Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \rightarrow \operatorname{Aut}(X)$ is the group homomorphism that defines the $G$-action. Under our assumption, the von Neumann algebra $L^{\infty}(X, \mu) \overline{\rtimes}_{\alpha} G$ is a factor. One can refer to this post for more details. Does anyone know references that discuss necessary conditions for the factor $L^{\infty}(X, \mu) \overline{\rtimes}_{\alpha} G$ to be hyperfinite?
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$\begingroup$ Assuming you meant $G$ to be a countable discrete group, the measure is a probability measure (or a finite measure in general - it doesn’t make any difference), and the action is measure-preserving, then it is a quite standard exercise to show the factor is hyperfinite iff $G$ is amenable. $\endgroup$– David GaoCommented Oct 13 at 2:21
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$\begingroup$ (And the easy direction is to show amenability of $G$ follows from hyperfiniteness of the factor. That is a direct consequence of the fact that a von Neumann subalgebra of a hyperfinite tracial von Neumann algebra is always hyperfinite. So if you just want necessary conditions, that’s already enough.) $\endgroup$– David GaoCommented Oct 13 at 2:29
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$\begingroup$ Thanks for your inputs. It would be fine if $G$ is assumed to be a countable discrete group and $\mu$ is a probability measure. However, I prefer not to only consider measure-preserving group action. If necessary, you could only consider the non-singular action (namely $g\mu\sim\mu$ for all $g\in G$), and references related to this case are also welcomed. $\endgroup$– Kaku SeigaCommented Oct 13 at 4:16
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$\begingroup$ It might be better to specifically say that in the post in that case, as otherwise people might assume the action is pmp, since that is the most well-studied case. Also, assuming the action is non-singular is necessary, as otherwise you don’t even have an action on $L^\infty(X, \mu)$. I don’t know of any answer to the question though, given that I’m only familiar with the pmp case. $G$ being amenable is still sufficient, that I know, but probably not necessary, I would guess. $\endgroup$– David GaoCommented Oct 13 at 4:57
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$\begingroup$ Got it. Will edit the post now. $\endgroup$– Kaku SeigaCommented Oct 13 at 13:10
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The hyperfiniteness you are asking about is equivalent to amenability of the action; it goes back to Zimmer and essentially was the motivation for his definition of amenable actions. As for the latter, there is a lot of equivalent defintions (the original definition of Zimmer not being the simplest one to work with). In a sense, the most constructive one is due to Renault (actually, in a more general groupoid context) and stems from Day's approximate characterization of amenable groups: a non-singular action $G \curvearrowright (X, \mu)$ is amenable if there is a sequence of maps $x\mapsto \lambda_n^x$ from $X$ to the space of probability measures on $G$ which is asymptotically equivariant in the sense that $\|\lambda_n^{gx}-g\lambda_n^x\|\to 0$ for any $g\in G$ and almost all $x\in X$.
Of course, any action of an amenable group is amenable. However (which was another motivation of Zimmer), there are numerous examples of amenable actions of non-amenable groups. The action space in these situations can usually be considered as a boundary of the group, be it in the topological category (for instance, the boundary action of a hyperbolic group) or in the measure category (for instance, the action on the Poisson boundary). The simplest example is the action of a finitely generated free group on its boundary (the space of infinite irreducible words $\equiv$ the space of ends $\equiv$ the Gromov boundary) endowed with any quasi-invariant measure.
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$\begingroup$ Thank you for your references. I came across the same definition of amenable group action in Ozawa's textbook (something called $C^*$-algebra ... finite dimensional approximation), and, if I recall, the definition in your answer is the one for a general amenable group action. It is new to me that $L^{\infty}(X, \mu) \overline{\rtimes}_{\alpha} G$ being hyperfinite is equivalent to the amenability of the group action. I currently do not have access to mathscinet but will look into papers written by Zimmer and Renault. $\endgroup$ Commented Oct 14 at 0:00