
## 3. Structural Properties of von Neumann Algebras

1. ### McDuff

#### Problem 3.1.

[C. Houdayer] Assume that $M\cong P_1\bar\otimes P_2$ is McDuff. Does it follow that $P_1$ or $P_2$ is McDuff?
• #### Problem 3.2.

[S. Popa] If $\beta_1^{(2)}(\Gamma)\neq 0$, then does it follow that $\Gamma$ is Cartan rigid.
• #### Problem 3.3.

[S. Raum] Let $G$ is a totally disconnected group and $K\subseteq G$ compact.
1. When do the $K$ invariant vectors in $L^2(K\backslash G) \subset L^2(G)$ generate $L^2(G)$.
2. Describe criteria for the factoriality of $L(K\backslash G / K)$
• #### Problem 3.4.

[S. Vaes]
1. Find two trace-scaling automorphisms of $L(\mathbb{F}_\infty)\bar\otimes \mathbb{B}(\ell^2)$ that are not outer conjugate.
2. Find two non-isomorphic type III$_\lambda$ factors with discrete core $L(\mathbb{F}_\infty)\bar\otimes \mathbb{B}(\ell^2)$.
• #### Problem 3.5.

[org.aimpl.user:rdesantiago@math.ucla.edu] Classify Cartan subalgebras of a II_$1$ factor $M$ when $M$ does not have a unique Cartan.

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