Abstract We show that unitary groups of II 1 {\mathrm{II}_{1}} factors and of properly infinite von Neumann algebras have uncountable cofinality and the Bergman property. In particular, we obtain a short alternative proof for the strong uncountable cofinality of U ⁢ ( ℓ 2 ⁢ ( ℕ ) ) {\mathrm{U}(\ell^{2}(\mathbb{N}))} , which was first proven by Ricard and Rosendal.

中文翻译：

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冯诺依曼代数酉群的强不可数共尾性

摘要 我们证明了 II 1 {\mathrm{II}_{1}} 因子的酉群和适当无穷的冯诺依曼代数具有不可数的共尾性和伯格曼性质。特别是，我们获得了 U ⁢ ( ℓ 2 ⁢ ( ℕ ) ) {\mathrm{U}(\ell^{2}(\mathbb{N}))} 的强不可数共尾性的简短替代证明，即Ricard 和 Rosendal 首先证明了这一点。