ A}.

As an immediate consequence of Lemma 1.2 we have:

Theorem 1.3(see [NR, Theorem 3.2]). For a locally convex-solid topologyτon Cb(X, E) (resp.ξonCb(X))we have:

τ^∧∨

=τ resp. ξ^∨∧

=ξ .

The strict topologies βz(X, E) onC_b(X, E), where z = t, τ, σ, g, phave been examined in [F], [KhC], [Kh], [KhO₁], [KhO₂], [KhO₃], [KhV₁], [KhV₂]. In this paper we will consider the strict topologiesβz(X, E), wherez=t, τ, σ. We will writeβz(X) instead ofβz(X,R).

Now we recall the concept of a strict topology on Cb(X, E). Let βX stand for the Stone- ˇCech compactification of X. For v ∈C_b(X), v denotes its unique continuous extension to βX. For a compact subsetQ of βX\X letC_Q(X) = {v ∈ C_b(X) : v |Q ≡ 0}. Let β_Q(X, E) be the locally convex topology on C_b(X, E) defined by the family of solid seminorms {̺v : v ∈ C_Q(X)}, where

̺v(f) = sup_x∈X|v(x)| kfk(x) forf ∈C_b(X, E).

Now letC be some family of compact subsets of βX\X. Thestrict topology β_C(X, E) onC_b(X, E) determined by C is the greatest lower bound (in the class of locally convex topologies) of the topologies β_Q(X, E), as Q runs overC (see [NR] for more details). In particular, it is known that β_C(X, E) is locally solid (see [NR, Theorem 4.1]).

The strict topologies βτ(X, E) and βσ(X, E) on C_b(X, E) are obtained by choosing the familyCτ of all compact subsets ofβX\X and the familyCσ of all zero subsets ofβX\X as C, resp. In view of [NR, Corollary 4.4] forz=τ, σwe have

βz(X)^∨=βz(X, E) and βz(X, E)^∧=βz(X).

The strict topologyβt(X, E) onC_b(X, E) is generated by the family{̺v :v∈ C₀(X)}, where C₀(X) denotes the space of scalar-valued continuous functions onX, vanishing at infinity. It is easy to show that

βt(X)^∨=βt(X, E) and βt(X, E)^∧=βt(X).

2. Topological dual ofCb(X, E) with locally solid topologies For a linear functional Φ onCb(X, E) let us put

|Φ|(f) = sup

|Φ(h)|:h∈Cb(X, E), khk ≤ kfk . The next theorem gives a characterization of the spaceC_b(X, E)^′. Theorem 2.1. We have

C_b(X, E)^′=

Φ∈C_b(X, E)^#:|Φ|(f)<∞ for all f ∈C_b(X, E) , whereC_b(X, E)^#denotes the algebraic dual ofC_b(X, E).

Proof:Indeed, by the way of contradiction, assume that for some Φ₀ ∈C_b(X, E)^′ we have |Φ₀|(f₀) = ∞ for some f₀ ∈ C_b(X, E). Hence there exists a sequence (hn) in C_b(X, E) such that khnk ≤ kf₀k and |Φ₀(hn)| ≥ n for all n ∈ N. Since kn⁻¹hnk_∞ → 0, we get n⁻¹Φ₀(hn) → 0, which is in contradiction with

|Φ₀(hn)| ≥n.

Next, assume by the way of contradiction that there exists a linear func- tional Φ₀ on C_b(X, E) such that |Φ₀|(f) < ∞ for all f ∈ C_b(X, E) and Φ₀ ∈/ C_b(X, E)^′. Then there exists a sequence (fn) inC_b(X, E) such thatkfnk∞= 1 and |Φ₀(fn)| > n³ for all n ∈ N. Since P_∞

n=1 1

n²kkfnkk∞ <∞ and the space (C_b(X),k·k_∞) is complete, there existsu₀∈C_b(X)⁺such thatP∞

n=1 1

n²kfnk=u₀. Letf0 =u0⊗e0 for some fixede0 ∈SE. Then _n¹2kf_nk ≤ kf₀k=u0. Hence for alln∈N,n < |Φ₀(fn/n²)| ≤ |Φ₀|(fn/n²)≤ |Φ₀|(f₀)<∞, which is impossible.

Thus the proof is complete.

Now we consider the concept of solidness inC_b(X, E)^′.

Definition 2.1. For Φ₁,Φ₂ ∈ C_b(X, E)^′ we will write |Φ₁| ≤ |Φ₂| whenever

|Φ₁|(f) ≤ |Φ₂|(f) for all f ∈ C_b(X, E). A subset A of C_b(X, E)^′ is said to be solid whenever |Φ₁| ≤ |Φ₂| with Φ₁ ∈ C_b(X, E)^′ and Φ₂ ∈ A implies Φ₁ ∈ A.

A linear subspaceI ofC_b(X, E)^′ will be called anideal wheneverI is solid.

Since the intersection of any family of solid subsets ofC_b(X, E)^′ is solid, every subsetAofC_b(X, E)^′ is contained in the smallest (with respect to the inclusion) solid set called thesolid hull ofAand denoted byS(A). Note that

S(A) =

Φ∈C_b(X, E)^′ :|Φ| ≤ |Ψ| for some Ψ∈A . Lemma 2.2. LetΦ∈C_b(X, E)^′. Then forf ∈C_b(X, E),

(∗) |Φ|(f) = sup

|Ψ(f)|: Ψ∈C_b(X, E)^′, |Ψ| ≤ |Φ| . Moreover, if Ais a subset of C_b(X, E)^′ then forf ∈C_b(X, E)we have

(∗∗) sup

|Φ|(f) : Φ∈A = sup

|Ψ(f)|: Ψ∈S(A)

= sup

|Ψ(f)|: Ψ∈conv S(A) .

