1. Ultraproducts

Embedability
Problem 1.1.
[J. Peterson] Let $M$ be a II$_1$ factor with Property (T) and $N$ a II$_1$ factor with Haagerup property. If $M\subseteq N^\omega $, prove there exists $N_n\subseteq N$, finite dimensional, so that $M\subset \displaystyle\prod_{\omega} N_n\subseteq N^\omega $.
Remark. [J. Peterson] This is analogous to Shalom’s theorem for groups:
If $\Gamma $ has property (T) and $\Lambda $ has the Haagerup property so that $\Gamma\hookrightarrow \Lambda^\omega $, then there exists a sequence of finite groups $\Lambda_n\leqslant \Lambda $ so that $\displaystyle\Gamma\hookrightarrow \prod_\omega \Lambda_n $.


Fundamental Groups of Ultraproducts of II$_1$ Factors
Problem 1.2.
Find a II$_1$ factor $P$ so that $\mathcal{F}(P^\omega)\neq \mathbb{R} $.
Remark. If this is true, I think it follows that there are uncountably many elementary equivalence classes of nonMcDuff II_1 factors. T

Remark. It seems open whether M^t always embeds in M^\omega. T


NonIsomorphic Ultrapowers
Problem 1.3.
[A. Ioana] Find full type II$_1$ factors $P_1$ and $P_2$ so that $P_1^\omega\not\cong P_2^\omega $.
Remark. [R. Boutonnet] Let $\{M_\alpha\}_{\alpha\in \{0,1\}^\mathbb{N}} $ be the family of McDuff II$_1$ factors with pairwise nonisomorphic ultrapowers (see BCI15). Does the same conclusion hold for the family $M_\alpha* \mathbb{Z} $? What about $M_\alpha*N $ where $N $ is some “rigid” factor?


NonIsomorphic Ultrapowers III
Problem 1.4.
[C. Houdayer] Construct type III factors with nonisomorphic ultrapowers. 
Problem 1.5.
Is $\mathcal{R}^\omega\cong L(G) $ for some group $G$.
Remark. This question doesn’t make complete sense to me. Something that I think was asked in line with this was whether M^{op} is elementarily equivalent to M for any II_1 factor. T


Isomorphic Ultrapowers
Problem 1.6.
[S. Vaes] Find a class of $II_1$ factors with isomorphic ultrapowers.
Remark. [T. Sinclair] By Sela’s work (2000), $\mathbb{F}_n^\omega\cong \mathbb{F}_m^\omega $. However, there is an obstruction in a naive application of this result.


Elementary Equivalence of the Free Group Factors
Problem 1.7.
[T. Sinclair] Is $L(\mathbb{F}_n) $ elementarily equivalent to $L(\mathbb{F}_\infty) $?
Cite this as: AimPL: Classification of group von Neumann algebras, available at http://aimpl.org/groupvonneumann.