A Radon-Nikodym type property for von Neumann algebras

Let M be a von Neumann algebra and ϕ be a normal semi-finite weight. Let $M_{\varphi }$ and $s_{\varphi }$ be the centralizer and the support of ϕ, respectively. Write $Z\left(M_{\varphi }\right)$ for the center of $M_{\varphi }$.

If ϕ satisfies the condition:

(1) $M_{\varphi }^{\prime }\cap M\subseteq M_{\varphi }$,

then the following “Radon-Nikodym type property” holds:

(2) for each normal semi-finite weight ψ, if $s_{\psi }\le s_{\varphi }$, $s_{\psi }\in M_{\varphi }$ and $M_{\varphi }\subseteq M_{\psi }$, then there is a self-adjoint positive operator h affiliated with $Z\left(M_{\varphi }\right)$ such that $\psi ={\varphi }_h$.

It is shown that if ϕ is strictly semi-finite, then $\left(2\right)\Rightarrow \left(1\right)$, and they are equivalent to:

(2) for every ^⁎-automorphism θ on M with $\theta {|}_{M_{\varphi }}={\mathrm{id}}_{M_{\varphi }}$, one has $\varphi \circ \theta =\varphi$;

(4) there exists a unique normal conditional expectation from M onto $M_{\varphi }$.

When M has separable predual and ϕ is both strictly semi-finite and faithful, Condition (1) is also equivalent to

(5) $M_{\varphi }$ contains a maximal abelian ^⁎-subalgebra of M.

Furthermore, if M has separable predual but has no type ${\mathrm{III}}_1$ part, then Condition (1) is satisfied for every strictly semi-finite weight ϕ. Moreover, a factor M with separable predual is not of type ${\mathrm{III}}_1$ if and only if the “Radon-Nikodym type property” as in Condition (2) holds for every strictly semi-finite weight $\varphi \in W\left(M\right)$.