tailieunhanh - Báo cáo toán học: "W*-dynamical systems and reflexive operator algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: W *- hệ thống động học và đại số nhà điều hành phản. | J. OPERATOR THEORY 8 1982 181-194 Copyright by INCREST 1982 0 -DYNAMICAL SYSTEMS AND REFLEXIVE OPERATOR ALGEBRAS JON KRAUS Let a be a ff-weakly continuous action of a locally compact abelian group G on a von Neumann algebra Ổ by -automorphisms. If a is inner and z is a positive semigroup in the dual group of G then to each inner unitary implementation of a we can associate via Stone s Theorem a commutative subspace lattice CSL sc c sc such that alg n Si. coincides with the algebra of -analytic operators of the action. Such a lattice SẾ is said to be G z -analytic. Loebl and Muhly have shown that every totally ordered CSL is R 0 oo -analytic. We extend their result and show that any width n CSL is R 0 oo -analytic. We also show that if sc is any CSL whose core S is totally atomic then there is a second countable compact abelian group G and a positive semigroup z in the dual group of G such that sc is G E -analytic. These results allow US to use spectral subspace techniques to study certain reflexive operator algebras. As an application we prove that if I cz sc I i ----- 1 2 are commuting CSL s with totally atomic cores then 1 algJS n st alg-S n alg 2 n where S -- sc sc sc. ỔỉỵÕịSỉĩ and the tensor product on the right hand side of 1 is the ơ-weak closure in Si of the algebraic tensor product. This result is related to Tomita s commutation theorem for tensor products of von Neumann algebras and to a result of Gilfeather Hopenwasser and Larson who show by other methods that 1 holds when Ổ and are type I factors and SSỵ and S . are totally ordered . 1. PRELIMINARIES AND NOTATION In this paper all Hilbert spaces will be assumed separable. Let H be a Hilbert space. We write B H for the algebra of all bounded operators on H. Projection will always mean self-adjoint projection and we identify projec- 182 JON KRAUS lions with their ranges. The projections in B H form a lattice under the operations A intersection and V closed linear span . A subspace lattice is a lattice of .