Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đại số mô hình cho các biện pháp điều hành tích cực Quý. | J. OPERATOR THEORY 7 1982 237-246 Copyright by INCREST 982 ALGEBRAIC MODELS FOR POSITIVE OPERATOR VALUED MEASURES SHIGERU ỈTOH 1. INTRODUCTION Dinculeanu and Foias 7 characterized the conjugate relation between probability measures in terms of the isomorphism of their algebraic models. Concerning other topics related to algebraic models we refer to Dinculeanu and Foias 6 and Foiaẹ 8 . Schreiber Sun and Bharucha-Reid 19 and Christensen and Bharucha--Reid 3 4 investigated algebraic models for measures induced by stochastic processes and for measures on Banach spaces. Christensen 5 gave the definition of algebraic models for positive operator valued measures and used it to prove an extension theorem 5 Theorem 3 for a consistent family of positive operator valued measures indexed by a directed set. In this paper we first introduce the concept of algebraic models for positive operator valued measures which is slightly different from that defined by Christensen 5 on separable Hilbert spaces. Then analogously to Dinculeanu and Foias 7 we give a characterization of the conjugate relation between positive operator valued measures by the isomorphism of their algebraic models. We also obtain a necessary and sufficient condition in order that a positive operator valued measure is conjugate to a spectral measure. 2. PRELIMINARIES Let H be a Hilbert space and Í2 j be a measurable space. Let B H be the set of bounded linear operators on H. Definition cf. Berberian 1 Definition 1 and Proposition 1 . A mapping E sđ - B H is called a normalized positive operator valued measure PO-measure if E satisfies the following conditions i For any Me sđ E M ỷ 0 238 SHIGERU ITOH ii E Q I the identity on H and 0 0 iii For any pairwise disjoint sequence Mn of sets in 32 EI I w-lim E M n l 7 i l where w-lim is the limit in the weak operator topology. If E M is an orthogonal projection for each M e sđ then E is called a spectral measure. Let 3ỉ 0 M e sđ Af 0 . For any 32 - measurable functions
