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

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 Mc Duff subfactors of L are open and closed in the Hausdorff metric d2 induced by the trace norm; then we show that the set of all hyperfinite von Neumann subalgebras of L is closed in d2. We also consider the connection of perturbation of operator algebras under d2 with the fundamental group and the generator problem of type II1 factors. When M is a finite von Neumann algebra with a normal faithful trace,the set of all von Neumann subalgebras B of M such that B  M is rigid is closed in the Hausdorff metric d2.