标题： Weyl群和冯·诺依曼代数的刚性 显示英文标题

标题： Weyl groups and rigidity of von Neumann algebras

摘要： 设$G$为一个中心平凡的非紧致半单代数群，$S < G$为一个极大分裂环，$H < G$为$S$在$G$中的中心化子，$\Gamma < G$为一个不可约格。 考虑与非奇异作用$\Gamma \curvearrowright G/H$关联的群测度空间冯·诺依曼代数$\mathscr M = \operatorname{L}(\Gamma \curvearrowright G/H)$，并将群冯·诺依曼代数$M = \operatorname{L}(\Gamma)$视为冯·诺依曼子代数$M \subset \mathscr M$。 We show that the group $\operatorname{Aut}_M(\mathscr M)$ of all unital normal $\ast$-automorphisms of $\mathscr M$ acting identically on $M$ is isomorphic to the Weyl group $\mathscr W_G$ of the semisimple algebraic group $G$. Our main theorem is a noncommutative analogue of a rigidity result of Bader-Furman-Gorodnik-Weiss for group actions on algebraic homogeneous spaces and moreover gives new insight towards Connes' rigidity conjecture for higher rank lattices. 显示英文摘要

摘要： Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $\Gamma < G$ an irreducible lattice. Consider the group measure space von Neumann algebra $\mathscr M = \operatorname{L}(\Gamma \curvearrowright G/H)$ associated with the nonsingular action $\Gamma \curvearrowright G/H$ and regard the group von Neumann algebra $M = \operatorname{L}(\Gamma)$ as a von Neumann subalgebra $M \subset \mathscr M$. We show that the group $\operatorname{Aut}_M(\mathscr M)$ of all unital normal $\ast$-automorphisms of $\mathscr M$ acting identically on $M$ is isomorphic to the Weyl group $\mathscr W_G$ of the semisimple algebraic group $G$. Our main theorem is a noncommutative analogue of a rigidity result of Bader-Furman-Gorodnik-Weiss for group actions on algebraic homogeneous spaces and moreover gives new insight towards Connes' rigidity conjecture for higher rank lattices.