Completely contractive factorizations of \(C^*\)-algebras
A map \(\phi\) :A\(\to B\) between \(C^*\)-algebras is said to be completely bounded if \(\sup_{n}\) \(\| \phi_ n\|\) is finite, where \(\phi_ n:A\otimes M_ n\to B\otimes M_ n\) is defined by \(\phi_ n(a_{ij})=(\phi (a_{ij}))\). If \(\sup_{n}\) \(\| \phi_ n\| \leq 1\) then \(\phi\) is said to be a complete contraction. If each \(\phi_ n\) is positive then \(\phi\) is called completely positive. Nuclearity for \(C^*\)-algebras can be characterized by the following property (among several): there exists a net of completely positive finite rank maps \(\phi\) \({}_{\alpha}:A\to A\) such that \(\lim_{\alpha}\| \phi_{\alpha}(x)-x\| =0\) for all \(x\in A.\) If the maps are only complete contractions, then the characterization no longer holds. There are examples of \(C^*\)-algebras for which the identity map is the point norm limit of finite rank complete contractions but which are not nuclear. The main result of this paper is that a \(C^*\)-algebra A is nuclear if and only if there exists a net \(\phi_{\alpha}:A\to A\) of complete contractions which converge in the point norm topology to the identity, and which factor through \(M_ n\).
scientific article; zbMATH DE number 3937832
scientific article