The correspondence associated to an inner completely positive map (Q1103826)

Let M be a \(\sigma\)-finite von Neumann algebra. A completely positive map \(\Phi\) : \(M\to M\) is inner if there is a sequence \(\{a_ i\}\subseteq M\) with \(\Phi (x)=\sum _{i}a\) \(*_ ixa_ i\) for \(x\in M\). If \(\Phi\) is any completely positive map on M then associated to \(\Phi\) is a correspondence \(H_{\Phi}\) from M to M. Inner completely positive maps are characterized as those maps for which \(H_{\Phi}\subseteq \sum ^{\oplus}L\) 2(M), where L 2(M) is the identity correspondence. For an inner completely positive map \(\Phi\), we give a complete invariant of the unitary equivalence class of the representation of \(M\otimes M^{op}\) associated to \(H_{\Phi}\); it is a decreasing sequence of central projections of M, \(\{d_ i\}\), such that \(H_{\Phi}\cong \Sigma ^{\oplus}d_ iL\) 2(M). The relation of the \(d_ i's\) to the \(a_ i's\) is as follows; \(d_ 1\) is the supremum of the central supports of the \(a_ i's\) and \(d_ n\) is the central projection over which at least n of the \(a_ i's\) are independent over the centre of M.