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A classification of real-line group actions with faithful Connes--Takesaki modules on hyperfinite factors

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OASW01 - Structure of operator algebras: subfactors and fusion categories

  We classify certain real-line-group actions on (type III ) hyperfinite factoers, up to cocycle conjugacy. More precisely, we show that an invariant called the Connes—Takesaki module completely distinguishs actions which are not approximately inner at any non-trivial point. Our classification result is related to the uniqueness of the hyperfinite type III 1 factor, shown by Haagerup, which is equivalent to the uniquness of real-line-group actions with a certain condition on the hyperfinite type II{\infty} factor. We classify actions on hyperfinite type III factors with an analogous condition. The proof is based on Masuda—Tomatsu's recent work on real-line-group actions and the uniqueness of the hyperfinite type III _1 factor.
 

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