The Effros–Maréchal Topology in the Space of von Neumann Algebras, II

We continue our study [9] of the Effros–Maréchal topology on the space vN(H) of all von Neumann algebras acting on a fixed separable Hilbert space H. In particular, we prove that factors of each of the types: II1, II∞, and (for fixed λ∈[0, 1]) IIIλ, form a dense subset of vN(H). Moreover, the density of type I-factors (or, equivalently, of injective factors) in vN(H), is proved to be equivalent to famous conjectures by A. Connes and E. Kirchberg.