A Riesz representation theory for completely regular Hausdorff spaces and its applications

1 Introduction and terminology

Throughout the paper let .E, || · ||_E) and (F, || · ||_F) be Banach spaces, and let E′ and F′ denote the Banach duals of E and F, respectively. By B_F′ and B_E we denote the closed unit ball In F and E, respectively. By 𝓛(E, F) we denote the space of all bounded linear operators from E to F. Given a locally convex space (Z, ξ) by (Z, ξ)′ or Zξ′ we will denote its topological dual. We denote by σ(Z, Zξ′) the weak topology on Z with respect to a dual pair 〈Z, Zξ′〉. Let us recall that (Z, ξ) is a generalized DF-space, if Z has a countable fundamental family {B_n : n ∈ ℕ} of absolutely convex ξ-bounded subsets and if ξ is the finest locally convex topology on Z which agrees with ξ on each B_n.

Assume that (X, 𝓣) is a completely regular Hausdorff space. By 𝓚(X) (resp. 𝓕(X) we will denote the family of all compact (resp. finite) sets In X. Let C_b(X, E) stand for the space of all bounded continuous functions f : X → E. By τ_c (resp. τ_s ) we denote the topology on C_b(X, E) of uniform convergence on all K ∈ 𝓚(X) (resp. on all M ∈ 𝓕(X)). By τ_u we denote the topology on C_b(X, E) of the uniform norm || · ||. By C_b(X, E)′ we denote the Banach dual of C_b(X, E). For f ∈ C_b(X, E) we will write f˜(t)=‖f(t)‖E for t ∈ X.

Let C_rc(X, E) denote the subspace of C_b(X, E) consisting of those functions h such that h(X) is relatively compact.

Let C_b(X) ⊗ E stand for the linear space spanned by the set of all functions of the form u ⊗ x, where u ∈ C_b(X), x ∈ E, and (u ⊗ x)(t) = u(t)x for t ∈ X.

By 𝓑o we denote the σ-algebraof Borel sets In X. By 𝓢(𝓑o, E) we denote the set of all E-valued 𝓑o-simple functions on X. Let B(𝓑o, E) stand for the Banach space of all totally 𝓑o-measurable functions g : X → E, equipped with the uniform norm || · || (see [1–3]). Then we have (see [3, Proposition 1]):

Different classes of strict topologies β_z, where z = σ, ∞, p, τ, t, on the spaces C_b(X), C_rc(X, E), C_b(X, E) are of importance In the topological measure theory and have been studied In numerous papers (see [4–15]). Then

In this paper, we will consider the strict topology β_t on C_b(X, E) that is also denoted by β(see [7, 8]) and by β_o (see [5, 11]). It is known that if X is locally compact, then the strict β_t coincides with the original topology β of Buck(see[4]).

Now we collect basic concepts and facts concerning the strict topology β and the weak strict topology δ on C_b(X, E).

Let 𝓥_β (resp. 𝓥_δ) stand for the set of all bounded functions υ : X → [0, ∞) such that for every ε > 0, {t ∈ X : υ(t) ≥ ∊} ∈ 𝓚(X) (resp. {t ∈ X : υ(t) ≥ ∊} ∈ 𝓕(X)). The strict topology β (resp. weak strict topology δ) on C_b(X, E) is generated by the family of seminorms: p_υ(f) = sup_t∈Xυ(t)||f(t)||_E for f ∈ C_b(X, E), where υ ∈ 𝓥_β (resp. υ ∈ 𝓥_δ). Then β (resp. δ) can be characterized as the finest locally convex topology on C_b(X, E) which coincides with τ_c (resp. τ_s) on τ_u-bounded subsets of C_b(X, E) (see [7, 11, 13]). The topologies δ, β and τ_u have the same bounded subsets(see [7, Theorem 3.4], [13]). This means that (C_b(X, E), β) and (C_b(X, E), δ) are generalized DF-spaces (see [11], [13, Corollary]); equivalently, β (resp. δ) coincides with the mixed topology γ[τ_u, τ_c] (resp. γ[τ_u, τ_s]) In the sense of Wiweger (see [16, 17] for more details). We have τ_s ⊂ δ ⊂ β ⊂ τ_u and τ_c ⊂ β and C_b(X) ⊗ E is β-dense In C_b(X, E) (see [11]). It is known that Cb(X,E)β′ (resp. Cb(X,E)δ′) is equal to the closure of Cb(X,E)τc′ (resp. Cb(X,E)τs′) In the Banach space (C_b(X, E)′, || · ||) (see [17, Proposition 1]). Moreover, a sequence (f_n) In C_b(X, E) is convergent to 0 for β (resp. δ) if and only if sup_n||f_n|| < ∞ and f_n → 0 for τ_c (resp. for τ_s) (see [17, Proposition 1]). Note that δ = β if and only if 𝓚(X) ⊂ 𝓕(X) (see [12, Proposition 3.2]).

