Classification of hyperfinite factors up to completely bounded isomorphism of their preduals
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Uffe Haagerup
Abstract
In this paper we consider the following problem: When are the preduals of two hyperfinite (= injective) factors ℳ and 
(on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if ℳ is semifinite and 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 IIIλ-factors, where 0 < λ ≦ 1, have cb-isomorphic preduals.
© Walter de Gruyter Berlin · New York 2009
Articles in the same Issue
- Gröbner geometry of vertex decompositions and of flagged tableaux
- The Diophantine equation aX4 – bY2 = 1
- C*-algebras and self-similar groups
- Conjugate varieties with distinct real cohomology algebras
- Classification of hyperfinite factors up to completely bounded isomorphism of their preduals
- Ricci flow of almost non-negatively curved three manifolds
- Linear dependence in Mordell-Weil groups
Articles in the same Issue
- Gröbner geometry of vertex decompositions and of flagged tableaux
- The Diophantine equation aX4 – bY2 = 1
- C*-algebras and self-similar groups
- Conjugate varieties with distinct real cohomology algebras
- Classification of hyperfinite factors up to completely bounded isomorphism of their preduals
- Ricci flow of almost non-negatively curved three manifolds
- Linear dependence in Mordell-Weil groups
