Weak containment of correspondences and approximate factorization of completely positive maps (Q752420)

Let M be a finite, \(\sigma\)-finite von-Neumann algebra and \(\phi: M\to M\) a completely positive normal map. With such a map one can associate a Hilbert space \(H_{\phi}\) called the correspondence associated to \(\phi\). Let \(\phi\), \(\psi: M\to M\) be two completely positive normal maps. The author shows that \(H_{\psi}\) is weakly contained in \(H_{\phi}\), if and only if \(\psi\) is a limit of maps which are compositions of \(\phi\) with inner maps. This means the following: Given \(\phi\), consider all maps of the form \(\theta (x)=\sum b^*_ i\phi (a^*_ ixa_ j)b_ j,\theta: M\to M\), for n-tuples \(a_ i,b_ i\in M\), \(i=1,2,...,n\), \(n=1,2,... \). The set of all finite sums of such maps is denoted by \(F_{\phi}.\) Theorems 3 and 4 in the paper can be united in the following statement: Given two completely positive normal maps \(\phi\),\(\psi: M\to M\) the correspondence \(H_{\psi}\) is weakly contained in \(H_{\phi}\) if and only if there is a bounded net \(\{\phi_{\alpha}\}\subset F_{\phi}\) such that \(\phi_{\alpha}(x)\to \psi (x)\) in \(\sigma (M,M_*)\) for each \(x\in M\).