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 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 III0-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 III1-factor. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic and more recently, Rosenthal, Sukochev and the first-named author proved that all hyperfinite type III lambda-factors, where 0 < lambda <= 1, have cb-isomorphic preduals.