Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hương vị các kỹ thuật quang phổ của Scott Brown. | Copyright by INCREST 1981 J. OPERATOR THEORY 6 1981 13-23 POSITIVE CONES AND -SPACES ASSOCIATED WITH A VON NEUMANN ALGEBRA HIDEKI KOSAKI 0. INTRODUCTION By using the Tomita-Takesaki theory Araki 2 introduced a one parameter family of positive cones p a e 0 1 2 associated with a von Neumann algebra- with a cyclic and separating vector f0. On the other hand Connes 5 Haagerup 9 and Hilsum 10 developed a theory of non-commutative ZAspaces for an arbitrary von Neumann algebra. In the present paper first we represent the algebra in question on a Hilbert space L Jf see consisting of certain operators and obtain simple relations between the positive cones and non-commutative -spaces. Based on this observation we consider certain problems concerning the positive cones. Among other results we obtain the necessary and sufficient conditions for a normal state to admit a representative vector in Px ae l 4 1 2 We shall freely use the standard results and notations of the relative modular theory as well as the Tomita-Takesaki theory which are found in 3 7 and 17 . Some results in the paper were obtained while the author stayed at the Centre de Physique Théorique . Marseille France University of Odense Denmark and University of Trondheim Norway and he would like to express his sincere gratitudes to Professor D. Kastler for his invitation and warm hospitality and Professors u. Haagerup and c. Skau for stimulating discussions on the present materials. At last but his most sincere gratitude goes to Professor M. Takesaki for discussions and constant encouragement. Also the author is indebted to the referee for improvement of the presentation of the paper. 1. PRELIMINARIES In this section we fix notations and collect some basic facts for later use. 14 HIDEKI KOSAKJ Standard form. 8 9 Let Ji be a von Neumann algebra. By making use of a crossed product and a dual action 18 we can construct a pair 0 0 which is unique up to isomorphism consisting of a semi-finite von .