If X is a locally compact Hausdorff space, by C_o(X, E) (resp. C_c(X, E)) we denote the space of all continuous functions f : X → E vanishing at Infinity (resp. with a compact support), equipped with the uniform norm.

For X being a compact Hausdorff space (resp. a locally compact Hausdorff space) and E, F being locally convex spaces (in particular, Banach spaces), the problem of an Integral representation of bounded linear operators from C(X, E) (resp. C_o(X, E) and C_c(X, E)) to F In terms of their representing measures m : 𝓑o → 𝓛(E, F″) has been studied by Foias and Singer [18], Dinculeanu [1, 2], Goodrich [19], Dobrakov [20] and Shuchat [3]. Different classes of operators on C(X, E) have been studied intensively; see [1, 2], [20–32]. The study of the relationship between operators and their representing operator measures is a central problem in the theory.

Linear operators from the spaces C_rc(X, E) and C_b(X, E), equipped with the strict topologies β_z (z = σ, ∞, τ) to a locally convex space (F, ξ) were studied by Katsaras and Liu ([33]), Aguayo-Garrido and Nova-Yanéz (see [34, 35]). In particular, Katsaras and Liu found an integral representation of weakly compact operators S : C_rc(X, E) → F and characterizations of (β_z, ξ)-continuous and weakly compact operators S : C_rc(X, E) → F for z = σ, τ (see [33]). Aguayo-Arrido and Nova-Yanéz derived a Riesz representation theorem for (β_z, ξ)-continuous and weakly compact operators T : C_b(X, E) → F for z = ∞, τ in terms of their representing operator Baire measures (see [35, Theorems 5 and 6]).

In [36] we have developed a general theory of continuous linear operators from C_b(X, E), equipped with the strict topologies β_z (where z = σ, ∞, p, τ, t) to a Banach space F. Next, in [37] In case C_b(X) ⊗ E is assumed to be β_σ-dense In C_b(X, E) (for example, if X has a σ-compact dense subset; resp. X is a D-space; resp. E is a D-space), we have studied different classes of (β_σ, || · ||_F)-continuous operators T : C_b(X, E) → F in terms of their representing Baire operator measures.

The main aim of this paper is to develop a Riesz Integral representation theory for (β, || · ||_F)-continuous operators T : C_b(X, E) → F, where X is supposed to be a completely regular Hausdorff space. As an application, we extend to "the setting of completely regular spaces (in particular, k-spaces)" the classical results concerning operators on the spaces C(X, E) and C_o(X, E), where X is a compact or a locally compact Hausdorff space. In Section ∈ we develop the duality theory of C_b(X, E), equipped with the strict topologies β and δ. In Section 3 using the device of embedding the space B(𝓑o, E) into Cb(X,E)β″ (= the bidual of (C_b(X, E), β)), the properties that C_rc(X, E) ⊂ B(𝓑o, E) and C_rc(X, E) is β-dense in C_b(X, E), we derive the integral representation of (β, || · ||_F)-continuous operators T : C_b(X, E) → F by the Riemann-Stieltjes type integrals with respect to the representing measures m : 𝓑o → 𝓛(E, F") (see Theorem 3.2 below). In Section 4 using the Dieudonné-Grothendieck type criterion for relative weak compactness in the Banach space M(X) for X being a k-space (see Theorem 3.5), we characterize strongly bounded operators T : C_b(X, E) → F (see Theorems 4.1 and 4.3 below). In Section 5 we establish a Radon-type extension theorem for strongly bounded operators T : C_b(X, E) → F (see Theorem 5.2 below). In Section 6 we extend and generalize some classical result of Brooks and Lewis [26] concerning weakly compact operators on spaces of continuous vector-valued functions (see Theorem 6.2). In Section 7 we study unconditionally converging operators T : C_b(X, E) → F. As an application, we derive that if X is a k-space and E is reflexive, then (C_b(X, E), β) has the V property of Pełczynski (see Corollary 7.9 below).

