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

TL;DR

This paper investigates when the preduals of hyperfinite factors are cb-isomorphic, revealing distinctions based on type and constructing examples of non-isomorphic preduals, thus advancing the classification of operator space structures.

Contribution

It provides a classification of hyperfinite factors' preduals up to cb-isomorphism, including new examples and characterizations for type III factors.

Findings

Preduals of semifinite and type III factors are not cb-isomorphic.

Constructed a family of hyperfinite type III$_0$-factors with non-cb-isomorphic preduals.

Characterized hyperfinite factors whose preduals are cb-isomorphic to the type III$_1$-factor predual.

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$_\lambda$-factors, where $0< \lambda\leq 1$, have cb-isomorphic preduals.