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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 , 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 .
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