Proof: Note first that|Φ|is a seminorm onC_b(X, E). To see that|Φ|(f₁+f₂)≤

|Φ|(f₁) +|Φ|(f₂) holds for f₁, f₂ ∈ C_b(X, E) with f₁, f₂ 6= 0, assume that h∈ C_b(X, E) andkhk ≤ kf₁+f₂k. Then forh_i = (kf_ik/(kf₁k+kf₂k))hfori= 1,2 we haveh=h₁+h₂ andkh_ik ≤ kf_ikfori= 1,2. Thus|Φ(h)| ≤ |Φ(h₁)|+|Φ(h₂)| ≤

|Φ|(h₁) +|Φ|(h₂)≤ |Φ|(f₁) +|Φ|(f₂). Hence|Φ|(f₁+f₂)≤ |Φ|(f₁) +|Φ|(f₂), as desired. Moreover, one can easily show that|Φ|(λf) =|λ| |Φ|(f) for allλ∈R.

For a fixed f₀ ∈ C_b(X, E) we define a functional Ψ₀ on the linear subspace L_f0 ={λf₀ :λ∈R}of C_b(X, E) by putting Ψ₀(λf₀) =λ|Φ|(f₀) forλ∈R. It is clear that Ψ₀ is a linear functional onL_f0 and |Ψ₀(λf₀)| =|Φ|(λf₀) for λ∈R. Then by the Hahn-Banach extension theorem there exists a linear functional Ψ onC_b(X, E) such that Ψ(f)≤ |Φ|(f) for allf ∈C_b(X, E) and Ψ(λf₀) = Ψ₀(λf₀) for allλ∈R. Since Ψ is linear and|Φ|(f) =|Φ|(−f) we get|Ψ(f)| ≤ |Φ|(f) for all f ∈C_b(X, E). To see that|Ψ| ≤ |Φ|letf ∈C_b(X, E) and takeh∈C_b(X, E) with khk ≤ kfk. Then|Ψ(h)| ≤ |Φ|(h)≤ |Φ|(f), so |Ψ|(f)≤ |Φ|(f). Thus |Ψ| ≤ |Φ|.

Moreover, Ψ(f₀) = Ψ₀(f₀) =|Φ|(f₀), so

|Φ|(f₀) = sup

|Ψ(f₀)|: Ψ∈Cb(X, E)^′, |Ψ| ≤ |Φ| .

Thus (∗) is shown. As a consequence of (∗) we easily obtain that (∗∗) holds.

We now introduce the concept of a solid dual system. Let I be an ideal of C_b(X, E)^′ separating the points ofC_b(X, E). Then the pairhC_b(X, E), Ii, under its natural duality

hf,Φi= Φ(f) for f ∈C_b(X, E), Φ∈I will be referred to as asolid dual system.

For a subsetAofC_b(X, E) and a subsetB ofI let us set A⁰=

Φ∈I:|hf,Φi| ≤1 for all f ∈A ,

0B=

f ∈C_b(X, E) :|hf,Φi| ≤1 for all Φ∈B . By making use of Lemma 2.2 we can get the following result.

Theorem 2.3. LethC_b(X, E), Iibe a solid dual system.

(i) If a subsetA of C_b(X, E)is solid, thenA⁰ is a solid subset of I.

(ii) If a subsetB of I is solid, then⁰B is a solid subset of Cb(X, E).

Proof: (i) Let|Φ₁| ≤ |Φ₂|with Φ₁ ∈I and Φ₂ ∈A⁰. Assume that f ∈A and leth∈C_b(X, E) withkhk ≤ kfk. Thenh∈A, becauseAis solid, so|Φ₂(h)| ≤1.

Hence|Φ₂|(f)≤1. Thus|Φ₁(f)| ≤ |Φ₁|(f)≤1, so Φ₁∈A⁰. This means thatA⁰ is a solid subset ofI.

(ii) Letkf₁k ≤ kf₂k with f₁ ∈C_b(X, E) and f₂ ∈ ⁰B. To see that f₁ ∈ ⁰B assume that Φ∈ B. SinceB is a solid subset of I, by Lemma 2.2 the identity

|Φ|(f₂) = sup{|Ψ(f₂)| : Ψ ∈B, |Ψ| ≤ |Φ|} holds. Thus for every Ψ∈ B with

|Ψ| ≤ |Φ|we have|Ψ(f₂)| ≤1, so|Φ|(f₂)≤1. Since|Φ(f₁)| ≤ |Φ|(f₁)≤ |Φ|(f₂)≤

1, we getf₁∈⁰B, as desired.

Theorem 2.4. Letτbe a locally solid topology onC_b(X, E). Then(C_b(X, E), τ)^′ is an ideal of Cb(X, E)^′.

Proof: To show that (C_b(X, E), τ)^′ ⊂ C_b(X, E)^′, by the way of contradiction assume that for some Φ₀∈(C_b(X, E), τ)^′ we have Φ₀ ∈/ C_b(X, E)^′, so in view of Theorem 2.1 we get|Φ₀|(f₀) =∞ for somef₀ ∈C_b(X, E). Hence there exists a sequence (hn) in C_b(X, E) such that kh_nk ≤ kf₀k and |Φ₀(hn)| ≥n for n∈N. Since n⁻¹f₀ →0 for τ, andτ is locally solid, we get n⁻¹hn → 0 for τ. Hence Φ₀(n⁻¹hn)→0, which is in contradiction with|Φ₀(hn)| ≥n.

