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1. Ultraproducts

    1. Embedability


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      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


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              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


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                          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


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                                  Problem 1.4.

                                  [C. Houdayer] Construct type III factors with non-isomorphic ultrapowers.


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                                      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


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                                              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


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                                                      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.