2 Duality of C_b(X, E) with the strict topologies

Recall that a countably additive scalar measure ν on 𝓑o is said to be a regular if its variation | ν | : 𝓑o → ℝ₊ is regular, i.e., for each A ∈ 𝓑o,

By M(X) we denote the space of all countably additive regular scalar Borel measures.

Let M(X, E′) denote the space of all countably additive measures μ : 𝓑o → E′ of bounded variation (|μ|(X) < ∞) such that for each x ∈ E, μ_x ∈ M(X), where μ_x(A) := μ(A)(x) for A ∈ 𝓑o. Then |μ| ∈ M(X) (see [8, Lemma 2.3]).

Assume that m : 𝓑o → 𝓛(E, F) is a vector measure. Then the semivariation m˜ of m on A ∈ 𝓑o is defined by m˜(A) : = sup || ∑m(A_i)(x_i)||_F, where the supremum is taken over all finite 𝓑o-partitions (Ai) of A and x_i ∈ B_F for each i. For each y′ ∈ F′ let

Then m_y′ : 𝓑o → E′ and for A ∈ 𝓑o, we have (see [1, § 4, Proposition 5])

By M(X, 𝓛(E, F)) we denote the space of all measures m : 𝓑o → 𝓛(E, F) such that m˜(X) < ∞ and for each y′ ∈ F′, m_y′ ∈ M(X, E′).

For A ∈ 𝓑o let 𝒟_A denote the collection of all 𝓐 = {A₁, …, A_n; t₁, …, t_n}, where (Ai)i=1n is a 𝓑o-partition of A and t_i ∈ A_i for i = 1, …, n. For 𝓐₁, 𝓐₂ ∈ 𝓓_A define 𝓐₁ ≥ 𝓐₂ if each set which appears in 𝓐₁ is contained in some 𝓐₂. in this way 𝓓_A becomes a directed set.

Assume that m ∈ M(X, 𝓛(E, F)). Then for f ∈ C_b(X, E) and 𝓐 = {𝓐₁,..., 𝓐_n; t₁,..., t_n} ∈ 𝓓_A we will write 𝓢𝓐(f):=∑i=1nm(Ai)(f(ti)). Following [19, Definition 2], we have a definition.

Then we have

Similarly, we get ‖S𝓐ε(f)−S𝓐2(f)‖F≤η2, and hence

It follows that (S_𝓐(f)) is a Cauchy net in (F, || · ||_F) and hence the integral ∫_A f dm := lim_𝓐S_𝓐(f) exists in F.

Note that for each y′ ∈ F' and 𝓐 = {A₁,..., A_n, t₁,..., t_n} ∈ 𝓓_A, we have y′(𝓢𝓐(f))=∑i=1nmy′(Ai)(f(ti)) and ∫_Af dm_y′ = lim_𝓐y′(𝓢_𝓐(f)). Hence, we get

In particular, if μ ∈ M(X, E′), then every f ∈ C_b(X, E) is μ-integrable over all A ∈ 𝓑o and one can easily show that

Note that the equation Φ_μ(f) = ∫_Xf dμ for f ∈ C_b(X, E), defines a β-continuous linear functional on C_b(X, E). Indeed, let (f_α) be a net in C_b(X, E) such that f_α → 0 for β. Then f˜α→0 for β in C_b(X) and since |μ| ∈ M(X), we get ∫Xf˜αd|μ|→0 (see [10, Lemma 4.2]). By (1) we obtain that Φ_μ(f_α) → 0, as desired.

Conversely, assume that Φ is a β-continuous linear functional on C_b(X, E). Then by [8, Theorem 3.2] there exists a unique μ ∈ M(X, E′) such that Φ(h) = ∫_X h dμ for h ∈ C_rc(X, E): Since C_rc(X, E) is β-dense in C_b(X, E), we obtain that Φ(f) = Φ_μ(f) =∫_X f dμ for f ∈ C_b(X, E).

