Spectral homomorphisms on a locally convex algebra ...

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Spectral homomorphisms on a locally convex algebra Cb (X ) Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65–516 Zielona Góra, Poland Received 16 June 2017; accepted 24 August 2017 Communicated by B. de Pagter

Abstract Let X be a completely regular Hausdorff space and Bo be the σ -algebra of Borel sets in X . Then the space Cb (X ) (resp. B(Bo)) of all bounded continuous (resp. bounded Bo-measurable) complex functions on X , equipped with the natural strict topology β is a locally convex algebra with the jointly continuous multiplication. It is shown that every (β, ξ )-continuous homomorphism from Cb (X ) to a complex sequentially complete locally convex algebra (A, ξ ) that maps 1 X to a unit 1 in A is a spectral homomorphism for a unique spectral measure m : Bo → A. As an application, we study continuous algebra homomorphisms from (Cb (X ), β) to the algebra L(F) of all bounded linear operators on a Banach space F, equipped with the strong operator topology. c 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Spaces of bounded continuous functions; Integration operators; Strict topologies; Locally convex algebras; Spectral measures

1. Introduction and preliminaries Throughout the paper we assume that (X, T ) is a completely regular Hausdorff space. By K we will denote the family of all compact sets in X . Let Bo stand for the σ -algebra of Borel sets in X . Let Cb (X ) (resp. B(Bo)) be the Banach space of all bounded continuous (resp. bounded Bo-measurable) complex functions on X , equipped with the topology τu of the uniform norm ∥ · ∥∞ . By S(Bo) we denote the space of all Bo-simple complex functions on X . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.indag.2017.08.006 c 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝

Please cite this article in press as: M. Nowak, Spectral homomorphisms on a locally convex algebra Cb (X ), Indagationes Mathematicae (2017), http://dx.doi.org/10.1016/j.indag.2017.08.006.

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Following [11] the strict topology β on B(Bo) is defined by the family of seminorms pw (v) := sup w(t)|v(t)| for v ∈ B(Bo), t∈X

where w runs over the family of all bounded functions w : X → [0, ∞) which vanish at infinity, i.e., for every ε > 0, there is K ∈ K such that supt∈X ∖K w(t) ≤ ε. In view of [11, Theorem 2.4] τc ⊂ β ⊂ τu on B(Bo) and β and τc coincide on any τu -bounded set in B(Bo), where τc denotes the compact-open topology on B(Bo). The topologies β and τu have the same bounded sets. If, in particular, X is compact, then β = τu . The strict topology β restricted to Cb (X ) (denoted by β again) has been studied intensively (see [5,6,10,11,14,20]). Then β can be characterized as the finest locally convex Hausdorff topology on Cb (X ) which coincides with τc on τu -bounded sets (see [20, Theorem 2.4], [19]). This means that (Cb (X ), β) is a generalized DF-space (see [19, Corollary]); equivalently, β coincides with the mixed topology γ [τu , τc ] in the sense of Wiweger ([23, (D), pp. 65–66], [6] for more details). It is known that (Cb (X ), β) is a locally convex algebra with the jointly continuous multiplication [16, Theorem 1]. Arguing verbatim as in the proof of [16, Theorem 1] one can show that also (B(Bo), β) is a locally convex algebra with the jointly continuous multiplication. When X is locally compact, β coincides with the strict topology of Buck [1] and the locally convex algebra (Cb (X ), β) has been studied in [2–4]. In particular, (Cb (R), β) is a complete A-convex algebra with identity which is not locally m-convex (see [4]). Hertz [12] developed harmonic analysis in the space (Cb (X ), β). Shuchat [22, Theorem 3] showed that every continuous homomorphism from the Banach algebra C(X ) (X is a compact Hausdorff space) to a locally convex algebra A that maps 1 X to a unit 1 in A is a spectral homomorphism for a unique spectral measure m : Bo → A. The aim of this paper is to extend this result to homomorphisms on the locally convex algebra Cb (X ), equipped with the natural strict topology β, where X is a completely regular Hausdorff space. As an application, we study continuous algebra homomorphisms from (Cb (X ), β) to the space L(F) of all bounded linear operators on a Banach space F, equipped with the strong operator topology (see [21, Theorem]). 2. Integral representation of operators on C b (X) Recall that a countably additive complex measure on Bo is called a Radon measure if its variation |µ| is regular, i.e., for each A ∈ Bo, |µ|(A) = sup{|µ|(K ) : K ∈ K, K ⊂ A} = inf{|µ|(O) : O ∈ T , O ⊃ A}. Let M(X ) denote the Banach space of all Radon measures, equipped with the total variation norm ∥µ∥ := |µ|(X ). The following characterization of the topological dual of (Cb (X ), β) will be of importance (see [14, Theorem 2], [11, Lemma 4.5], [10, Theorem 1]). Theorem 2.1. For a linear functional Φ on Cb (X ) the following statements are equivalent: (i) Φ is β-continuous. (ii) There exists a unique µ ∈ M(X ) such that ∫ Φ(u) = Φµ (u) = u dµ for u ∈ Cb (X ). X



