Corrigendum to “Functional decompositions on vector-valued function spaces via operators” [J. Math. Anal. Appl. 389 (2012) 1173–1190]

J. Math. Anal. Appl. 404 (2013) 192–194

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Corrigendum

Corrigendum to ‘‘Functional decompositions on vector-valued function spaces via operators’’ [J. Math. Anal. Appl. 389 (2012) 1173–1190] Titarii Wootijirattikal a,b , Sing-Cheong Ong c,∗ , Jitti Rakbud d a

Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand b Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand c

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

d

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand

article

info

Article history: Received 11 December 2012 Available online 12 March 2013 Submitted by Stephen Power

abstract This is a correction of the proof of the assertion in the appendix of the manuscript ‘‘Functional decompositions on vector-valued function spaces via operators’’ JMAA 389, 1173–1190. © 2013 Elsevier Inc. All rights reserved.

We are very much indebted by an anonymous reviewer for pointing out, with the example ρn = δ1/n − δ0 on C [0, 1], that our second reduction in the proof for Lemma 19 on [2, p. 1188] is invalid. Regrettably, this oversight of ours invalidated our ‘‘proof’’ of the conclusion in Proposition 20. The purpose of this note is to correct the proof. Let A be a C ∗ -algebra with identity 1. A linear functional ρ on A is a positive functional if ρ(a) ≥ 0 for all positive elements a ∈ A. The positivity of a linear functional ρ is equivalent to its boundedness (∥ρ∥ < ∞), and in that case its norm is ∥ρ∥ = ρ(1) [1, Theorem 4.3.2]. A state is a positive linear functional of norm 1. The state space, s(A), of A is the set of all states on A, endowed with the weak∗ topology inherited as a subspace of the dual space, A# , of A. With the relative weak∗ topology, the state space is a compact Hausdorff space [1, p. 257]. For a (not necessarily bounded) linear functional f on A, the conjugate linear functional is defined by f ∗ (x) = (f (x∗ )) for all x ∈ A. A linear functional ρ on A is hermitian if ρ = ρ ∗ . Each linear functional f on A has a unique canonical decomposition f = (ℜf ) + i(ℑf ) as a linear combination of hermitian linear functionals, its real and imaginary parts, ℜf = 12 (f + f ∗ ) and ℑf = 2i (f ∗ − f ) [1, p. 255]. Each bounded hermitian linear functional ρ has a unique decomposition + − ρ = ρ+ − ρ −as the  difference of two positive linear functionals ρ , ρ (its positive and negative parts) such that ∥ρ∥ = ρ +  + ρ −  [1, Th 4.3.6, p. 259] (a Hahn–Jordan decomposition analogue). Lemma 1. Let {ργ } be a bounded net of bounded hermitian linear functionals on A that is weak∗ convergent to 0. Then there are a subnet {ργβ } and a positive linear functional ψ such that (ργβ )+ → ψ and (ργβ )− → ψ , in the weak∗ topology.

 +  − − ργ as the unique difference of two positive functionals as above. Assume without   loss of generality that ργ  ≤ 1 for all γ . First note that the set of positive linear functionals, [A# ]+ 1 , on A that are in the Proof. Decompose each ργ = ργ

DOI of original article: http://dx.doi.org/10.1016/j.jmaa.2011.12.057. Corresponding author. E-mail addresses: [email protected], [email protected] (T. Wootijirattikal), [email protected], [email protected] (S.-C. Ong), [email protected], [email protected] (J. Rakbud).

∗

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.02.031

T. Wootijirattikal et al. / J. Math. Anal. Appl. 404 (2013) 192–194

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∗ closed unit ball [A# ]1 is compact in the weak∗ topology. (For if {ψγ } ⊆ [A# ]+ 1 weak converges to a bounded linear functional ψ , then for each positive element a ∈ A, we have ψ(a) = limγ ψγ (a) ≥ 0. Thus ψ is a positive linear functional, ∗ # and [A# ]+ 1 is closed in the weak topology, and hence it is compact as a closed subset of the compact [via Alaoglu] set [A ]1 .)

