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Factorization through matrix spaces for finite rank operators between \(C^*\)-algebras. (English) Zbl 0947.46053
The authors present factorization results involving the completely bounded norm \(\|.\|_{cb}\) and the Haagerup decomposable norm \(\|.\|_{dec}\). Here is the main theorem:
Let \(A\) and \(B\) be two \(C^*\)-algebras and let \(M_n\) be the \(C^*\)-algebra of all \(n\times n\) complex matrices. Suppose \(u:A\to B\) is a finite rank bounded operator. Then:
(a) For every \(\varepsilon> 0\) there exists an integer \(n\geq 1\) and two linear maps \(\alpha: A\to M_n\), \(\beta: M_n\to B\) such that \(u= \beta\alpha\) and \(\|\alpha\|_{cb}\|\beta\|_{dec}\leq (1+\varepsilon)\|u\|_{dec}\).
(b) If \(A\) is a von Neumann algebra and \(u\) is \(w^*\)-continuous, then \(\alpha\) in (a) can be taken \(w^*\)-continuous, too.
The paper contains also some related results and comments on the properties of the Haagerup decomposable norm.

MSC:
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
47L99 Linear spaces and algebras of operators
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47B49 Transformers, preservers (linear operators on spaces of linear operators)
46A32 Spaces of linear operators; topological tensor products; approximation properties
46L06 Tensor products of \(C^*\)-algebras
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