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 $G \curvearrowright (Y \times Y, \mu \times \mu)$?

4$\begingroup$ If the $X$ action is ergodic, then the $Y$ action must be weakmixing and hence the $X$ action must be weakmixing. (That is you cannot find a square root like this if the $X$ action is ergodic but not weakmixing). $\endgroup$ – Anthony Quas Nov 24 '15 at 6:16

$\begingroup$ 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. $\endgroup$ – Vladimir Nov 24 '15 at 8:04

2$\begingroup$ 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) $\endgroup$ – Anthony Quas Nov 24 '15 at 9:10

$\begingroup$ Thanks again Anthony. And yes  my choice of title is a little confusing. $\endgroup$ – Vladimir Nov 24 '15 at 15:15

2$\begingroup$ Perhaps "Cartesian square root" would be a better terminology. $\endgroup$ – Lee Mosher Nov 26 '15 at 14:51