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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]
R.C. Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958) 95–104. A. Cochran, Topological algebras and Mackey topologies, Proc. Amer. Math. Soc. 30 (1) (1971) 115–119. A. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc. 41 (2) (1973) 473–479. A.C. Cochran, R. Keown, C.R. Wiliams, On a class of topological algebras, Pacific J. Math. 34 (1) (1970) 17–25. J.B. Conway, The strict topology and compactness in the space of measures, II, Trans. Amer. Math. Soc. 126 (1967) 474–486. J.B. Cooper, The strict topology and spaces with mixed topologies, Proc. Amer. Math. Soc. 30 (3) (1971) 583–592. J. Diestel, J.J. Uhl, Vector Measures, Amer. Math. Soc., Providence, RI, 1977 Math. Surveys 15. R.E. Edwards, Functional Analysis, Theory and Applications, Holt,Rinehart and Winston, New York, 1965. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. D. Fremlin, D.J.H. Garling, R.G. Haydon, Bounded measures on topological spaces, Proc. Lond. Math. Soc. 25 (3) (1972) 115–136. R. Giles, A generalization of the strict topology, Trans. Amer. Math. Soc. 161 (1971) 467–474. C. Hertz, The spectral theory of bounded functions, Trans. Amer. Math. Soc. 94 (1960) 181–232. J. Hoffmann-Jörgënsen, Vector measures, Math. Scand. 28 (1971) 5–32. J. Hoffmann-Jörgënsen, A generalization of the strict topology, Math. Scand. 30 (1972) 313–323. S.S. Khurana, A topology associated with vector measures, J. Indian Math. Soc. 45 (1981) 167–179. S.S. Khurana, Strict topologies as topological algebras, Czechoslovak Math. J. 51 (126) (2001) 433–437. M. Nowak, Integral representation of continuous operators with respect to strict topologies, Results Math. 72 (2017) 843–863. T.V. Panchapagesan, Applications of a theorem of Grothendieck to vector measures, J. Math. Anal. Appl. 214 (1997) 89–101. J. Schmets, J. Zafarani, Strict topologies and (gDF)-spaces, Arch. Math. 49 (1987) 227–231. F.D. Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc. 168 (1972) 311–336. A. Shuchat, Vector measures and the spectral theorem, in: Vector and Operator Measures and Applications, (Proc. Sympos. Alta, Utah, 1972), Academic Press, New York, 1973, pp. 339–341.
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–
[22] A. Shuchat, Spectral measures and homomomorphisms, Rev. Roumaine Math. Pures Appl. 23 (6) (1978) 939–945. [23] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961) 47–68.
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.