TAILIEUCHUNG - Báo cáo toán học: "Operator systems and their application to the Tomita-Takesaki theory "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hệ thống điều hành và ứng dụng lý thuyết Tomita-Takesaki. | J. OPERATOR THEORY 2 1979 169-209 Copyright by INCREST 1979 OPERATOR SYSTEMS AND THEIR APPLICATION TO THE TOMITA-TAKESAKI THEORY s. L. WORONOWICZ 0. INTRODUCTION Let M be a von Neumann algebra and M be the predual of M. Already in the elementary theory of von Neumann algebras one meets many natural operations involving elements of M and M . The best known example is the multiplication of normal functionals by elements of the algebra M X M 3 3 A a - Aae M for details see 3 . Another example is provided by the polar decomposition of normal functionals. Denoting by phase a the unitary appearing in the polar decomposition of a e M we introduce a natural mapping phase . We shall prove in this paper that the basic operations investigated in the Tomita-Takesaki theory such as the action of the modular groups and the Radon-Nikodym derivatives can be expressed by the elementary mappings mentioned above. To this end we consider the similar problem in the purely algebraic setting where the faithful states are replaced by selfadjoint positive nondegenerate operators. Let a family of selfadjoint positive nondegenerate operators acting on a Hilbert space H. We consider the following two families of mappings B H 3 A - ÁỊAA 6 B H where a 1 e A 16 R and B H 3 A - Phase d ada 4- bAp e BỢT where a b are complex numbers with positive real parts Phase Xdenotes the unitary factor appearing in the polar decomposition of X and the addition of operators denoted by 4- is understood in the sense explained in 4 Ch VI . The mappings 0 1 are typical for the Tomita-Takesaki theory where as 0 2 are closely related to the elementary operations involving elements of M and M described in the be-gining. 170 s. L. WORONOWICZ The main result of this paper asserts that the families 0 1 and 0 2 are connected each mapping belonging to the first family can be expressed by mappings of the second family and vice versa. To make this statement precise we consider systems of the form S M 4a 6

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