To see that (C_b(X, E), τ)^′ is an ideal of C_b(X, E)^′ assume that |Φ₁| ≤ |Φ₂| with Φ₁ ∈ C_b(X, E)^′ and Φ₂ ∈ (C_b(X, E), τ)^′. Let fα τ

−→ 0 and ε > 0 be given. Then there exists a net (hα) in C_b(X, E) such thatkhαk ≤ kfαkfor each α and |Φ₂|(fα) ≤ |Φ₂(hα)|+ε. Clearly hα τ

−→ 0, because τ is locally solid, so Φ2(hα) → 0. Since |Φ₁(fα)| ≤ |Φ₁|(fα) ≤ |Φ₂|(fα) ≤ |Φ₂(fα)|+ε, we get Φ1(fα)→0, so Φ1∈(Cb(X, E), τ)^′, as desired.

Theorem 2.5. For a Hausdorff locally convex topology τ onC_b(X, E) the fol- lowing statements are equivalent:

(i) τ is locally solid;

(ii) (Cb(X, E), τ)^′ is an ideal of Cb(X, E)^′ and for every τ-equicontinuous subsetAof (Cb(X, E), τ)^′ its solid hullS(A)is alsoτ-equicontinuous.

Proof: (i) =⇒(ii) By Theorem 2.4 (Cb(X, E), τ)^′ is an ideal ofCb(X, E)^′, and thus we have the solid dual system hC_b(X, E),(C_b(X, E), τ)^′i. Assume that a subset A of (C_b(X, E), τ)^′ is equicontinuous. Hence A ⊂ V⁰ for some solid τ- neighbourhood V of zero. Hence S(A)⊂S(V⁰) =V⁰ (see Theorem 2.3). This means thatS(A) is aτ-equicontinuous subset of (C_b(X, E), τ)^′.

(ii) =⇒(i) LetB_τbe a local base at zero forτconsisting of absolutely convex,τ- closed sets. Assume thatV isτ-neighbourhood of zero. Then there existsU ∈ B_τ

such that U ⊂ V. Moreover, the polar set U⁰ is a τ-equicontinuous subset of (C_b(X, E), τ)^′. By our assumption S(U⁰) is alsoτ-equicontinuous. Hence there existsW ∈ Bτ such thatW ⊂⁰S(U⁰). Since the set⁰S(U⁰) is solid inCb(X, E), S(W)⊂⁰S(U⁰)⊂⁰(U⁰) = abs convU^τ =U ⊂V. This shows thatτ is locally

solid, as desired.

For each Φ∈C_b(X, E)^′ let ϕ_Φ(u) = sup

Φ(h)

:h∈C_b(X, E), khk ≤u for u∈C_b(X)⁺. One can easily show thatϕ_Φ:C_b(X)⁺→R⁺is an additive and positively homo- geneous mapping (see [KhO₁, Lemma 1]), soϕ_Φ has a unique positive extension to a linear mapping fromC_b(X) toR(denoted byϕ_Φ again) and given by

ϕ_Φ(u) =ϕ_Φ(u⁺)−ϕ_Φ(u⁻) for all u∈C_b(X)

(see [AB, Lemma 3.1]). Hence ϕ_Φ = |ϕ_Φ| holds on C_b(X)⁺. Since C_b(X)^′ = C_b(X)^∼ (the order dual of C_b(X)) (see [AB₂, Corollary 12.5]), we get ϕ_Φ ∈ C_b(X)^′. Moreover, we have:

ϕ_Φ kfk

=|Φ|(f) for f ∈Cb(X, E) and

ϕ_Φ(u) =|Φ|(u⊗e0) for u∈Cb(X)⁺. The following lemma will be useful.

Lemma 2.6. (i)Assume thatLis an ideal of C_b(X)^′. Then the set Cb(X, E)^′_L:=

Φ∈Cb(X, E)^′:ϕ_Φ ∈L is an ideal of C_b(X, E)^′.

(ii)Assume thatI is an ideal of C_b(X, E)^′. Then the set C_b(X)^′_I:=

ϕ∈C_b(X)^′:|ϕ| ≤ϕ_Φ for some Φ∈I is an ideal of C_b(X)^′ andC_b(X, E)^′_C

b(X)^′I

=I.

Proof: (i) We first show that C_b(X, E)^′_L is a linear subspace of C_b(X, E)^′. Assume that Φ₁,Φ₂ ∈ C_b(X, E)^′_L, i.e., ϕ_Φ1, ϕ_Φ2 ∈ L. It is easy to show that ϕ_Φ1+Φ2(u)≤(ϕ_Φ1 +ϕ_Φ2)(u) for u∈C_b(X)⁺, so ϕ_Φ1+Φ2 ∈L, i.e., Φ₁+ Φ₂ ∈ C_b(X, E)^′_L. Next, let Φ ∈ C_b(X, E)^′_L and λ ∈ R. Then ϕ_Φ ∈ L and since ϕ_λΦ=λϕ_Φ, we getλΦ∈C_b(X, E)^′_L.

To show that C_b(X, E)^′_L is solid in C_b(X, E)^′, assume that |Φ₁| ≤ |Φ₂| with Φ₁ ∈ C_b(X, E)^′ and Φ₂ ∈C_b(X, E)^′_L, i.e., ϕ_Φ2 ∈L. Then for u∈C_b(X)⁺ we haveϕ_Φ1(u) =|Φ₁|(u⊗e₀)≤ |Φ₂|(u⊗e₀) =ϕ_Φ2(u). Henceϕ_Φ1 ∈L, becauseL is an ideal ofC_b(X)^′. Thus Φ₁∈C_b(X, E)^′_L, as desired.

