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

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) ?