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If we assume only that φ ∈ E1 (R) . Proof. If φk is a sequence of elementary operators converging pointwise to φ, then for each functional ρ ∈ R∗ the sequence (φ∗k (ρ)) converges to φ∗ (ρ) ∈ R∗ in the weak* topology, hence, since R is a Grothendieck space [19], the convergence is ∗∗ in the weak topology of R∗ . This means that the maps φ∗∗ converge to k on R ∗∗ ∗∗ φ in the point-weak* topology. Note that each φk is an elementary operator on Approximation by Elementary Complete Contractions 33 R∗∗ , hence a module map over the center Z of R∗∗ , and that ρz ∈ R∗ for each z ∈ R∗∗ and ρ ∈ R∗ .

Let A be a unital C ∗ -algebra. n. in E1 (A) , then φ is in the pointwise closure of a net of elementary completely positive contractions σk . If, in addition, φ is unital, σk can be chosen to be unital. Proof. Let h = φ(1). 3). 1 there is a net ψk = a∗k bk in E1 (A) converging pointwise to ψ such that ak , bk ≤ 1. 1) the net of completely positive contractions θk = a∗k ak also converges pointwise to ψ. Hence the net of maps σk := hθk h = c∗k ck , where ck := ak h, converges to hψh = φ. If φ(1) = 1,we may replace each σk by the unital map τk (x) = σk (x) + 1 − c∗k ck x 1 − c∗k ck .

137 (2009), 2375–2385. [17] B. Magajna, Uniform approximation by elementary operators, Proc. Edinburgh Math. Soc. 52 (2009), 731–749. I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2002. [19] H. Pﬁtzner, Weak compactness in the dual of a C ∗ -algebra is determined commutatively, Math. Ann.