We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.

中文翻译：

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冯诺依曼代数的 Banach-Mazur 稳定性

我们开始研究冯诺依曼代数相对于 Banach-Mazur 距离的扰动。我们首先证明类型分解是连续的，即如果两个冯诺依曼代数是接近的，那么它们各自的每个类型的和是接近的。然后我们证明，在其Hochschild 上同调群上的一些消失条件下，冯诺依曼代数是Banach-Mazur 稳定的，即任何足够接近的冯诺依曼代数实际上是Jordan ∗-同构的。这些消失的条件可能是空的。