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n We will use the following topological result: if (K i )i=1 are disjoint compact sets in X , then n there are disjoint open sets (Oi )i=1 with K⏐i ⊂ Oi for i ⏐ = 1, . . . , n, and one can choose u i ∈ Cb (X ) with 0 ≤ u i ≤ 1 X such that u i ⏐ K ≡ 1 and u i ⏐ X ∖O ≡ 0 (see [9, Theorem 3.1.6 i i and Theorem 3.1.7]). The following result will be useful in the proof of Theorem 3.1.

Lemma ∫ 2.2. For every v ∈ B(Bo) there exists a net (u α ) in Cb (X ) with ∥u α ∥∞ ≤ ∥v∥∞ such that X |v − u α |d|µ| → 0 for every µ ∈ M(X ). Proof. Let v ∈ B(Bo). Assume that µ ∈ M(X ) and ε > 0. Then there exists a sequence in S(Bo) ∫ such that sn (t) → v(t) for t ∈ X and |sn (t)| ≤ |v(t)| for t ∈ X . Hence ∫ X |sn (t) − v(t)|d|µ| → 0. ∑k ε It follows that there exists s = i=1 ai 1 Ai ∈ S(Bo) such that X |s − v|d|µ| ≤ 2 and ∥s∥∞ ≤ ∥v∥∞ . By the regularity of |µ| one can choose K i ∈ K with K i ⊂ Ai such that |µ| (Ai ∖ K i ) ≤ 4n(aεi +1) for i = 1, . . . , n. Then there exist pairwise disjoint Oi ∈ T with K i ⊂ Oi such that |µ| (Oi ∖ K i ) ≤ 4n(aεi +1) for i = 1, . . . , n. For i = 1, . . . , n one can choose ⏐ ⏐ u i ∈ Cb (X ) with 0 ≤ u i ≤ 1 X and u i ⏐ K ≡ 1 and u i ⏐ X ∖O ≡ 0. Then for i = 1, . . . , n, i

i

|u i − 1 Ai | ≤ |u i − 1 K i | + |1 K i − 1 Ai | ≤ 1 Oi ∖K i + 1 Ai ∖K i . ∑n Define u = i=1 ai u i ∈ Cb (X ). Then ∫ n ∫ ∑ |u − s| d|µ| ≤ |1 Ai − u i | |ai | d|µ| X

≤

n ∑ i=1

i=1

X

|µ| (Oi ∖ K i ) |ai | +

n ∑ ε |µ| (Ai ∖ K i )| |ai | ≤ . 2 i=1

It follows that ∫ ∫ ∫ ε ε |u − v| d|µ| ≤ |u − s| d|µ| + |s − v| d|µ| ≤ + = ε. 2 2 X X X Note that ∥u∥∞ ≤ ∥s∥∞ ≤ ∥v∥∞ . Thus there exists a uniformly bounded net (u α ) in Cb (X ), indexed by σ (Cb (X ), M(X ))-neighborhoods of 0 in Cb (X ), directed by inclusion, such that ∫ |v − u α | d|µ| → 0 for every µ ∈ M(X ) and ∥u α ∥∞ ≤ ∥v∥∞ . ■ X Assume that (E, ξ ) is a complex locally convex Hausdorff space. By E ξ′ we denote the topological dual of (E, ξ ). Let E ξ′′ be the bidual of (E, ξ ). Let E denote the family of all ξ equicontinuous subsets of E ξ′ . Then ξ is generated by the family of seminorms { p D : D ∈ E}, where p D (e) := sup{|e′ (e)| : e′ ∈ D} for e ∈ E. The so-called natural topology ξE on E ξ′′ is generated by the family of seminorms {q D : D ∈ E}, where q D (e′′ ) := sup{|e′′ (e′ )| : e′ ∈ D} for e′′ ∈ E ξ′′ (see [8, §8.7] for more details). Let i E : E → E ξ′′ stand for the canonical injection, that is, i E (e)(e′ ) = e′ (e) for e ∈ E and ′ e ∈ E ξ′ . Let j E : i E (E) → E stand for the left inverse of i E , i.e., j E (i E (e)) = e for e ∈ E. Please cite this article in press as: M. Nowak, Spectral homomorphisms on a locally convex algebra Cb (X ), Indagationes Mathematicae (2017), http://dx.doi.org/10.1016/j.indag.2017.08.006.

