tailieunhanh - Báo cáo toán học: "Abelian operator alegbras and tensor products "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tàiAbel alegbras điều hành và các sản phẩm tensor. | J. OPERATOR THEORY 14 5985 391 -407 g Copyright by INCREST 1985 ABELIAN OPERATOR ALGEBRAS AND TENSOR PRODUCTS JON KRAUS One of the central results in the theory of tensor products of von Neumann algebras is Tomita s commutation formula if JI and -J are von Neumann algebras then 1 . JT y . It was observed in 15 that if we let and j5 2 denote the projection lattices of JI and Ji respectively then 1 can be rewritten as 2 alg i alg 2 alg 2 . This version of Tomita s formula makes sense for any pair of reflexive algebras algJz and alg 5 2- It remains an open question whether the tensor product formula 2 is valid when alg and alg 2 are arbitrary reflexive algebras. However 2 has been verified in a number of special cases 15 17 19 20 21 22 23 In particular it is known that if is a commutative subspace lattice that is either completely distributive 23 or finite width 19 then 2 is valid for and any subspace lattice J 2. One of the main results of this paper is that if T is a subnormal operator acting on a Hilbert space Jf or if T is a BCP -operator on or more generally if T e Ak then 2 is valid when lat T and Jỉ 2 is any subspace lattice. The proofs of the results concerning the tensor product formula 2 in 19 22 23 and this paper all make use of slice maps. If Ji and Ar are von Neumann algebras and p is in the predual J of JI then the right slice map Rọ see . 33 is the unique Ơ-weakly continuous linear map from Ji . K such that RẶẠ B p A B A G JI The left slice maps L .Jt Ar - Jt ý are similarly defined. A ff-weakly closed subspace y of 5 . f the algebra of bounded operators on is said to have Property s 22 if whenever is a ơ-weakly closed subspace of a von Neumann 392 JOS KRAUS algebra J we have A e f xj. r RỢ A e 3 for all Ọ e n - .f -ff. It is shown in 22 that if algj has Property Sff then 2 is valid for any subspace lattice 5 2. Thus if every ff-weakly closed subspace if has Property Sff then the tensor product formula 2 is valid for all pairs of subspace lattices .