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

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