Let $M$ be a type $II\_1$ factor von Neumann algebra, and let $G$ be a discrete group acting on $M$ which is free and ergodic. Is the crossed product von Neumann algebra $M \rtimes G$ type $II\_1$ factor? 



types of crossed product von Neumann algebras
One assumption that would guarantee that the crossed product is a factor is that the action be properly outer. Incidentally, I am not sure what you mean by ``ergodic'' in this setting.


The crossed product is automatically a finite von Neumann algebra (if $M$ is any von Neumann algebra and $G$ is a discrete group, then the projection from $L^2(M\rtimes G)$ onto $L^2(M)$ gives rise to a conditional expectation $E:M\rtimes G\to M$; if $M$ has a $G$-invariant trace it is immediate to verify that $\tau\circ E$ is a trace). Jon Bannon's comment does not apply in this case since you assume $M$ to be type II$\_1$ *factor* so that the trace on $M$ is unique and thus preserved by any automorphism of $M$. There are no trace-scaling automorphisms of II$\_1$ factors; they only exist for factors of type II$\_\infty$. Of course if $M$ is not a factor it is possible for $G$ to act non-trivially on the center $Z(M)$ and this can lead to crossed products which are not finite (e.g. take $M$ abelian). 