(ii) To prove thatC_b(X)^′_Iis an ideal ofC_b(X)^′ assume that|ϕ₁| ≤ |ϕ₂|, where ϕ₁ ∈C_b(X)^′ and ϕ₂ ∈C_b(X)^′_I. Then |ϕ₂| ≤ϕ_Φ for some Φ∈I, so|ϕ₁| ≤ϕ_Φ, and this means thatϕ₁∈C_b(X)^′_I.

To show thatI⊂C_b(X, E)^′_C

b(X)^′I, assume that Φ∈I. Thenϕ_Φ ∈C_b(X)^′_I, so Φ∈Cb(X, E)^′_C

b(X)^′I.

Now, we assume that Φ ∈ C_b(X, E)^′_C

b(X)^′I

, i.e., Φ ∈ C_b(X, E)^′ and ϕ_Φ ∈ C_b(X)^′_I. It follows that there exists Φ₀ ∈Isuch thatϕ_Φ≤ϕ_Φ0. Hence for every f ∈ C_b(X, E) we have|Φ|(f) = ϕ_Φ(kfk) ≤ ϕ_Φ0(kfk) = |Φ₀|(f). Thus Φ∈ I,

becauseI is an ideal ofC_b(X, E)^′.

LetAbe a subset ofC_b(X, E)^′_τ. ThenS(A)⊂C_b(X, E)^′_τ asC_b(X, E)^′_τ is solid (by Theorem 2.4). Hence

S(A) =

Φ∈Cb(X, E)^′_τ : |Φ| ≤ |Ψ| for some Ψ∈A .

In view of Lemma 2.2 for a subsetAofC_b(X, E)^′ andf ∈C_b(X, E) we have:

(+) sup

|Φ|(f) : Φ∈A = sup

ϕ_Φ(kfk) : Φ∈A

= sup

|Ψ(f)|: Ψ∈S(A) .

Theorem 2.7. Letτ be a locally convex-solid Hausdorff topology onC_b(X, E).

Then for a subsetAof C_b(X, E)^′ the following statements are equivalent:

(i) Aisτ-equicontinuous;

(ii) conv (S(A))isτ-equicontinuous;

(iii) S(A)isτ-equicontinuous;

(iv) the subset{ϕ_Φ: Φ∈A}of C_b(X)^′ isτ^∧-equicontinuous.

Proof: (i) =⇒ (ii) In view of Theorem 2.4 we have a solid dual system hC_b(X, E), Cb(X, E)^′_τi. LetAbeτ-equicontinuous. Then by Theorem 1.1 there is a convex solidτ-neighbourhoodV of zero such thatA⊂V⁰. Hence conv (S(A))⊂ conv (S(V⁰)) = V⁰ (see Theorem 2.3), and this means that conv (S(A)) is still τ-equicontinuous.

(ii) =⇒(iii) It is obvious.

(iii) =⇒(iv) Assume that the subsetS(A) ofC_b(X, E)^′isτ-equicontinuous. Let {ρ_α:α∈ A}be a family of solid seminorms onC_b(X, E) that generatesτ. Given ε >0 there existα₁, . . . , αn∈ Aandη >0 such that sup{|Ψ(f)|: Ψ∈S(A)} ≤ε

whenever ραi(f) ≤ η for i = 1,2, . . . , n. To show that {ϕ_Φ : Φ ∈ A} is τ^∧- equicontinuous, it is enough to show that sup{|ϕ_Φ(u)| : Φ ∈A} ≤ε whenever ρ^∧_αi(u) ≤ η for i = 1,2, . . . , n. Indeed, let u ∈ C_b(X) and ρ^∧_αi(u) ≤ η for i = 1,2, . . . , n. Then ραi(u⊗e₀) ≤ η (i = 1,2, . . . , n), so sup{|Ψ(u⊗e₀)| : Ψ ∈ S(A)} ≤ ε. Hence, in view of (+) we obtain that sup{ϕ_Φ(|u|) : Φ ∈ A} ≤ ε, becauseku⊗e₀k=|u|. But|ϕ_Φ(u)| ≤ϕ_Φ(|u|), and the proof is complete.

(iv) =⇒ (i) Assume that the set {ϕ_Φ : Φ ∈ A} is τ^∧-equicontinuous. Let {ρα:α∈ A}be a family of solid seminorms onC_b(X, E) that generatesτ. Given ε >0 there existα1, . . . , αn∈ Aandη >0 such that sup{|ϕ_Φ(u)|: Φ∈A} ≤ε whenever u ∈ C_b(X) and ρ^∧_αi(u) ≤ η for i = 1,2, . . . , n. Let f ∈ C_b(X, E) with ραi(f) ≤ η for i = 1,2, . . . , n. Since ρ^∧_αi(kfk) = ραi(kfk ⊗e₀) = ραi(f) (i= 1,2, . . . , n), sup{|ϕ_Φ(kfk)|: Φ ∈A} ≤ε. But |Φ(f)| ≤ |Φ|(f) =ϕ_Φ(kfk), so sup{|Φ(f)|: Φ∈A} ≤ε. This means thatA isτ-equicontinuous.

Corollary 2.8. Letτ be a locally convex-solid topology onC_b(X, E). Then for Φ∈C_b(X, E)^′ the following statements are equivalent:

(i) Φisτ-continuous;

(ii) ϕ_Φ isτ^∧-continuous.

Corollary 2.9. Let ξ be a locally convex-solid topology on C_b(X). Then for Φ∈C_b(X, E)^′ the following statements are equivalent:

(i) Φisξ^∨-continuous;

(ii) ϕ_Φ isξ-continuous.

