When are crossed product von Neumann algebras injective?

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?

• Yes, it's also amenability of the action. Although the proofs are more technical. I think this was shown originally by Zimmer. Apr 11, 2020 at 12:48

• Thats a tricky question. Brown-Ozawa is more centered in the case of $C^\ast$-algebras than von Neumann algebras. The book of Zimmer on ergodic theory of semisimple groups is measure-theoretical but doesn't center around von Neumann algebras directly. Apr 11, 2020 at 23:17