On the properties of some sets of von Neumann algebras under perturbation

Abstract

Let ℒ be a type II1 factor with separable predual and τ be a normal faithful tracial state of ℒ. We first show that the set of subfactors of ℒ with property Γ, the set of type II1 subfactors of ℒ with similarity property and the set of all McDuff subfactors of ℒ are open and closed in the Hausdorff metric d 2 induced by the trace norm; then we show that the set of all hyperfinite von Neumann subalgebras of ℒ is closed in d 2. We also consider the connection of perturbation of operator algebras under d 2 with the fundamental group and the generator problem of type II1 factors. When Open image in new window is a finite von Neumann algebra with a normal faithful trace, the set of all von Neumann subalgebras Open image in new window of Open image in new window such that Open image in new window is rigid is closed in the Hausdorff metric d 2.