Remark. For the equivalence (i)⇐⇒(iv) of Theorem 2.7 for the strict topologies βz(X, E) (z=σ, τ, t,∞, g) see [KhO₃, Lemma 2].

Corollary 2.10. (i)Letξbe a locally convex-solid topology onC_b(X). Then C_b(X), ξ′ =n

ϕ∈C_b(X)^′ :|ϕ| ≤ϕ_Φ for some Φ∈ C_b(X, E), ξ^∨′o .

(ii)Letτ be a locally convex-solid topology onCb(X, E). Then C_b(X), τ^∧′ =n

ϕ∈C_b(X)^′:|ϕ| ≤ϕ_Φ for some Φ∈ C_b(X, E), τ′o .

Proof: (i) Let ϕ∈ (C_b(X), ξ)^′. Define a linear functional Φ₀ on the subspace C_b(X)(e₀) (={u⊗e₀:u∈C_b(X)}) ofC_b(X, E) by putting Φ₀(u⊗e₀) =ϕ(u) for u∈C_b(X). Let{pα:α∈ A}be a family of Riesz seminorms generatingξ. Since ϕ∈(C_b(X), ξ)^′, there existc >0 andα₁, . . . , αn∈ Asuch that foru∈C_b(X)

Φ0(u⊗e0) =

ϕ(u)

≤c max

1≤i≤npαi(u) =c max

1≤i≤np^∨_αi(u⊗e0).

This means that Φ₀ ∈(C_b(X)(e₀), ξ^∨|C_b(X)(e₀))^′, so by the Hahn-Banach ex- tension theorem there is Φ ∈ (C_b(X, E), ξ^∨)^′ such that Φ(u⊗e₀) = ϕ(u) for all u ∈ C_b(X). We shall now show that |ϕ| ≤ ϕ_Φ, i.e., |ϕ|(u) ≤ ϕ_Φ(u) for all u∈C_b(X)⁺. Indeed, let u∈C_b(X)⁺ be given and let v ∈C_b(X) with |v| ≤u.

Then we have|ϕ(v)|=|Φ(v⊗e₀)| ≤ϕ_Φ(u), so|ϕ| ≤ϕ_Φ, as desired.

Next, assume thatϕ ∈C_b(X)^′ with |ϕ| ≤ ϕ_Φ for some Φ ∈(C_b(X, E), ξ^∨)^′. In view of Corollary 2.9, ϕ_Φ ∈ (C_b(X), ξ)^′ and since (C_b(X), ξ)^′ is an ideal of C_b(X)^′, we conclude thatϕ∈(C_b(X), ξ)^′.

(ii) It follows from (i), because (τ^∧)^∨=τ.

It is well known that ifL is aσ-Dedekind complete vector-lattice and ifH is a relativelyσ(L^∼_n, L)-compact subset of L^∼_n (resp. a relativelyσ(L^∼_c , L)-compact subset ofL^∼_c ), then the set conv (S(H)) is still relativelyσ(L^∼_n, L)-compact (resp.

relatively σ(L^∼_c , L)-compact) (see [AB, Corollary 20.12, Corollary 20.10]) (here L^∼_n andL^∼_c stand for the order continuous dual and theσ-order continuous dual ofLresp.).

Now, we shall show that this property holds in (C_b(X, E)^′_βz, σ(C_b(X, E)^′_βz, C_b(X, E))) forz=σ, τ, t.

Recall that a completely regular Hausdorff spaceXis called aP-space if every G_δset inX is open (see [GJ, p. 63]).

The following result will be of importance.

Theorem 2.11. LetHbe a norm-bounded andσ(C_b(X, E)^′_βz,C_b(X, E))-compact subset of C_b(X, E)^′_βz, wherez=σ(resp.z=τ andX is a paracompact space ; resp.z=τ andX is a P-space). Then H isβz(X, E)-equicontinuous.

Proof: See [KhO₁, Theorem 5] for z = σ; [Kh, Theorem 6.1] for z = τ and

[KhC, Lemma 3] forz=t.

Now we are ready to state our main result.

Theorem 2.12. LetH be a norm bounded subset of C_b(X, E)^′_βz, wherez=σ (resp. z = τ and X is a paracompact space ; resp. z = t and X is a P-space).

Then the following statements are equivalent:

(i) H is relatively countablyσ(C_b(X, E)^′_βz, C_b(X, E))-compact;

(ii) H isβz(X, E)-equicontinuous;

(iii) conv (S(H))is relativelyσ(C_b(X, E)^′_βz, C_b(X, E))-compact;

(iv) S(H)is relatively σ(C_b(X, E)^′_βz, C_b(X, E))-compact;

(v) H is relativelyσ(C_b(X, E)^′_βz, C_b(X, E))-compact.

Proof: (i) =⇒(ii) See Theorem 2.11.

(ii) =⇒(iii) In view of Theorem 2.7 the set conv (S(H)) isβz(X, E)-equiconti- nuous, i.e., there is a neighbourhood of 0 forβz(X, E) such that conv (S(H))⊂V⁰

(= the polar set with respect to the dual pairhC_b(X, E), C_b(X, E)^′_βzi). Then by the Banach-Alaoglu’s theorem the setV⁰isσ(C_b(X, E)^′_βz, C_b(X, E))-compact, so the set conv (S(H)) is relativelyσ(Cb(X, E)^′_βz, Cb(X, E))-compact.

(iii) =⇒(iv) =⇒(v) =⇒(i) It is obvious.