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Let Cb (X )′′β denote the bidual of (Cb (X ), β). Then one can embed B(Bo) into Cb (X )′′β by the mapping π : B(Bo) → Cb (X )′′β , where for v ∈ B(Bo), ∫ π (v)(Φµ ) := v dµ for µ ∈ M(X ) X

(see [17, §3]). Assume that T : Cb (X ) → E is a (β, ξ )-continuous linear operator. Let Tˆ := T ′′ ◦ π : B(Bo) → E ξ′′ , where T ′′ : Cb (X )′′β → E ξ′′ denotes the biconjugate operator of T . Then Tˆ is a (β, ξE )-continuous linear operator (see [17, Theorem 3.1]). For A ∈ Bo let m(A) ˆ := Tˆ (1 A ). Hence mˆ : Bo → E ξ′′ is a vector measure having the ξE -bounded range and will be called a representing measure of T (see [13,18]). For every e′ ∈ E ξ′ , let ′ mˆ e′ (A) := m(A)(e ˆ ) for A ∈ Bo.

From the general properties of the operator Tˆ it follows immediately that Tˆ (Cb (X )) ⊂ i E (E) and T (u) = j E (Tˆ (u)) for u ∈ Cb (X ). Recall that a completely regular Hausdorff space (X, T ) is a k-space if a subset A of X is closed whenever A ∩ K is compact for all compact sets in X . In particular, every locally compact Hausdorff space, every metrizable space and every space satisfying the first countability axiom is a k-space (see [9, Chap. 3, § 3]). Definition 2.1. A ξ -countably additive measure m : Bo → E is called a Radon measure if m is regular, i.e., for every D ∈ E, A ∈ Bo and ε > 0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that p D (m(B)) = supe′ ∈D |m e′ (B)| ≤ ε for every B ∈ Bo with B ⊂ O ∖ K (equivalently; supe′ ∈D |m e′ |(O ∖ K ) ≤ ε). The following general result characterizes (β, ξ )-continuous linear operators T : Cb (X ) → E in terms of their representing measures. Theorem 2.3. Assume that X is a k-space and (E, ξ ) is a sequentially complete lcHs. Let T : Cb (X ) → E be a (β, ξ )-continuous linear operator and mˆ be its representing measure. Assume that m(A) ˆ ∈ i E (E) for every A ∈ Bo and m = j E ◦ mˆ : Bo → E. Then the following statements hold: (i) m : Bo → E is Radon measure. (ii) Tˆ (B(Bo)) ⊂ i E (E). (iii) Every v ∈ B(Bo) is m-integrable (see [14, § 6]) and ∫ ˆ T (v) := j E (T (v)) = v dm. X

