TAILIEUCHUNG - Báo cáo toán học: "T-theorem for L^p-spaces associated with a von Neumann algebra "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: T-định lý cho L ^ p-không gian liên kết với một đại số von Neumann. | J. OPERATOR THEORY 7 1982 267-277 Copyright by INCREST 1982 T-THEOREM FOR ZASPACES ASSOCIATED WITH A VON NEUMANN ALGEBRA HIDEKI KOSAKI 0. INTRODUCTION Following the development of the Tomita-Takesaki theory 13 Araki 1 introduced a family of positive cones 0 a -1- associated with a von Neumann algebra JI admitting a cyclic and separating vector See . . In 9 10 we observed that the cones are closely related to the positive parts of associated non-commutative -spaces 1 p oo 2 8 10 11 In fact the surjectivity of the map ị 6 p - ũị eJ i is exactly the validity of a T-theorem for the -space with a ------- - which will be made precise in 2. Also for finite von 2 2p Neumann algebras this T-theorem holds always. The main purpose of the paper however is to prove the converse. Namely we shall prove that the validity of this T-theorem for the -space 2 p oo implies the finiteness of the algebra. in question. For p co this is known as the -theorem due to Dye Murray-von Neumann 5 . For a factor of type either 1 or Illi 0 z 1 we shall prove a slightly stronger result. We shall freely use the basic facts and notations of relative modular theory 3 13 and non-commutative -spaces 2 8 10 11 . The author is indebted to Dr. Christian Skau for some ideas in the paper. Also the present research was done while he was at the University of California Los Angeles and he thanks Professor Masamichi Takesaki for constant encouragement. 1. PRELIMINARIES In this section we collect some basic definitions and properties partially to fix our notations. 268 H1DEKI KOSAK1 Let Z J be a standard form 6 . For each ự. e Ji j there correspond a unique implementing vector in which will be denoted by Q that is Ip x co .x .v ự í c xe Ji. For a faithful p and an arbitrary Ip the closure of the densely-defined closable operator s p x v e JZệ v - x xỳ e JiQ 1 admits the polar decomposition Jzi . Here the positive self-adjoint operator Jộv. is known as the relative modular operator. In case p Ip Ay is exactly the .

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