 +   +  that weak∗ converges to a positive linear functional ψ1 . By further ργ has a subnet ργα α  −   +  weak∗ converge and ργβ dropping down to a subnet {ργβ }β of the net {ργα }α , we may assume that both ργβ β β  +  −   ∗ weak converges to ψ1 − ψ2 . Since − ργβ to positive linear functionals ψ1 and ψ2 respectively. Thus ργβ = ργβ {ργβ }β is a subnet of {ργ }, which weak∗ converges to 0, we have ψ1 = ψ2 , and this is ψ as asserted.  It follows that the net

Lemma 2. Let {ψγ } be a bounded net of positive linear functionals that weak∗ converges to a bounded  linear  functional ψ . Then ψ is a positive linear functional. If ψ ̸= 0, there is a net (of states) {ϕγ } ⊆ s(A) such that ψγ = ψγ  ϕγ for all γ and {ϕγ } weak∗ converges to ϕ = [∥ψ∥]−1 ψ . Proof. Positivity of ψ is proven in the proof of Lemma 1. Now assume ψ ̸= 0 and ψγ  ≤ M for all γ . Since ψγ (1) →



  −1 −1   ψ if ψγ = ψ(1) = ∥ψ∥ ̸= 0 ([1, Theorem ̸ 0 and ϕγ = ϕ   4.3.2, p.256]) and ϕ = ∥ψ∥ ψ ∈ s(A), we may set ϕγ = ψγ   if ψγ = 0. Then ψγ = ψγ  ϕγ for all γ , with ϕγ ∈ s(A) for all γ . Simple argument, such as the following, shows that ϕγ     ∥ψ∥ weak∗ converges to ϕ . Since ψγ  = ψγ (1) → ψ(1) = ∥ψ∥ > 0, there is a γ0 such that ψγ  ≥ 2 > 0 for all γ ≽ γ0 .      ϵ∥ψ∥ ϵ∥ψ∥ Let x ∈ A and ϵ > 0. There is γ1 such that ψγ  − ∥ψ∥ < 2(∥x∥+1) and ψγ (x) − ψ(x) < 2 for all γ ≥ γ1 . Choose γ2 ≽ γ0 and γ2 ≽ γ1 . Then, for γ ≽ γ2 , we have γ ≽ γ1 , γ ≽ γ0 , and hence ψγ ̸= 0 and      ψ (x) ψ(x)   ψ (x) ψ (x)   ψ (x) ψ(x)    γ γ γ      γ ϕγ (x) − ϕ(x) =    −  −  ≤   − +  ψγ  ∥ψ∥   ψγ  ∥ψ∥   ∥ψ∥ ∥ψ∥    ψγ (x)     1  ψγ (x) − ψ(x)   ∥ψ∥ − ψγ  + = ∥ψ∥ ∥ψ∥ ψγ  ∥x ∥ ϵ ∥ψ∥ 1 ϵ ∥ψ∥ ≤ · + · < ϵ. ∥ψ∥ 2(∥x∥ + 1) ∥ψ∥ 2   −1 This shows that ϕγ = ψγ  ψγ weak∗ converges to ∥ψ∥−1 ψ = ϕ .  We denote by Z a Banach space of complex valued functions on a nonempty set S and by AS the set of all functions mapping S to A. Lemma 3. Let x ∈ AS satisfy (i) ϕ ◦ x ∈ Z for all ϕ ∈ s(A), and (ii) the map ϕ → ϕ ◦ x is weak∗ to norm continuous from s(A) to Z. Then

  ∥x∥ := sup ∥ϕ ◦ x∥Z : ϕ ∈ s(A) < ∞ and the space of all such x is a Banach space with the norm ∥x∥. The finiteness of ∥x∥ follows readily from compactness of s(A) and the continuity assumption. Since we are not going to use the fact that all such functions form a Banach space, we omit the proof. Lemma 4. Letx ∈ AS satisfy (i) ϕ ◦ x ∈ Z for all ϕ ∈ s(A) and (ii) the map ϕ → ϕ ◦ x is weak∗ to norm continuous from s(A) to Z. Suppose ρβ is a bounded net of positive linear functionals on A that weak∗ converges to a bounded linear functional ρ .

Then ρ is a positive linear functional, and limβ ρβ ◦ x − ρ ◦ x = 0.





Proof. Positivity 2 again,   of ρ follows from∗ Lemma 2. If ρ ̸= 0, then, by Lemma  there  is a net {ϕβ } ⊆ s(A) of states such −1    ∥ρ∥ that ρβ = ρβ ϕβ and {ϕβ } weak converges to the state ϕ := ρ . Since ρβ  = ρβ (1) → ρ(1) = ∥ρ∥ ̸= 0, there is

 −1

a β0 such that ρβ ̸= 0 for all β ≽ β0 . Thus for β ≽ β0 , ϕβ (x) = ρβ  hence ϕβ → ϕ in the weak∗ topology. Thus by our hypothesis on x,



   ρβ (x) → ∥ρ∥−1 ρ (x) = ϕ(x) for all x ∈ A, and

limβ ϕβ ◦ x − ϕ ◦ x = 0,





and hence

     ρβ ◦ x − ρ ◦ x = (ρβ  ϕβ ) ◦ x − (∥ρ∥ ϕ) ◦ x Z   Z   ≤ (ρβ  ϕβ ) ◦ x − (∥ρ∥ ϕβ ) ◦ xZ + (∥ρ∥ ϕβ ) ◦ x − (∥ρ∥ ϕ) ◦ xZ       = ρβ  − ∥ρ∥ ϕβ ◦ xZ + ∥ρ∥ ϕβ ◦ x − ϕ ◦ xZ     β ≤ ρβ (1) − ρ(1) ∥x∥ + ∥ρ∥ ϕβ ◦ x − ϕ ◦ xZ −→ 0.

