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Let an action of a group $\Gamma$ on a manifold $M$ such that $L^{∞}(M)⋊Γ$ is a type $III$ factor.

André Henriques posted here the following comment :

I don't know the literature, so I can't point to a reference, but here's how things go:

Given an (let's say a.e. smooth) action of a group $Γ$ on a manifold $M$, you can form the bundle of densities $Ω^{top}_{>0}M$, which is a principal bundle with structure group $\mathbb{R}_{>0}$.
The action of $Γ$ on $M$ induces an action on $Ω^{top}_{>0}M$, and the von Neumann algebra $L^{∞}(M)⋊Γ$ is a type $III_{1}$ factor iff the action of $Γ$ on $Ω^{top}_{>0}M$ is ergodic.
If that action is not ergodic, the von Neumann algebra $L^{∞}(Ω^{top}_{>0}M)^{Γ}$ is equipped with an action of $\mathbb{R}_{>0}$ (coming from the action on $Ω^{top}_{>0}M$). This corresponds to a action of $\mathbb{R}_{>0}$ on some measure space $X$. If that action is transitive, it is equivalent to $\mathbb{R}_{>0}$ acting on $\mathbb{R}_{>0}/\mathbb{Z}^{λ}$ for some $λ∈(0,1)$, and the factor $L^{∞}(M)⋊Γ$ is of type $III_{λ}$. Otherwise, $L^{∞}(M)⋊Γ$ is of type $III_{0}$.

Is there a reference for this result (or something close to it) ?

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This question has nothing to do with manifolds - you are just talking about the classification of type III actions with quasi-invariant measure in terms of their Radon-Nikodym cocycles. It should be contained, for instance, in the old "Indian" book of Klaus Schmidt.

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Thank you for the answer. Is it the following book ? Cocycles of ergodic transformation groups. Lecture Notes in Mathematics, Vol. 1, MacMillan (India) 1977. PDF. Where is this result on the book, which page ? Is it proved only for the hyperfinite case, or in general ? –  Sébastien Palcoux Sep 27 '13 at 19:16
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Yes - that's the one I meant, but he doesn't discuss this stuff there. Another standard reference is the book by Hamachi and Osikawa MR0617740, which surely covers it, but I don't have it with me right now. You can find these statements, for instance, in Section 2 of faculty.uml.edu/vprasad/Abelian%20cocycles.pdf (in particular, p. 191). –  R W Sep 27 '13 at 20:36
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