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Classification of subfactors and their endomorphisms. (English) Zbl 0865.46044

Regional Conference Series in Mathematics. 86. Providence, RI: American Mathematical Society (AMS). x, 110 p. (1995).
Classification results for inclusions of weakly separable von Neumann factors of finite Jones index in terms of the standard invariant are presented. The main theorem 5.1 states that under a suitable condition, the inclusion of \(N\) by \(M\) is isomorphic to the tensor product of the model type \(\text{II}_1\) inclusion (constructed from the standard invariant) with \(M\). The main assumptions are of two types. First the graph of the subfactor is assumed to be strongly amenable, i.e. to be ergodic and to satisfy a Følner-type condition. Second, the inclusion is to be approximately inner and centrally free. The condition on the graph is automatically satisfied when it is finite, i.e. the inclusion has finite depth. The approximately inner and centrally free properties are satisfied when \(N\) and \(M\) are hyperfinite, provided that the modular group of \(M\) is “independent” from the standard invariant, a situation that is automatically satisfied, for example, for a finite depth inclusion of hyperfinite type \(\text{III}_1\) factors. Thus the main application of this general result is that a finite depth inclusion of hyperfinite type \(\text{III}_1\) factors with finite index is completely determined by the standard invariant and is isomorphic to the tensor product of the standard type \(\text{II}_1\) inclusion with \(M\).
Reviewer: H.Araki (Kyoto)

MSC:

46L35 Classifications of \(C^*\)-algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
46L37 Subfactors and their classification
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