3. Weak-star compactness in some spaces of vector measures

In this section we consider criteria for relative weak-star compactness in some spaces of vector measures Mz(X, E^′) for z = σ, τ, t. In particular, by mak- ing use of Theorem 2.11 we show that if a subset H of Mz(X, E^′) is relatively σ(Mz(X, E^′), C_b(X, E))-compact, then the set conv (S(H)) is still relatively σ(Mz(X, E^′), C_b(X, E))-compact (here S(H) stand for the solid hull of H is Mz(X, E^′)). We start by recalling some notions and results concerning the topo- logical measure theory (see [V], [S], [Wh]).

LetB(X) be the algebra of subsets ofX generated by the zero sets. LetM(X) be the space of all bounded finitely additive regular (with respect to the zero sets) measures onB(X). The spaces of allσ-additive, τ-additive and tight members of M(X) will be denoted by Mσ(X), Mτ(X) and Mt(X) respectively (see [V], [Wh]). It is well known thatMz(X) forz =σ, τ, tare ideals ofM(X) (see [Wh, Theorem 7.2]).

Theorem 3.1 (A.D. Alexandroff ; [Wh, Theorem 5.1]). For a linear functional ϕ:C_b(X)→R the following statements are equivalent.

(i) ϕ∈C_b(X)^′.

(ii) There exists a uniqueµ∈M(X)such that ϕ(u) =ϕµ(u) =

Z

X

udµ for all u∈C_b(X).

Moreover,µ≥0if and only if ϕµ(u)≥0for allu∈C_b(X)⁺.

ByM(X, E^′) we denote the set of all finitely additive measuresm:B(X)→E^′ with the following properties:

(i) For everye∈E, the functionme:B(X)→Rdefined byme(A) =m(A)(e), belongs toM(X).

(ii) |m|(X)<∞, where forA∈B(X)

|m|(A) = supn

n

X

i=1

m(Bi)(ei) :

n

[

i=1

Bi=A, Bi∈B(X), Bi∩Bj =∅ fori6=j, e_i∈B_E, n∈No

.

Forz=σ, τ, tlet Mz(X, E^′) =

m∈M(X, E^′) :me∈Mz(X) for everye∈E .

It is well known that |m| ∈M(X) (resp.|m| ∈Mz(X) forz =σ, τ, t) whenever m∈M(X, E^′) (resp.m∈Mz(X, E^′) forz=σ, τ, t) (see [F, Proposition 3.9]).

Now we are ready to define the notion of solidness inM(X, E^′).

Definition 3.1. For m₁, m₂ ∈ M(X, E^′) we will write |m₁| ≤ |m₂| whenever

|m₁|(B)≤ |m₂|(B) for everyB ∈ B(X). A subsetH ofM(X, E^′) is said to be solid whenever|m₁| ≤ |m₂| with m₁ ∈M(X, E^′) andm₂ ∈ H imply m₁ ∈H. A linear subspaceI ofM(X, E^′) will be called an ideal ofM(X, E^′) wheneverI is a solid subset ofM(X, E^′).

Proposition 3.2. Mz(X, E^′) (z=σ, τ, t)is an ideal of M(X, E^′).

Proof: Let |m₁| ≤ |m₂|, where m₁ ∈ M(X, E^′) and m₂ ∈ Mz(X, E^′). Then

|m₁| ∈ M(X) and |m₂| ∈ Mz(X), and since Mz(X) is an ideal of M(X) we conclude that|m₁| ∈Mz(X). For eache∈Ewe have|(m₁)e|(B)≤ kek_E|m₁|(B) forB∈B(X), so (m1)e∈Mz(X), i.e.,m1∈Mz(X, E^′).

Since the intersection of any family of solid subsets ofM(X, E^′) is solid, every subsetHofM(X, E^′) is contained in the smallest (with respect to inclusion) solid set called thesolid hull ofH and denoted byS(H). Note that

S(H) =

m∈M(X, E^′) :|m| ≤ |m^′| for some m^′∈H .

Now we recall some results concerning a characterization of the topological duals of (C_b(X, E), βz(X, E)) in terms of the spacesMz(X, E^′) (z=σ, τ, t).

Theorem 3.3. Assume thatβz(X, E)is the strict topology onCb(X, E), where z=σandCb(X)⊗E is dense in(Cb(X, E), βσ(X, E)) (resp.z=τ; resp.z=t).

Then for a linear functionalΦonCb(X, E)the following statements are equivalent.

(i) Φisβz(X, E)-continuous.

(ii) There exists a uniquem∈Mz(X, E^′)such that Φ(f) = Φm(f) =

Z

X

fdm for every f ∈C_b(X, E).

(iii) The functionalϕ_Φ isβz(X)-continuous.

Moreover,kΦmk=|m|(X)form∈Mz(X, E^′).

Proof: (i) ⇐⇒ (ii) See [Kh, Theorem 5.3] for z = σ; [Kh, Corollary 3.9] for z=τ; [F1, Theorem 3.13] forz=t.

(ii)⇐⇒(iii) It follows from Corollary 2.8, becauseβz(X, E)^∧=βz(X).

Lemma 3.4. Assume thatm∈Mz(X, E^′), wherez=σandC_b(X)⊗Eis dense in(C_b(X, E), βσ(X, E)) (resp.z=τ; resp.z=t). Then

ϕ_Φm(u) = Z

X

ud|m|=ϕ_|m|(u) for all u∈C_b(X).

Proof: Let u ∈ C_b(X)⁺ and m ∈ Mz(X, E^′). Then for h ∈ C_b(X, E) with khk ≤uby [F₂, Lemma 3.11] we have

|Φm(h)|= Z

X

hdm ≤

Z

X

khkd|m| ≤ Z

X

ud|m|=ϕ_|m|(u).

Hence

ϕ_Φm(u) =|Φm|(u⊗e₀) = sup

|Φm(h)|:h∈C_b(X, E),khk ≤u ≤ϕ_|m|(u).