Now we can state the following characterization of (β, || · ||_F)-continuous functionals on C_b(X, E).

We will use the following topological result: if (Ki)i=1n are disjoint compact sets in X, then there are disjoint open sets (Oi)i=1n with K_i ⊂ O_i for i = 1, …, n, and one can choose υ_i ∈ C_b(X) with 0 ≤ υ_i ≤ 𝟙_X such that υ_i|_Ki. ≡ 1 and υ_i|_X\Oi. ≡ 0 (see [38, Theorem 3.1.6 and Theorem 3.1.7]).

The following lemma will be useful.

where the supremum is taken over finite disjointly supported collections {u₁, …, u_n} in C_b(X) with ||u_i|| = 1 and supp u_i ⊂ O and {x₁, …, x_n} ⊂ B_E.

Proof. Let A ∈ 𝓑o. Then for f ∈ C_b(X, E), ||f|| ≤ 1, by (1) we have |∫Af dμ|≤∫Af ˜d|μ|≤|μ|(A). On the other hand, let ∊ > 0 be given. Then there exist a finite 𝓑o-partition (Ai)i=1n of A and x_i ∈ B_E, i = 1, …, n such that

By the regularity of μ_xi ∈ M(X) for i = 1,...,n, we can choose K_i ∈ 𝓚(X), K_i ⊂ A_i such that |μxi|(Ai\Ki)≤ε3n for i = 1, …, n. Choose pairwise disjoint O_i ∈ 𝓣 with K_i ⊂ O_i for i = 1, …, n such that |μxi|(Oi\Ki)≤ε3n. Then for i = 1,...,n we can choose υ_i ∈ C_b(X) with 0 ≤ υ_i ≤ 𝟙_X, υ_i|_Ki ≡ 1, υ_i|_X\Oi≡ 0.

Define ho=∑i=1n(υi⊗xi).Then ||h_o|| = 1 and

Hence we get,

and hence |μ|(A) ≤ | ∫_A h_od μ| +∊. Thus the proof of (2) is complete.

Assume that O ∈ 𝓣. Let V_i = O_i ∩ O for i = 1,...,n. Then |μxi|(Vi\Ki)≤|μxi|(Oi\Ki)≤ε3n for i = 1,...,n. For i = 1,...,n choose μ_i ∈ C_b(X) with 0 ≤ μ_i, ≤ 𝟙_X, μ_i|_Ki. ≡ 1, μ_i|_X\Vi≡ 0. Let h0=∑i=1n(ui⊗xi). Then ||h_o|| = 1 and supp h_o ⊂ O, and hence by (2), |μ|(O) ≤ |∫_O h₀dμ |+ ∊. Note that ∫Oh0 dμ=∑i=1n∫Xui dμxi, where supp μ_i are pairwise disjoint and supp μ_i ⊂ O for i = 1, …,n. Thus (3) holds. □

From Lemma 2.5 it follows that if μ ∈ M(X, E′), then

Proof. (i) ⇔ (ii) See [15, Lemma 2].

(ii) ⇔ (iii) It follows from Lemma 2.5. □

Schmets and Zafarani developed the duality theory of C_b(X), equipped with the weak strict topology δ (see [12, Theorem 4.1 and Proposition 4.2]). Now we extend these results to the vector-valued setting.

For t ∈ X and x′ ∈ E′, let δ_t,_x′(f) := x′(f(t)) for f ∈ C_b(X, E). Then δ_t,x′ is a τ_s-continuous linear functional and ||δ_t,_x′|| = ||x′||_E′. By Theorem 2.4 there exists a unique μ_t,x′ ∈ M(X, E′) such that

Then |μ_{t, x′}|(X) = ||δ_t,x′|| and using Lemma 2.5, we get |μ_{t, x′}|(X \ {t}) = 0. Hence |μ_{t, x′}|({t}) = ||x'||_E' and μ_t,x'({t}) = x′. It follows that μ_{t, x'} is concentrated on {t}, that is, for A ∈ 𝓑o,

