
## 1. Ultraproducts

1. ### 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 $M\subset \displaystyle\prod_{\omega} N_n\subseteq N^\omega$.
1. 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 $\mathcal{F}(P^\omega)\neq \mathbb{R}$.
1. Remark. If this is true, I think it follows that there are uncountably many elementary equivalence classes of non-McDuff II_1 factors. -T
• Remark. It seems open whether M^t always embeds in M^\omega. -T
• ### Non-Isomorphic Ultrapowers

#### Problem 1.3.

[A. Ioana] Find full type II$_1$ factors $P_1$ and $P_2$ so that $P_1^\omega\not\cong P_2^\omega$.
1. Remark. [R. Boutonnet] Let $\{M_\alpha\}_{\alpha\in \{0,1\}^\mathbb{N}}$ be the family of McDuff II$_1$ factors with pairwise non-isomorphic 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?
• ### Non-Isomorphic Ultrapowers III

#### Problem 1.4.

[C. Houdayer] Construct type III factors with non-isomorphic ultrapowers.
• #### Problem 1.5.

Is $\mathcal{R}^\omega\cong L(G)$ for some group $G$.
1. 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.
1. 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.