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Classification of hyperfinite factors up to completely bounded isomorphism of their preduals. (English) Zbl 1178.46056

Let \(\mathcal{M}\) and \(\mathcal{N}\) be hyperfinite (=injective) factors on separable Hilbert spaces. The present paper is devoted to the following problem: When are the preduals of \(\mathcal{M}\) and \(\mathcal{N}\) cb-isomorphic (i.e., isomorphic as operator spaces)?
E.Christensen and A.Sinclair proved in [“Completely bounded isomorphisms of injective von Neuman algebras”, Proc.Edinb.Math.Soc., II.Ser.32, No.2, 317–327 (1989; Zbl 0651.46059)] that all infinite-dimensional hyperfinite factors with separable preduals are cb-isomorphic. But if one turns to preduals of von Neumann algebras, the situation is rather different. In [“Banach embedding properties of non-commutative \(L^p\)-spaces” (Mem.Am.Math.Soc.776) (2003; Zbl 1045.46039)], H.P.Rosenthal, F.A.Sukochev and the first-named author proved that if \(\mathcal{M}\) is a \(II_1\)-factor and \(\mathcal{N}\) is a properly infinite von Neumann algebra, then their preduals are not cb-isomorphic. They also proved that all hyperfinite type \(III_{\lambda}\)-factors, where \(0 < \lambda \leq 1\), have cb-isomorphic preduals.
In the present work, the authors show that if \(\mathcal{M}\) is semifinite and \(\mathcal{N}\) is type \(III,\) then their preduals are not cb-isomorphic. Moreover, they construct a one-parameter family of hyperfinite type \(III_0\)-factors with mutually non cb-isomorphic preduals, and give a characterization of those hyperfinite factors \(\mathcal{M}\) whose preduals are cb-isomorphic to the predual of the unique hyperfinite type \(III_1\)-factor.

MSC:

46L36 Classification of factors
46L07 Operator spaces and completely bounded maps
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