
## 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.