  In this paper we consider the following problem: When are the preduals of two
hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces)
cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is
semifinite and $\N$ 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 $\M$ 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 $0< \lambda\leq 1$, have cb-isomorphic preduals.
