tailieunhanh - Báo cáo toán học: "Topological direct integrals of left Hilbert algebras. I "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trực tiếp tích phân tô pô đại số Hilbert . | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 29-45 TOPOLOGICAL DIRECT INTEGRALS OF LEFT HILBERT ALGEBRAS. I NORBERT RIEDEL INTRODUCTION In some papers it was shown see 8 17 how the theory of standard von Neumann algebras may be used in order to improve some results in the separable theory of direct integrals which goes back to J. von Neumann. In this paper we are concerned in the following question. Using the theory of standard von Neumann algebras is it possible to develop a non separable theory of direct integrals such that within this theory some of the more delicate problems may be solved including the questions on the types of the associated von Neumann algebras. This question is also motivated by mathematical physics where decompositions of representations of non separable c -algebras are considered see 3 and in this context standard von Neumann algebras play a natural role see 19 . Using Hilbert algebras only the author started to develop such a theory in 11 . We also want to mention the paper of H. Halpern 7 where the theory of standard von Neumann algebras has been used in order to decompose type III von Neumann algebras with a cyclic and separating vector. This paper is divided into three sections. First we introduce the notion of a continuous field of left Hilbert algebras and of the corresponding direct integral with respect to a positive Radon measure. Moreover we investigate the behavior with respect to direct integral decomposition of some characteristic operators. In Section 2 we prove that the type of the semifinite portion of the associated von Neumann algebras reduces to the components. Throughout in Section 2 we assume a countability condition to be satisfied which is the analogue of H8 in 11 . We will treat the corresponding problem for the type-III-portion in a subsequent paper. Finally in Section 3 we show that the theory applies to the central decomposition of a state p on some c -algebra sđ with unit if p satisfies the .