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) ?