(iv) For every e′ ∈ E ξ′ and v ∈ B(Bo), ∫ e′ (T (v)) = v dm e′ , X ′

where m (A) = e (m(A)) for A ∈ Bo. e′



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(v) T : B(Bo) → E is (β, ξ )-continuous. (vi) T (vn ) → 0 in ξ whenever (vn ) is a uniformly bounded sequence in B(Bo) such that vn (t) → 0 for every t ∈ X . Proof. (i)–(v) See [17, Theorems 3.1 and 3.5]. (vi) Assume that (vn) is a uniformly bounded sequence in B(Bo) such that vn (t) → 0 for every t ∈ X . Let D ∈ E be given. By (i) the family {m e′ : e′ ∈ D} is uniformly countably ′ additive, and it follows that uniformly ∫countably additive. Hence by ∫ the family {|m e′ | : e ∈ D} is [15, Theorem 1] supe′ ∈D X |vn | d|m e′ | → 0. Since |e′ (T (vn ))| ≤ X |vn | d|m e′ | for e′ ∈ D, we get p D (T (vn )) → 0, i.e., (vi) holds. ■ 3. Spectral homomorphisms on (C b (X), β) Assume that (A, ξ ) is a complex locally convex algebra with a unit 1. Multiplication is only assumed to be separately continuous and it is clearly then separately weakly continuous. Recall that a Radon measure m : Bo → A is said to be spectral if m(X ) = 1 and m(A1 ∩ A2 ) = m(A1 ) · m(A2 ) for A1 , A2 ∈ Bo. Now we can state our main result. Theorem 3.1. Assume that X is a k-space and (A, ξ ) is a complex sequentially complete locally convex Hausdorff algebra with a unit 1. Let T : Cb (X ) → A be a (β, ξ )-continuous algebra homomorphism such that T (1 X ) = 1, and mˆ be its representing measure. If m(A) ˆ ∈ i E (E) for A ∈ Bo, then the following statements hold: (i) The extension operator T : B(Bo) → A is a (β, ξ )-continuous algebra homomorphism. (ii) m = j E ◦ mˆ : Bo → E is a spectral measure. (iii) T is a σ -homomorphism, i.e., T (vn ) → 0 in ξ whenever (vn ) is a uniformly bounded sequence in B(Bo) such that vn (t) → 0 for every t ∈ X . (iv) For each v ∈ B(Bo), T (v) ∈ A is the limit in (A, ξ ) of a sequence of linear combinations of idempotents. (v) If a ∈ A commutes with the range of T , then a commutes with the range of T . Proof. (i) In view of Theorem 2.3 T is a (β, ξ )-continuous operator. We will show that T is multiplicative. Let v ∈ B(Bo) and w ∈ B(Bo). Then according to Lemma 2.2 there exist nets (vα ) and (wδ ) in Cb (X ) with ∥vα ∥∞ ≤ ∥v∥∞ and ∥wδ ∥∞ ≤ ∥w∥∞ such that for every µ ∈ M(X ), ∫ ∫ |vα − v| d|µ| −→ 0 and |wδ − w| d|µ| −→ 0. α

X

δ

X

Hence for a fixed α, we have π (vα wδ ) −→ π (vα w) in σ (Cb (X )′′β , Cb (X )′β ) because for every δ µ ∈ M(X ), ⏐∫ ⏐ ⏐ ⏐ |π(vα wδ )(Φµ ) − π (vα w)(Φµ )| = ⏐⏐ (vα wδ − vα w) dµ⏐⏐ ∫ ∫ X ≤ |vα wδ − vα w| d|µ| ≤ ∥v∥∞ |wδ − w| d|µ| −→ 0. X ′′

Cb (X )′′β

X ′′ Aξ is

Since the mapping T : → (T ′′ ◦ π )(B(Bo)) ⊂ i A (A) and jA : i A (A)

δ

(σ (Cb (X )′′β , Cb (X )′β ), σ (A′′ξ , A′ξ ))-continuous with → A is (σ (i A (A), A′ξ ), σ (A, A′ξ ))-continuous, we


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get T (vα wδ ) = jA (T ′′ (π(vα wδ ))) −→ jA (T ′′ (π (vα w))) = T (vα w) δ

σ (A, A′ξ ).

in we get

On the other hand, since the multiplication in A is separately weakly continuous,

T (vα )T (wδ ) −→ T (vα )T (w) in σ (A, A′ξ ). δ

But T (vα wδ ) = T (vα )T (wδ ), so T (vα w) = T (vα )T (w) for every α. Similarly, we can obtain that T (vα w) −→ T (vw) and T (vα )T (w) −→ T (v)T (w) α

in

α

σ (A, A′ξ ).

It follows that T (vw) = T (v)T (w). (ii) It follows from (i) and Theorem 2.3. (iii) It follows from Theorem 2.3. (iv) It is obvious.

