Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Liên tục tuyệt đối và disjointness của các quốc gia trong C *- động học hệ thống. | J. OPERATOR THEORY 11 1984 319-331 g Copyright by INCREST 1984 ABSOLUTE CONTINUITY AND DISJOINTNESS OF STATES IN C -DYNAMICAL SYSTEMS FUMIO HI Al INTRODUCTION The notions of absolute continuity and disjointness of measures are fundamental in the measure theory. These notions and the Radon-Nikodym theorem are extended in several ways to the noncommutative case on von Neumann algebras or c -algebras. In this paper the absolute continuity and disjointness of states of c -algebras are discussed. In Section 1 we consider some types of absolute continuity of states. For states p and ìị of a c -algebra A we say that ỷ is absolutely continuous resp. quasi-absolutely continuous with respect to p if the support resp. the central support of ý is contained in that of p where the supports the central supports are taken for the normal extensions of p and ự to the von Neumann algebra induced by a certain representation of A. Let a C -dynamical system A G t be given. For invariant states p and lỷ it is shown that ị is absolutely continuous with respect to p if and only if ìỊ is in the norm closure of the invariant states dominated by cp. If A is Cz-abelian then the absolute continuity of invariant states is equivalent to that of their maximal representing measures. We show that A is -central if and only if the absolute continuity and quasi-absolute continuity are identical for each invariant states. We also present the relation between the absolute continuity and the relative entropy of states. The Kubo-Martin-Schwinger KMS condition was introduced to describe thermodynamical equilibrium states of a quantum system. Combined with the-Tomita-Takesaki theory 22 the KMS condition has played a vital role in the ana lysis of operator algebras. The lowest energy states of a quantum system are formulated by ground states which are regarded as limits of KMS states when the inverse temperature tends to oo. After the concept of passivity given by Pusz and Woro-nowicz 19 from the .
