Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tản, từ, hệ thống động lực, và abelianness tiệm cận. | J. OPERATOR THEORY 13 1985 237-253 Copyright by INCREST 1985 DISSIPATIONS DERIVATIONS DYNAMICAL SYSTEMS AND ASYMPTOTIC ABELIANNESS AK1TAKA KISHĨMOTO and DEREK w. ROBINSON 1. INTRODUCTION Let 21 G a be a C -dynamical system where th is a simple c -algebra with identity and G is compact and abelian. Next let 5 be a linear operator from the G -finite elements 2lF of 21 into 2L There have been many recent investigations of this situation with the extra assumption that Ỏ commutes with a see for example 1 and the references therein . The principal aim of these investigations was to characterize those 5 which generate C0-groups of -automorphisms of 2Í or Co-semigroups of completely positive maps. In this paper we study the same questions without the assumption that Ô and a commute. Instead we assume that 21 is asymptotically abelian with respect to an automorphism T and that á and 5 commute with T. Somewhat surprisingly this latter assumption leads to similar but even stronger conclusions. Related results have previously been given by Takesaki 8 and Longo and Peligrad 9 . For example if Ỗ is a -derivation then 5 automatically vanishes on the fixed point algebra 21 of a Ô is closable and its closure Ỗ generates a group of -automorphisms j which commutes with a. Similarly if 5 is a -dissipation for which 5 2I 0 then Ỏ is closable and 5 generates a Co-semigroup of completely positive contractions ft which commutes with á. In both cases acts by multiplication on the spectral subspaces 2I y of 21 . c v í lx X 6 2I y . If Ỗ is a -derivation then i p y A y is a homomorphism of G the dual of G into R. If Ỗ is a -dissipation then fl is a negative definite function. The key point in deriving these results is the observation that asymptotic abelianness defines a natural topology on the multiple tensor products of 21 with itself. This structure is analyzed in Section 2 and is exploited in Section 3 to obtain the generator results. In Section 3 we also examine almost periodic .
