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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.

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Please clarify what R is supposed to be in your question - given that R sometimes denotes the hyperfinite II_1 factor. –  Yemon Choi Jan 10 '11 at 0:21
    
$R$ is von Neumann algebra –  alg1oper Jan 11 '11 at 23:08
    
$\sigma_\phi$ is modular automorphisms group associated with weight $\phi$ –  alg1oper Jan 11 '11 at 23:10
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