On the other hand, in view of [Kh, Theorem 2.1] we have ϕ_|m|(u) =

Z

X

ud|m|= sup

|Φm(g)|:g∈C_b(X)⊗E, kgk ≤u , soϕ_|m|(u)≤ϕ_Φm(u). Thusϕ_|m|(u) =ϕ_Φm(u) for allu∈C_b(X).

Lemma 3.5. Assume that m1, m2 ∈ Mz(X, E^′), wherez = σ and Cb(X)⊗E is dense in (Cb(X, E), βσ(X, E))(resp. z =τ; resp. z =t). Then the following statements are equivalent:

(i) |m₁| ≤ |m₂|, i.e.,|m₁|(B)≤ |m₂|(B)for everyB ∈B(X);

(ii) ϕ_|m1|(u)≤ϕ_|m2|(u)for everyu∈Cb(X)⁺; (iii) |Φm1|(f)≤ |Φm2|(f)for everyf ∈C_b(X, E).

Proof: (i)⇐⇒(ii) It easily follows from Theorem 3.1.

(ii) =⇒(iii) In view of Lemma 3.4 we get

|Φm1|(f) =ϕ_Φm1 kfk

=ϕ_|m1| kfk

≤ϕ_|m2| kfk

=ϕ_Φm2 kfk

= Φm2

(f).

(iii) =⇒(ii) By Lemma 3.3 foru∈C_b(X)⁺ ande₀∈S_E we have ϕ_|m1|(u) =ϕ_Φm1(u) =

Φm1

(u⊗e₀)

≤ Φm2

(u⊗e₀) =ϕ_Φm2(u) =ϕ_|m2|(u).

Lemma 3.6. Assume thatH ⊂Mz(X, E^′), wherez=σandC_b(X)⊗Eis dense in(C_b(X, E), βσ(X, E)) (resp.z=τ; resp.z=t), and letΦ_H ={Φm:m∈H}.

Thenconv (S(Φ_H)) = Φ_{conv (S(H))}.

Proof: Assume that Φ∈conv (S(ΦH)). Then Φ =Pn

i=1αiΦmi = Φ^Pn_i=1αimi, where mi ∈ Mz(X, E^′) and αi ≥ 0 for i = 1,2, . . . , n with Pn

i=1αi = 1, and

|Φmi| ≤ |Φ_m′

i| for some m^′_i ∈ H and i = 1,2, . . . , n. In view of Lemma 3.5

|m_i| ≤ |m^′_i|, i.e., mi ∈ S(H) for i = 1,2, . . . , n and Pn

i=1αimi ∈ conv (S(H)).

This means that Φ∈Φconv (S(H)).

Assume that Φ∈Φ_{conv (S(H))}. Then Φ = Φ^Pn

i=1αimi =Pn

i=1α_iΦmi, where mi ∈ Mz(X, E^′) and αi ≥ 0 for i = 1,2, . . . , n with Pn

i=1αi = 1, and |m_i| ≤

|m^′_i| for some m^′_i ∈ H and i = 1,2, . . . , n. By Lemma 3.5 |Φ_mi| ≤ |Φ_m′ i| for

i= 1,2, . . . , n, so Φ∈conv (S(Φ_H)).

Corollary 3.7. Assume that m₀ ∈Mz(X, E^′), wherez =σ andC_b(X)⊗E is dense in(C_b(X, E), βσ(X, E)) (resp.z=τ; resp.z=t)and lete∈SE. Then for everyu∈C_b(X)⁺ we have:

Z

X

ud|m₀|= supn Z

X

udme

:m∈Mz(X, E^′), |m| ≤ |m₀|o .

Proof: Letm₀ ∈Mz(X, E^′) ande∈S_E. Assume that Φ∈C_b(X, E)^′ and|Φ| ≤

|Φm0|. Since Φm0 ∈C_b(X, E)^′_βz (see Theorem 3.3), by making use of Theorem 2.4 we get Φ∈ C_b(X, E)^′_βz. Hence in view of Theorem 3.3 and Lemma 3.5 we see that Φ = Φm for somem∈Mz(X, E^′) with|m| ≤ |m₀|.

Moreover, it is easy to observe that for everym ∈M(X, E^′) andu∈C_b(X)

we have: Z

X

(u⊗e) dm= Z

X

udme.

Thus in view of Lemma 3.4, Lemma 2.2 and Lemma 3.5 we get:

Z

X

ud|m₀|=ϕ_Φm0(u) =|Φm0|(u⊗e)

= sup

|Φ(u⊗e)|: Φ∈C_b(X, E)^′, |Φ| ≤ |Φm0|

= sup

|Φm(u⊗e)|:m∈Mz(X, E^′), |m| ≤ |m₀|

= supn Z

X

(u⊗e) dm

:m∈Mz(X, E^′), |m| ≤ |m₀|o

= supn Z

X

udme

:m∈Mz(X, E^′), |m| ≤ |m₀|o .

To state our main result we recall some definitions (see [Wh, Definition 11.13, Definition 11.23, Theorem 10.3]).

A subsetAofMσ(X) (resp.Mτ(X)) is said to beuniformlyσ-additive (resp.

uniformlyτ-additive) if wheneverun(x)↓0 for everyx∈X,un∈C_b(X)⁺ (resp.

uα ↓ 0 for every x∈ X, uα ∈ C_b(X)⁺), then sup{|R

Xundµ| : µ ∈ A} −→

n 0 (resp. sup{|R

Xuαdµ|:µ∈A} −→

α 0).

A subset AofMt(X) is said to beuniformly tight if givenε >0 there exists a compact subsetK ofX such that sup{|µ|(X\K) :µ∈A} ≤ε.

