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

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

2007-06-23

arxiv.org/abs/0706.3463v2

Mathematics

Operator Algebras

30 pages

Scientific paper

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.

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