tailieunhanh - Báo cáo toán học: "Weakly closed ideales of nest algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Lý tưởng về đại số tổ yếu đóng. | J. OPERATOR THEORY 7 1982 . 219-235 Copyright by INCREST 1982 WEAKLY CLOSED IDEALS OF NEST ALGEBRAS J. A. ERDOS and s. c. POWER INTRODUCTION Let be a nest algebra of operators on a Hilbert space H. We obtain a description of all weakly closed two-sided ideals of 33 more generally we describe all weakly closed two-sided j -submodules of In the course of the proof we identify the finite rank operators of any norm-closed jaZ-submodule of jSf ZZ and show that such modules have the same closure in the weak strong ultraweak or ultrastrong topologies. A simple and more direct proof of the main result for the special case of the Volterra nest algebra is sketched at the end of Section 1. Some of the properties of such ideals and modules are investigated. The predual of any such weakly closed module is found. For certain cases the perturbed modules ữu X where X denotes the compact operators on FT are described in a fashion analogous to the description of s X in 3 . For any module of this type the commutant C s V of sđ modulo is described thus giving a description of the first Hochschild cohomology spaces with coefficients in these modules. We find that li is of the form ểv where is a subalgebra of the core of j . For the case when 6U is an algebra we show that AlgLatX This appears to be a new example of a reflexive algebra. Throughout will denote a complete nest of projections on a Hilbert space H. The nest algebra Algtf of s is denoted by J . The terminology and notation concerning nest algebras used in this paper are standard and may be found in 7 2 6 . All Hilbert spaces considered will be complex and the term projection will always mean orthogonal projection. The rank one operator X x e f will be denoted by e f. The terms module and ideal will be used to mean two-sided module and two-sided ideal. t. MODULES AND IDEALS Suppose that E E is an order homomorphism of ễ into itself that is E F implies that E F . Then the set le Z - Ẽ XE 0 for all E 6 ể 220 J. A. ERDOS and s. .