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?
Best Answer
The papers of Zimmer are good references, but they only cover the commutative case (the case of an action on a measure space). For actions of (locally compact) groups on general von Neumann algebras, amenability is defined and studied by Claire Anantharaman-Delaroche in the following papers:
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann.(French. English summary)Math. Scand.45(1979), no.2, 289–304. https://www.mscand.dk/article/view/11844
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann. II.(French. English summary)[Amenable action of a locally compact group on a von Neumann algebra. II] Math. Scand.50(1982), no.2, 251–268. https://www.mscand.dk/article/view/11958
In the first paper it is already proved that the crossed product $M\bar\rtimes_\alpha\Gamma$ if injective if and only if $M$ is injective and the action $\alpha$ is amenable. Here $\Gamma$ is a discrete group. For general locally compact groups only one direction of this holds.
For actions of (discrete) groups on $C^*$-algebras, amenability is defined and studied in the follow up paper:
Claire Anantharaman-Delaroche, Systèmes dynamiques non commutatifs et moyennabilité.(French)[Noncommutative dynamical systems and amenability] Math. Ann.279(1987), no.2, 297–315. https://link.springer.com/article/10.1007/BF01461725
In this paper it is proved that for an action $\alpha$ of a (discrete) group $\Gamma$ on a $C^*$-algebra $A$, the crossed product $A\rtimes_\alpha \Gamma$ is nuclear if and only if $A$ is nuclear and $\alpha$ is amenable.
Amenable actions of locally compact groups were only defined and studied recently in the preprint (still not published):
https://arxiv.org/abs/2003.03469 Amenability and weak containment for actions of locally compact groups on $C^*$-algebras, by Alcides Buss, Siegfried Echterhoff, Rufus Willett
Further references, and historical background, can be found in that preprint.