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4. C$^*$ Algebras

    1. C$^*$ Super-Rigidity for Non-Discrete Groups Acting on Trees

      Problem 4.1.

      [S. Raum] Does $C^*_r(G_1)\cong C^*_r(G_2) $ imply that $G_1\cong G_2 $?
        1. Remark. [Sven Raum] A more instance of this problem is: can the finite permutation group F be recovered from the reduced group C*-algebra of a Burger-Moses group U(F)?
            • Problem 4.2.

              [S. Raum] Find torsion-free simple groups such that $\Gamma_1\not\cong \Gamma_2 $ and $C^*_r(\Gamma_1)\cong C^*_r(\Gamma_2) $.
                • Problem 4.3.

                  Is $C^*_r(\mathbb{F}_\infty) $ finitely generated?
                    • Problem 4.4.

                      [S. Raum] Is $\left(\mathbb{Z}/2\mathbb{Z} \right)^{(\mathbb{F}_2)}\rtimes (\mathbb{F}_2\times\mathbb{F}_2) $ C$^*$ super-rigid?
                        1. Remark. [S. Raum] While the resulting C$^*$ algebra does not have a unique trace, all traces can be described.
                            • Problem 4.5.

                              [S. Vaes] Prove cocycle superrigidity for Bernoulli actions of property (T) groups with totally disconnected target groups.

                                  Cite this as: AimPL: Classification of group von Neumann algebras, available at http://aimpl.org/groupvonneumann.