Banach–Mazur stability of von Neumann algebras

Abstract

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.