# Cartesian square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such that $G \curvearrowright (X,\nu)$ is isomorphic to the diagonal action $G \curvearrowright (Y \times Y, \mu \times \mu)$?

• If the $X$ action is ergodic, then the $Y$ action must be weak-mixing and hence the $X$ action must be weak-mixing. (That is you cannot find a square root like this if the $X$ action is ergodic but not weak-mixing). – Anthony Quas Nov 24 '15 at 6:16
• Thanks! For amenable groups (at least) it seems like it should be possible to take the square root of i.i.d. actions. I wonder if more can be said. – Vladimir Nov 24 '15 at 8:04
• The terminology "square root" is slightly difficult because Ornstein used it to mean (for $\mathbb Z$ actions) a transformation $S$ such that $S^2=T$. For i.i.d., this seems straightforward as you say (just using entropy) – Anthony Quas Nov 24 '15 at 9:10
• Thanks again Anthony. And yes - my choice of title is a little confusing. – Vladimir Nov 24 '15 at 15:15
• Perhaps "Cartesian square root" would be a better terminology. – Lee Mosher Nov 26 '15 at 14:51