Noncommutative ergodic theory of higher rank lattices · 2022-ana-204

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Question 1

Determine whether the following holds. Let $G$ be a connected simple real Lie group with trivial center and real rank at least $2$, and let $\Gamma<G$ be a lattice. For any discrete group $H$, let $\mathrm L(H)=\lambda_H(H)''$ be the von Neumann algebra generated by the left regular representation $\lambda_H$ of $H$ on $\ell^2(H)$. Then, for every discrete group $\Lambda$, an isomorphism of von Neumann algebras $\mathrm L(\Gamma)\cong\mathrm L(\Lambda)$ implies an isomorphism of groups $\Gamma\cong\Lambda$.

Source reference: Jesse Peterson, “Problems on rigidity in von Neumann algebras,” in Open problems, Vanderbilt University online problem list, 2024, Problem R.1 (Connes's rigidity conjecture for higher-rank lattices), https://math.vanderbilt.edu/peters10/problems.html.

Question 2

Determine whether the following holds. For $i\in\{1,2\}$, let $G_i$ be a connected simple real Lie group with trivial center and real rank at least $2$, and let $\Gamma_i<G_i$ be a lattice. For a discrete group $H$, let $\mathrm L(H)=\lambda_H(H)''$ be the von Neumann algebra generated by the left regular representation $\lambda_H$ of $H$ on $\ell^2(H)$. If $\mathrm L(\Gamma_1)\cong\mathrm L(\Gamma_2)$ as von Neumann algebras, then $G_1\cong G_2$ as Lie groups.

Source reference: Cyril Houdayer, “Noncommutative ergodic theory of higher rank lattices,” Proceedings of the International Congress of Mathematicians 2022, Vol. 4, EMS Press, 2023, pp. 3202–3223, Section 5 (Conjecture), DOI 10.4171/ICM2022/39, https://doi.org/10.4171/ICM2022/39.

Question 3

Determine whether, for all integers $m,n\geq 3$, the group von Neumann algebras $\mathrm L(\operatorname{PSL}_m(\mathbb Z))$ and $\mathrm L(\operatorname{PSL}_n(\mathbb Z))$ are isomorphic only if $m=n$, where $\mathrm L(H)=\lambda_H(H)''$ denotes the von Neumann algebra generated by the left regular representation of a discrete group $H$ on $\ell^2(H)$.

Source reference: Jesse Peterson, “Problems on rigidity in von Neumann algebras,” in Open problems, Vanderbilt University online problem list, 2024, Problem R.4, https://math.vanderbilt.edu/peters10/problems.html.

Question 4

Determine whether the following holds. For every countable discrete group $\Gamma$ with Kazhdan's property $(T)$ such that every nonidentity conjugacy class in $\Gamma$ is infinite, there are only finitely many group-isomorphism classes of discrete groups $\Lambda$ satisfying $\mathrm L(\Lambda)\cong\mathrm L(\Gamma)$, where $\mathrm L(H)=\lambda_H(H)''$ denotes the von Neumann algebra generated by the left regular representation of a discrete group $H$ on $\ell^2(H)$.

Source reference: Jesse Peterson, “Problems on rigidity in von Neumann algebras,” in Open problems, Vanderbilt University online problem list, 2024, Problem R.6, https://math.vanderbilt.edu/peters10/problems.html.

Question 5

For each integer $d\geq 2$, determine whether there exists $p\in[1,\infty]\setminus\{2\}$ such that the noncommutative $L^p$-space $L^p(\mathrm L(\operatorname{SL}_{2d-1}(\mathbb Z)))$, equipped with its canonical operator-space structure, has the operator approximation property; that is, whether its identity map is a stable point-norm limit of finite-rank maps. Here $\mathrm L(H)=\lambda_H(H)''$ is the group von Neumann algebra of a discrete group $H$.

Source reference: Mikael de la Salle, “Kakeya conjecture and High-Rank Lattice von Neumann algebras,” arXiv preprint, 2026, Question 1.1, arXiv:2602.14623, https://arxiv.org/abs/2602.14623.