Now we are in position to prove our desired result.

Theorem 3.8. For a subset H of Mz(X, E^′), where z =σ and C_b(X)⊗E is dense in (C_b(X, E), βσ(X, E)) (resp. z =τ and X is paracompact; resp.z = t andX is aP-space)the following statements are equivalent.

(i) H is relativelyσ(Mz(X, E^′), C_b(X, E))-compact.

(ii) conv (S(H))is relativelyσ(Mz(X, E^′), C_b(X, E))-compact.

(iii) The set {|m| : m ∈ H} in Mz(X)⁺ is uniformlyσ-additive for z = σ, (resp. uniformlyτ-additive forz=τ; resp. uniformly tight forz=t).

Proof: (i) =⇒ (ii) It is seen that H is relatively σ(Mz(X, E^′), C_b(X, E))- compact if and only if Φ_H is relatively σ(C_b(X, E)^′_βz, C_b(X, E))-compact.

Hence by Theorem 2.12 and Lemma 3.6 the set Φconv (S(H)) is still relatively σ(C_b(X, E)^′_βz, C_b(X, E))-compact. This means that conv(S(H)) is relatively σ(Mz(X, E^′), C_b(X, E))-compact.

(ii) =⇒(i) It is obvious.

(i)⇐⇒(iii) In view of Theorem 2.12H is relativelyσ(Mz(X, E^′),Cb(X, E))- compact if and only if Φ_H is βz(X, E)-equicontinuous; hence in view of Theo- rem 2.7 and Lemma 3.4 the subset{ϕ_|m|:m∈H} of (C_b(X), βz(X))^′ isβz(X)- equicontinuous. It is known that the subset{ϕ_|m| :m∈H} of (C_b(X), βz(X))^′ isβz(X)-equicontinuous if and only if the set {|m|:m∈H}in Mz(X)⁺ is uni- formlyσ-additive forz=σ(see [Wh, Theorem 11.14]) (resp. uniformlyτ-additive for z =τ (see [Wh, Theorem 11.24]); resp. uniformly tight for z =t (see [Wh,

Theorem 10.7])).

4. A Mackey-Arens type theorem for locally convex-solid topologies on C_b(X, E)

LetIbe an ideal of C_b(X, E)^′ separating points ofC_b(X, E). For each Φ∈Ilet us put

ρ_Φ(f) =|Φ|(f) for f ∈C_b(X, E).

One can show that ρ_Φ is a solid seminorm onC_b(X, E) (see the proof of Lem- ma 2.2). We define the absolute weak topology |σ|(C_b(X, E), I) onC_b(X, E) as

the locally convex-solid topology generated by the family{ρ_Φ : Φ∈I}. In view of Lemma 2.2 we have

ρ_Φ(f) =|Φ|(f) = sup

|Ψ(f)|: Ψ∈I, |Ψ| ≤ |Φ| .

This means that|σ|(C_b(X, E), I) is the topology of uniform convergence on sets of the form{Ψ∈I:|Ψ| ≤ |Φ|}=S({Φ}), where Φ∈I.

Assume that L is an ideal of C_b(X)^′ separating the points of C_b(X). For each ϕ ∈ L the function pϕ(u) = |ϕ|(|u|) for u ∈ C_b(X) defines a Riesz semi- norm on C_b(X). The family {pϕ : ϕ∈ I} defines a locally convex-solid topol- ogy |σ|(C_b(X), L) on C_b(X), called the absolute weak topology generated by L (see [AB]).

Recall that |σ|(C_b(X), L)^∨ is the locally convex-solid topology on C_b(X, E) generated by the family{p^∨_ϕ:ϕ∈L}, wherep^∨_ϕ(f) =pϕ(kfk) forf ∈C_b(X, E).

We shall need the following result.

Lemma 4.1. LetI be an ideal ofC_b(X, E)^′ separating the points ofC_b(X, E).

Then

|σ| C_b(X, E), I

=|σ| C_b(X), C_b(X)^′_I∨

whereC_b(X)^′_I={ϕ∈C_b(X)^′:|ϕ| ≤ϕ_Φ for some Φ∈I}.

Proof: Letϕ∈C_b(X)^′, i.e.,|ϕ| ≤ϕ_Φ for some Φ∈I. Then for f ∈C_b(X, E) we have

p^∨_ϕ(f) =pϕ kfk

=|ϕ| kfk

≤ϕ_Φ kfk

=|Φ|(f) =ρ_Φ(f).

This means that|σ|(C_b(X), C_b(X)^′_I)^∨ ⊂ |σ|(C_b(X, E), I).

Next, let Φ∈I. Then forf ∈C_b(X, E) we have ρ_Φ(f) =|Φ|(f) =ϕ_Φ kfk

=pϕΦ kfk

=p^∨_ϕΦ(f).

This shows that |σ|(C_b(X, E), I)⊂ |σ|(C_b(X), C_b(X)^′_I)^∨, and the proof is com-

plete.

Now we are ready to state the main result of this section.

Theorem 4.2. LetIbe an ideal of C_b(X, E)^′ separating the points of C_b(X, E).

Then

C_b(X, E),|σ| C_b(X, E), I′

=I.

Proof: To see that (C_b(X, E),|σ|(C_b(X, E), I))^′ ⊂ I assume that Φ ∈ (Cb(X, E),|σ|(C_b(X, E), I))^′. In view of Lemma 2.6 we have to show that Φ ∈ C_b(X, E)^′_C

b(X)^′I

, that is Φ ∈ C_b(X, E)^′ and ϕ_Φ ∈ C_b(X)^′_I. In fact, we know