A generalization of \(\varphi\)-conditional expectation and operator valued weight (Q1328879)

Summary: Let \(M\) be a von Neumann algebra and \(N\) a von Neumann subalgebra of \(M\). For any normal faithful semifinite weights \(\varphi\) and \(\psi\) on \(M\) and \(N\), respectively, we construct a normal map \(E: M_ +\to \widehat N_ +\), which is the \(\varphi\)-conditional expectation if \(\psi= \varphi|_ N\), and is the operator valued weight if \(\sigma^ \psi_ t= \sigma^ \varphi_ t|_ N\) \((\forall t\in \mathbb{R})\).