Let C(X, E) stand for the space of all continuous functions f : X → E, equipped with the topology τ_s of simple convergence. It is known that (C(X, E), τ_s)' = Lin{δ_{t, x′} : t ∈ X, x' ∈ E'}, where δ_t,x'(f) = x'(f(t)) for f ∈ C(X, E) (see [39], [40, Section 1]). Since C_b(X, E) is dense in (C(X, E), τ_s) (see [41, Theorem 1.5.3]), we get

Proof. (i) ⇒ (ii) Assume that (i) holds. Then there exists a sequence (t_n) in X such that υ(t)=∑n=𝟙∞υ(tn)1{tn}(t) for t ∈ X, where υ(t_n) → 0. Let

Note that L ⊂ c_o(E), where c_o(E) is equipped with the supremum norm || · ||_∞. Define a functional Ψ_υ on L by Ψ_υ((υ(t_n)f(t_n))) := Φ(f). Then Ψ_υ is a || · ||_∞-continuous linear functional and ||Ψ_υ|| ≤ a. Hence by the Hahn-Banach theorem there exists a || · ||_∞-continuous linear extension Ψ_υ on c_o(E) of Ψ_υ such that ||Ψ_υ|| = ||Ψ_υ||. It follows that there exists (xn′) in ℓ¹(E′) such that Ψυ¯((xn))=∑n=1∞xn′(xn) for (x_n) ∈ c_o(E) and ‖Ψυ¯‖=∑n=1∞‖xn′‖E′≤a. Then for f ∈ C_b(X,E),

(ii) ⇒ (i) Assume that (ii) holds. Then for f ∈ C_b(X, E),

Proof. (i) ⇒ (ii) Assume that Φ∈Cb(X,E)δ′. Then there exist υ ∈ 𝓥_δ and a > 0 such that |Φ(f)| ≤ ap_υ(f) for f ∈ C_b(X, E). By Lemma 2.8 there is a sequence (t_n) in X and (xn′)∈ℓ1(E′) with ∑n=1∞‖xn′‖E′≤a such that Φ(f)=∑n=1∞υ(tn)xn′(f(tn)) for f ∈ C_b(X, E). Since

we see that μ=∑n=1∞υ(tn)μtn,xn′∈M(X,E′). It follows that for f ∈ C_b(X, E),

Let ∊ > 0 be given. Choose n_∊ ∈ ℕ such that ∑n=nε+1∞υ(tn)‖xn′‖E′≤ε. Hence for M_∊ = {t₁,..., t_n} by Lemma 2.5, we get

This means that μ ∈ M_d(X, E′) and Φ(f) = ∫_X f dμ for f ∈ C_b(X, E).

(ii) ⇒ (i) Assume that (ii) holds. Then we can choose a sequence (M_n) in 𝓕(X) such that M_n ⊂ M_n+1 for n ∈ ℕ and |μ|(X \ M_n) ≤ 2^–2n. Let υ(t)=∑n=1∞2−n1Mn(t) for t ∈ X. Then υ ∈ 𝓥_δ. Assume that f ∈ C_b(X, E) and p_υ(f) = sup_t∈Xυ(t)||f(t)||_E ≤ 1. Then sup{||f(t)||_E : t ∈ M_n+1 \ M_n} ≤ 2ⁿ and sup{||f(t)||_E : t ∈ M₁} ≤ 1. Note that |μ|(X\∪n=1∞Mn)=0, so ∫X\∪n=1∞Mnf˜d|μ|=0. Hence

It follows that |Φ_μ(f) ≤ (|μ|(X) + 1)p_υ(f) for f ∈ C_b(X, E), i.e.,Φμ∈Cb(X,E)δ′. □

Proof (i) ⇒ (ii) Assume that (i) holds. Then sup_μ∈𝓜 |μ|(X) = sup_μ∈𝓜 ||Φ_μ|| < ∞ because δ ⊂ β ⊂ τ_u. Moreover, there exist υ ∈ 𝓥_δ and a > 0 such that sup_μ∈𝓜 |∫_Xf dμ| ≤ ap_υ(f) for all f ∈ Cb(X, E). Hence in view of the proof of Lemma 2.8 there exists a sequence (t_n) in X with υ(t_n) → 0 and for each μ ∈ 𝓜 there exists (xμ,n′)∈ℓ1(E′) with ∑n=1∞‖xμ,n′‖E′≤a such that

