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

Let a"' be a type II1 factor with separable predual and tau be a normal faithful tracial state of a"'. We first show that the set of subfactors of a"' with property I", the set of type II1 subfactors of a"' with similarity property and the set of all McDuff subfactors of a"' 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 a"' 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 is a finite von Neumann algebra with a normal faithful trace, the set of all von Neumann subalgebras of such that is rigid is closed in the Hausdorff metric d (2).