Operator valued weights without structure theory

Abstract:

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If ${\mathcal {N}}\subset {\mathcal {M}}$ are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight $T \colon {\mathcal {M}}_{+} \rightarrow \widehat {{\mathcal {N}}_{+}}$ if and only if there exist fns weights $\tilde \varphi$ on ${\mathcal {M}}$ and $\varphi$ on ${\mathcal {N}}$ satisfying $\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \forall x \in {\mathcal {N}} , t \in \mathbb {R}$. In fact, $T$ can be chosen such that $\tilde \varphi = \varphi \circ T$; $T$ is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.