I am looking for a good reference for existence of the trace on $R\rtimes_{\sigma_\phi}{\mathbb R} $ for $\phi$ being f.s.n. weight. For the $\phi$ being f. strictly s.n. weight it is probably van Daele book For the integrable $\phi$  Connes Takesaki flow of weights The Haagerups paper on Operator valued weights II implies lack of operator valued weight for the case when R is type III.