Let ∊ > 0 be given. Choose n_∊ ∈ ℕ such that υ(tn)≤εa for n ≥ n_∊. Hence for each μ ∈ 𝓜, ∑n=nε+1∞υ(tn)‖xμ,n′‖E′≤ε. Let f ∈ C_b(X, E) with ||f|| ≤ 1 and f|_M∊ ≡ 0, where M_∊ = {t₁, …, t_n∊). Then for every μ ∈ 𝓜,

(ii) ⇒ (iii) Assume that (ii) holds. By Lemma 2.5 for μ ∈ 𝓜 and M ∈ 𝓕(X),

It follows that (iii) holds.

(iii) ⇒ (i) Assume that (iii) holds. Then there exists a sequence (M_n) in 𝓕(X) such that M_n ⊂ M_n+1 and sup{|μ|(X \ M_n) : μ ∈ 𝓜} ≤ 2^–2n. Let υ(t):=∑n=𝟙∞2−n1Mn(t) for t ∈ X. Then υ ∈ 𝓥_δ. Assume that f ∈ C_b(X, E) and p_υ(f) = sup_t∈Xυ(t)||f(t)||_E ≤ 1. Then sup{||f(t)||_E : t ∈ M_n+1\ M_n} ≤ 2ⁿ, sup{||f(t)_E : t ∈ M₁} ≤ 1 and for every μ ∈ 𝓜, we get

Let sup_μ∈𝓜 |μ|(X) = a. Then for f ∈ C_b(X, E) and every μ ∈ 𝓜, we have |Φ_μ (f)| ≤ (a + 1)p_υ (f), and it follows that (i) holds. □

3 A Riesz representation theory of operators onCb(X, E)

Let Cb(X,E)β″ denote the bidual of (C_b(X, E), β). Since β-bounded subsets of C_b(X, E) are τ_u-bounded, the strong topology β(Cb(X,E)β′,Cb(X,E)) in Cb(X,E)β′ coincides with the norm topology in C_b(X, E)′ restricted to Cb(X,E)β′. Hence Cb(X,E)β″=(Cb(X,E)β′,‖ ⋅ ‖)′.

Assume that T : C_b(X, E) → F is a (β, || · ||_F)-continuous operator. Let T′:F′→Cb(X,E)β′ and T″:Cb(X,E)β″→F″ stand for the conjugate and the biconjugate operators of T, respectively, i.e., T′(y′) := y′ ◦ T for y′ ∈ F', and T"(Ψ)(y′) := Ψ(y′ ◦ T) for Ψ∈Cb(X,E)β″(=(Cb(X,E)β′,‖ ⋅ ‖)′) and y′ ∈ F′.

Then one can embed B(𝓑o, E) into Cb(X,E)β″ by the mapping π:B(𝓑o,E)→Cb(X,E)β″, where for g ∈ B(𝓑o, E),

and (I) ∫_X g dμ denotes the so-called immediate integral (see [1, 2]). Then

and hence π is bounded and ||π(g)|| ≤ ||g||. One can easily show that

where ‖Ψ‖=sup{|Ψ(Φ)|:Φ∈Cb(X,E)β′,  ‖Φ‖≤1}. Then T and T″ are bounded operators, and we define a bounded operator by:

Define a measure m : 𝓑o → 𝓛(E, F") (called the representing measure of T) by

For each y′ ∈ F′, let

Then m_y′: 𝓑o → E′ and by [1, § 4, Proposition 5] for A ∈ 𝓑o we have,

Then T^ admits an integral representation by the so-called immediate integral (I) ∫_X g dm, developed by Foias and Singer [18] and Dinculeanu [1, § 6], [2, § 1], that is,

where (s_n) is a sequence in 𝓢(𝓑o, E) such that ||s_n – g || → 0. Then for y′ ∈ F′,

Let i_F : F → F″ denote the canonical embedding, i.e., i_F(y)(y′) = y′(y) for y ∈ F, y′ ∈ F′. Moreover, let j_F : i_F(F) → F stand for the left inverse of i_F, that is, j_F ◦ i_F = id_F.

From the general properties of the operator T^ for h ∈ C_rc(X, E) we have,

Hence for each y′ ∈ F′, we get