Let $\alpha$ be a continuous action of a discrete group $\Gamma$ on a von Neumann algebra $\mathcal{M}$. We can build the corresponding crossed product von Neumann algebra $\mathcal{N}:=\mathcal{M} \overline{\rtimes}_\alpha \Gamma$.
It is well-known that in the $C^\ast$-algebraic setting nuclearity of the corresponding crossed product can be characterized by the amenability of the action. The analogeous question for von Neumann algebras would be the question for injectivity.
I'm hence wondering when $\mathcal{N}$ is an injective von Neumann algebra. Are there any results about that, maybe even a characterization of injective von Neumann algebras arising from the crossed product construction? If no, what about the case where $\mathcal{M}$ is abelian?
