By Nadia Boudi, Martin Mathieu (auth.), Raúl E. Curto, Martin Mathieu (eds.)
This quantity includes solicited articles through audio system on the workshop starting from expository surveys to unique learn papers, each one of which conscientiously refereed. all of them undergo witness to the very wealthy arithmetic that's attached with the examine of simple operators, may perhaps or not it's multivariable spectral conception, the invariant subspace challenge or tensor items of C*-algebras.
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Additional info for Elementary Operators and Their Applications: 3rd International Workshop held at Queen's University Belfast, 14-17 April 2009
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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