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  If ρ = 0, then ρβ  = ρβ (1) → ρ(1) = 0. Since ∥x∥ = supϕ∈s(A) ∥ϕ ◦ x∥Z ,           ρβ ◦ x − ρ ◦ x = ρβ ◦ x = (ρβ  ϕβ ) ◦ x = ρβ  ϕβ ◦ x Z Z Z   β ≤ ρβ  ∥x∥ −→ 0.   Therefore we have limβ ρβ ◦ x − ρ ◦ xZ = 0 in all cases.  Proposition 5. Let x ∈ AS have the properties that ϕ ◦ x ∈ Z for all ϕ ∈ s(A) and that ϕ → ϕ ◦ x is weak∗ to norm continuous from s(A) to Z. Then f ◦ x ∈ Z for all f ∈ A# and the map f → f ◦ x is weak∗ to norm continuous from [A# ]1 to Z. Proof. Let {fα } be a net in [A# ]1 that weak∗ converges to f . We show that limα ∥fα ◦ x − f ◦ x∥Z = 0, by showing that every subnet of {fα ◦ x} has a subnet that converges to f ◦ x in Z. First note that since the net {(fα − f )} weak∗ converges to 0, and (fα −f )◦x = fα ◦x−f ◦x, we may assume, without loss of generality, that {fα } weak∗ converges to 0. We need only to show that limα ∥fα ◦ x∥ = 0 under this more restrictive assumption. Write each fα = (ℜfα ) + i(ℑfα ) in terms of its real and imaginary parts of bounded hermitian linear functionals [1, p. 255]. Then, for each self-adjoint a ∈ A, fα (a) = (ℜfα )(a) + i(ℑfα )(a) with (ℜfα )(a) and (ℑfα )(a) being real [1, p. 255]. Thus (ℜfα )(a) → 0 and (ℑfα )(a) → 0. Since this is true for all self-adjoint a ∈ A, (ℜfα ) → 0 and (ℑfα ) → 0 in the weak∗ topology. We note also that ∥(ℜfα )∥ ≤ ∥fα ∥ ≤ 1 and likewise ∥(ℑfα )∥ ≤ 1 for all α . Thus {(ℜfα )} and {(ℑfα )} are bounded nets of hermitian bounded linear functionals, and so is each of their subnets. We show that every subnet of {fα ◦ x} contains a subnet that converges to 0 in the norm of Z. To that end, let {fαβ } be a subnet of {fα }. By Lemma 1 there are a subnet {gη } of {fαβ } and positive linear functionals ψ1 , ψ2 such that (ℜgη )+ → ψ1 , (ℜgη )− → ψ1 , (ℑgη )+ → ψ2 , and (ℑgη )− → ψ2 in the weak∗ topology. By Lemma 4 lim (ℜgη )+ ◦ x − ψ1 ◦ xZ = 0,

lim (ℜgη )− ◦ x − ψ1 ◦ xZ = 0,

lim (ℑgη )+ ◦ x − ψ2 ◦ xZ = 0,

lim (ℑgη )− ◦ x − ψ2 ◦ xZ = 0.





η



η







η





η

Thus

    gη ◦ x = (ℜgη )+ ◦ x − (ℜgη )− ◦ x + i[(ℑgη )+ ◦ x] − i[(ℑgη )− ◦ x] Z Z     ≤  (ℜgη )+ ◦ x − ψ1 ◦ x − (ℜgη )− ◦ x − ψ1 ◦ x Z     +  (ℑgη )+ ◦ x − ψ2 ◦ x − (ℑgη )− ◦ x − ψ2 ◦ x Z     ≤ (ℜgη )+ ◦ x − ψ1 ◦ xZ + (ℜgη )− ◦ x − ψ1 ◦ xZ     + (ℑgη )+ ◦ x − ψ2 ◦ xZ + (ℑgη )− ◦ x − ψ2 ◦ xZ → 0.       This shows that every subnet of the net fγ ◦ x in Z has, in turn, a subnet gη ◦ x such that limη gη ◦ xZ = 0; we may thus conclude that limα ∥fα ◦ x∥Z = 0.  References [1] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. I, Academic Press, New York, 1983. [2] T. Wootijirattikal, S.-C. Ong, J. Rakbud, Functional decompositions on vector-valued function spaces via operators, J. Math. Anal. Appl. 389 (2012) 1173–1190.