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

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tích phân tô pô đại số trực tiếp của Hilbert trái. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 . 213-229 TOPOLOGICAL DIRECT INTEGRALS OF LEFT HILBERT ALGEBRAS. II NORBERT RIEDEL 1. INTRODUCTION This paper is a continuation of 7 For some continuous field of left Hilbert algebras 2lJ ei3 A see 7 we assume the following condition to be satisfied C For any X ye A the function c t A l T is continuous on Í2 X R. By 7 condition C is stronger than condition L8 in 7 . We do not know whether C follows from the other properties of A or not. However condition C is satisfied in the case of the central decomposition of a KMS-state which was considered in 7 Section 3. Pursuing the investigations of Section 2 in 7 we mainly intend to prove the following theorem. . Theorem. Let A be a continuous field ợ . s which is defined on the locally compact space Q such that condition C is satisfied. Suppose that 21 contains a unit ẽị for any i e .Q and the vector field ị C ị is contained in A. Let 21 be the direct integral of 2L ieij A with respect to some Radon measure p on Q see 7 . If s F s tAe maximal central projection in i 2I SL such that the von Neumann algebra ỗ 2l jSf 2Iỉ ộ is of type III e Ỉ2 then the following identity holds s d see 5 p. 195 . In order to prove Theorem we need some tools which will be developed in Section 2 and Section 3. First in Section 2 we gather some fundamental properties of crossed products of von Neumann algebras with cyclic an separating vectors. Next in Section 3 we introduce the covariant continuous field of 1. H. a. s J 612 L which is associated with 2IÍ ỈSO A . Now the theory of M. Takesaki in 12 13 and the results of H. Halpern in 3 allow to reduce our problem to a corresponding 214 NORBERT RIEDEL problem in the separable case. This will be done in Section 4. Finally we achieve the proof by an application of the results of c. Lance in 4 . 2. SOME FACTS ABOUT CROSSED PRODUCTS Let SI be a . with unit e and letJiT be the completion of SI. Furthermore let crJieR .