(v) Let a ∈ A and aT (u) = T (u)a for every u ∈ Cb (X ). Then by Lemma 2.2 for every v ∈ B(Bo) there exists a net (u α ) in Cb (X ) with ∥u α ∥∞ ≤ ∥v∥∞ such that π (u α ) → π (v) in σ (Cb (X )′′β , Cb (X )′β ). Then T (u α ) → T (v) in σ (A, A′ξ ). Hence aT (v) = σ (A, A′ξ ) − lim aT (u α ) = σ (A, A′ξ ) − lim T (u α )a = T (v)a. α

α

■

Assume that (F, ∥ · ∥ F ) is a complex Banach space and L(F) denotes the Banach space of all bounded linear operators U : F → F, equipped with the norm ∥ · ∥ of the uniform operator topology UOT. By SOT and WOT we denote the strong operator topology and the weak operator topology on L(F), respectively. It is known that (L(F), SOT) is a quasicomplete locally convex Hausdorff algebra with the separately (but not jointly) continuous multiplication, and the unit I (the identity operator). Theorem 3.2. Assume that X is a k-space and a complex Banach space F contains no isomorphic copy of co . Let T : Cb (X ) → L(F) be a (β, SOT)-continuous algebra homomorphism such that T (1 X ) = I . Then the following statements hold: (i) T maps τu -bounded sets in Cb (X ) onto relatively WOT-compact sets in L(F). (ii) The extension operator T : B(Bo) → L(F) is a (β, SOT)-continuous algebra homomorphism and maps τu -bounded sets onto relatively WOT-compact sets in L(F). (iii) The representing measure m of T is a spectral measure. ∫ (iv) T is a σ -homomorphism, i.e., for every x ∈ F, ∥( X vn dm)(x)∥ F → 0 whenever (vn ) is a uniformly bounded sequence in B(Bo) such that vn (t) → 0 for every t ∈ X . (v) T : B(Bo) → L(F) is a bounded operator between Banach spaces B(Bo) and L(F) and ∫ for every v ∈ B(Bo), T (v) = X v dm as the integral in (L(F), UOT). Proof. (i) For every x ∈ F, let Tx (u) := T (u)(x) for u ∈ Cb (X ). Then Tx : Cb (X ) → F is a (β, ∥ · ∥ F )-continuous linear operator, and by [17, Corollary 4.6] Tx is weakly compact. Assume that (u n ) is a uniformly bounded sequence in Cb (X ) such that u n (t) → 0 for every t ∈ X . By [17, Theorem 4.2] we get ∥T (u n )(x)∥ F = ∥Tx (u n )∥ F → 0, i.e., T (u n ) → 0 in SOT and hence T maps τu -bounded sets onto relatively WOT-compact sets in L(F). (ii) It follows from (i) and [17, Corollary 4.3]. (iii)–(iv) See Theorem 3.1. Please cite this article in press as: M. Nowak, Spectral homomorphisms on a locally convex algebra Cb (X ), Indagationes Mathematicae (2017), http://dx.doi.org/10.1016/j.indag.2017.08.006.


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(v) For x ∈ F, define T x (v) := T (v)(x) for v ∈ B(Bo). Then for A ∈ Bo m x (A) := T x (1 A ) = T (1 A )(x) = m(A)(x). Since∑ T x : B(Bo) → F is bounded, we have that ∥m x ∥(X ) < ∞, where ∥m x ∥(X ) = sup ∥ εi m x (Ai )∥ F , and the supremum is taken over all finite Bo-partitions of X and scalars εi with |εi | ≤ 1. Let ∥m∥(A) stand for ∑ the semivariation of m on A ∈ Bo for the uniform operator topology, i.e., ∥m∥(A) := sup ∥ εi m(Ai )∥, where the supremum is taken over all finite Bo-partitions of A and εi are scalars with |εi | ≤ 1. The uniform bounded principle implies that ∥m∥(X ) < ∞. Let Tm : B(Bo) → L(F) be the corresponding (∥ · ∥∞ , ∥ · ∥)-bounded integration operator, i.e., for every v ∈ B(Bo), ∫ ∫ sn dm, v dm = UOT − lim Tm (v) := X

X

where (sn ) is a sequence in S(Bo) such that ∥sn − v∥∞ → 0. Then ∥Tm ∥ = ∥m∥(X ) (see [7, Theorem 13, p. 6]). Since the operators T and Tm are (∥ · ∥∞ , SOT)-continuous, for every x ∈ F, we have ∥T (sn )(x) − T (v)(x)∥ F → 0 and ∥Tm (sn )(x) − Tm (v)(x)∥ F → 0. But T (sn )(x) = Tm (sn )(x) for n ∈ N, so T (v)(x) = Tm (v)(x) for every x ∈ F. Thus T (v) = Tm (v), so T is a (∥ · ∥∞ , ∥ · ∥)-bounded operator. ■ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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