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I asked a question over on Math.Stackexchange with the same title, but I didn't get any activity over there, which made me think that the question would be better suited for MathOverflow. I suppose you members of MathOverflow will be the judge.

Let $G$ be a countable, infinite conjugacy class (ICC) group (i.e., if $g \in G \setminus \{1\}$, then the conjugacy class of $g$ is infinite). This will ensure that the group von Neumann algebra $L(G)$ is a so-called $II_1$ factor von Neumann algebra.

Recall that a $II_1$ factor von Neumann algebra $M$ is said to be a McDuff factor provided it tensorially absorbs the hyperfinite $II_1$ factor; i.e., $M \cong M \otimes R$, where $R$ is the hyperfinite $II_1$ factor (one way of constructing $R$ is to take a discrete, amenable, ICC group $\Gamma$ and $L(\Gamma)$ will be the hyperfinite $II_1$ factor).

In Inner amenability, property Gamma, McDuff II_1 factors and stable equivalence relations, Deprez and Vaes define a McDuff group as a group $G$ which admits a free ergodic probability measure preserving action $\alpha$ on a measure space $(X, \mu)$ such that the crossed product $II_1$ factor $L^{\infty}(X, \mu) \rtimes_{\alpha} G$ is a McDuff factor.

I am relatively new to this subject, so I hope my question isn't too elementary, but what is the connection/relation between these two notions? More specifically, what is the connection (if any) between $L(G)$ being a McDuff factor and $L^{\infty}(X, \mu) \rtimes_{\alpha} G$ being a McDuff factor?

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    $\begingroup$ In arxiv.org/abs/1205.5123, Kida constructs an icc group $G$ that is a McDuff group (in the sense of admitting a free, ergodic, probability measure preserving action with the crossed product being a McDuff $II_1$ factor), but such that $L(G)$ does not even have property Gamma. Certainly, $L(G)$ is not a McDuff $II_1$ factor. $\endgroup$ Dec 19 '20 at 16:21
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    $\begingroup$ The converse is not clear to me and I think that it would require new ideas to construct an icc group such that $L(G)$ is a McDuff $II_1$ factor and yet, $G$ is not a McDuff group. The difficulty comes from the fact that most concrete constructions of groups $G$ with $L(G)$ being a McDuff $II_1$ factor are such that this McDuff property is witnessed by concrete central sequences living in smaller and smaller "parts" of $G$. Often you can construct an action of $G$ where these "smaller and smaller parts" asymptotically act trivially, implying that the crossed product is a McDuff $II_1$ factor. $\endgroup$ Dec 19 '20 at 16:26

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