tailieunhanh - Báo cáo toán học: "Tomita-Takesaki theory for Jordan Algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tomita Takesaki lý thuyết đại số Jordan. | Copyright by INCREST 1984 J. OPERATOR THEORY 11 1984 348 364 TOMITA-TAKESAKI THEORY FOR JORDAN ALGEBRAS UFFE HAAGERUP and HARALD HANCHE-OLSEN 1. INTRODUCTION The present paper is an attempt to generalize the Tomita-Takesaki Theory to JBW-algebras the Jordan algebra analogues of JF -algebras cf. 19 . If Jt is a iv -algebra and p is a normal faithful state on Ji it is not possible to distinguish ơi and ƠV_Ị solely in terms of the Jordan product X o y -ị- xy yx . Indeed if 2 one computes the modular automorphism group of p considered as a state on the opposite algebra . op . the vector space Ji equipped with the product x y - yx one gets ơ1_t instead of ơi. However it turns out that the one parameter family PĨ ơ- can be generalized in a natural way to normal states on JBW-algebras cf. Theorem and Proposition . The construction of pl makes use of the full structure theory of JBW-algebras. We reduce the general case to the following two cases I There exists a trace Ĩ on the JBW-algebra M such that p x r xo for some invertible h e M . II There exists a von Neumann algebra Ji and an involutive anti-automorphism 0 of JI such that M is isomorphic to the self-adjoint part of Ji9 x e M I p x x . In the case of jps-algebras the give a very useful characterization of the modular automorphism group al- No analogue of the seems to be available for pl. However it is possible to give a characterization of pl which does not involve structure theory. We prove that the one parameter family Pt pl is characterized by the following five conditions 344 UFFE HAAGERUP and HARALD HANCHE-OLSEN i The map t - pt x is w -continuous for all xe M ii Each p is normal positive and preserves the unit . . 1 _ iii p0 - idM and psp Ý ps ps_t s t e R iv p p ữ b p ữop ố a b 6 v The bilinear form on M defined by 00 s x J ị p p ứ oí cosh ff -1d -CO is a self-polar form on M in the sense of Connes 6 and Woronowicz 26 . It follows from this result that for .