Classification of hyperfinite factors up to completely bounded isomorphism of their preduals

In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\text{M}$ and $\mathbb{N}$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\text{M}$ is semifinite and $\mathbb{N}$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III${}_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\text{M}$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III${}_1$-factor. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic. More recently Rosenthal, Sukochev and the first-named author proved that all hyperfinite type III${}_{\lambda }$-factors, where $0<\lambda \le 1$, have cb-isomorphic preduals.