tailieunhanh - Báo cáo toán học: "Factors of type III_1, property $L'_\lambda$ and closure of inner automorphisms "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các yếu tố của loại III_1, tài sản $ L'_ \ lambda $ và đóng cửa bên trong automorphisms. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 189-211 FACTORS OF TYPE 1IIX PROPERTY L Ả AND CLOSURE OF INNER AUTOMORPHISMS A. CONNES INTRODUCTION Uffe Haagerup has succeeded to prove that any hyperfinite factor of type IirT has a trivial bicentralizer II 12 . This result together with an unpublished result of the author announced in 3 solves the longstanding problem of classification of hyperfinite factors of type lllj. ft proves that they are all isomorphic to the Araki-Woods factor 7 cf. I and completes the classification of hyperfinite factors cf. 3 for a review . Our aim in this paper is to give the details of the proof of the implication M hyperfinite HL with trivial bicentralizer M isomorphic to . Uffe Haagerup has found another proof of this implication and his proof is more direct than ours. Thus the only excuse for presenting our proof is that it gives a new characterization of property L cf. Part II and of the closure of In13 for M a factor of type III cf. Part III . Also it is nice to know that the automorphism approach can be made to work in all cases. The idea of our proof is the following. By previous results 4 one knows that any hyperfinite factor N of type III 2 1 is isomorphic to the Araki-Woods factor 7 first analyzed by R. Powers 13 1 . Let then T and ơ ơ í log 2 be a modular automorphism of a given hyperfinite factor M of type IIIj it follows easily that N R where N Mxi z is the crossed product of M by ơ ơỹ. Let 0 be the dual action of S on N one has cf. 14 M K Nx gSỵ R .xigSi. This reduces the original problem to the classification of certain actions of S1 on R in fact of those actions for which mod ớ t Vie S . Here mod a for a e AutN is the action of a on the flow of weights of N. Since V is of type IIIẤ the automorphisms of its flow of weights are parametrized by s1. í 90 A. CONNES Now given two actions Ỡ and Ớ as above one can form 0 Õ where ÕỊ 0Lt V í G s1. This tensor product a ớ ỡ is an action of s1 on R Rị X R such that 1 7 A XI