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

Uffe Haagerup; Magdalena Musat

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arXiv · 0706.3463

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

Uffe Haagerup·Magdalena Musat

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Abstract

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$_λ$-factors, where $0< λ\leq 1$, have cb-isomorphic preduals.

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BibTeX RIS

Uffe Haagerup, Magdalena Musat. 2007-08-08. Classification of hyperfinite factors up to completely bounded isomorphism of their preduals. https://arxiv.org/abs/0706.3463

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