tailieunhanh - Báo cáo toán học: "Prime actions of compact abelian groups on the hyperfinite type II_1 factor "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Thủ tướng Chính phủ hành động của các nhóm nhỏ gọn abelian trên các yếu tố II_1 loại hyperfinite. | J. OPERATOR THEORY 9 1983 181-186 Copyright by INCREST 1983- PR1ME ACTIONS OF COIMPACT ABELIAN GROUPS ON THE HYPERFINITE TYPE II FACTOR V. F. R. JONES 1. This note is intended as an appendix to Ocneanu s paper on amenable group actions. It seems likely that using Ocneanu s results one may obtain a satisfactory classification of all actions of a separable compact abelian group on the hyperfinite type III factor R. Finite group actions are classified in 6 . Here we restrict ourselves to prime actions . ones whose fixed point algebra is a factor because the classification is easy to understand and using Ocneanu s work and Takesaki duality easy to prove. An understanding of abelian actions will certainly be important for the classification of classical group actions as the maximal torus is a compact abelian group. In this paper action will mean faithful action. An ergodic action is certainly prime and we shall show that our classification in this case is the same as that of 8 . The author would like to thank M. Takesaki for the elegant way of avoiding excessive use of cocycles in the proof of the theorem and A. Ocneanu for many interesting discussions of his work. The work for this paper was done at the Timisoara Conference in June 1980 and at lhe University of Geneve. 2. The characteristic invariant was introduced in 5 and 6 but for an abelian group acting on a factor it takes a particularly simple form which we shall now describe. Let M be a factor A a discrete abelian group and a A - Aut M an action. Let N A be the subgroup of A acting by inner automorphisms and choose for each h G N a unitary vh in M with 01ft - Ad vh. Then since M is a factor there are scalars Ẳ g h gT ze C z 1 with otg Vi Ả g h vh. The function 2 A X N - T so defined is independent of the choice of the lift s and is a homomor 182 V. F. R. JONES phism in both variables . is bilinear for the Z-module structure . The restriction of z to N X N is antisymmetric . A z h 1 for h e N. With this