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](https://mathoverflow.net/questions/141577/a-non-hyperfinite-type-iii-factor-from-an-action-of-the-free-group-on-the-circle#comment365395_141577) 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) ?



Reference request for a type III action of a group on a manifold
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.


