tailieunhanh - Báo cáo toán học: "A Radon-Nikodym theorem for positive linear functionals on *-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một định lý Radon-Nikodym cho functionals tích cực trên đại số tuyến tính *-. | J. OPERATOR THEORY 10 1983 77-86 Copyright by INCREST 1983 A RADON-NIKODYM THEOREM FOR POSITIVE LINEAR FUNCTIONALS ON -ALGEBRAS ATSUSHI INOUE 1. INTRODUCTION Noncommutative Radon-Nikodym theorem for von Neumann algebras has been investigated in detail 7 15 19 but the study for algebras of unbounded operators seems to be hardly done except to 9 In this paper we shall develop a Radon-Nikodym theorem in the context of unbounded operator algebras obtained by positive linear functionals on -algebras. For a positive linear functional or more generally for a positive invariant sesquilinear form p on a -algebra jaZ the well-known GNS-construction yields a quartet tĩợ Sỉ v .w where Ểdy is a dense subspace in a Hilbert space 7t Z is a closed ớ -algebra on Sy and Ầy is a linear map of jaZ into 3 v satisfying for each x ye The Gudder s Radon-Nikodym theorem Ị9 asserts that if a positive linear functional ỷ on a -algebra ỉđ with identity e is strongly absolutely continuous with respect to a positive linear functional J9 on sđ that is the map ýlựx - Ẳ x is closable then there exists a positive self-ad-joint operator H on such that ự x HẰẶxyHẢẶèy for each xe sđ. However the relation between the Radon-Nikodym derivative H and the ỡ -algebra Ti jaZ seems to be vague. With this view we look again at a Radon-Nikodym theorem for -algebras and obtain the result that a positive invariant sesquilinear form ự on a -algebra ỉđ is strongly absolutely continuous with respect to a positive invariant sesquilinear form p if and only if there exists a sequence Hn of positive operators in the Powers commutant nv stfy of the ơ -algebra 7tv jaZ such that a converges for each X ye b Hnl2Av x converges in for each xe sd c ty x y lim pK x y lim f Ấ x pv T for each X y e sđ. n- X n n- co Furthermore we shall apply this result to the spatial theory for unbounded operator algebras. 78 ATSl SHl . 2. POSITIVE INVARIANT SESQUILINEAR FORMS Let 12 be a dense subspace of a Hilbert space . By J S we .