Factorization through matrix spaces for finite rank operators between \(C^*\)-algebras
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.
scientific article; zbMATH DE number 1425256
scientific article