TAILIEUCHUNG - Báo cáo toán học: "Perturbations of reflexive operator algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn của đại số nhà điều hành phản thân. | J. OPERATOR THEORY 15 1986 289-305 Copyright by INCREST 1986 PERTURBATIONS OF REFLEXIVE OPERATOR ALGEBRAS KENNETH R. DAVIDSON This paper is a sequel to a joint paper with Man Duen Choi 4 in which perturbation results for finite dimensional operator algebras are discussed. Here we are concerned with infinite dimensional reflexive algebras. The starting point is the extension of a theorem about algebras with finite distributive subspace lattice to algebras containing a maximal ơ-complete Boolean algebra of idempotents. Specifically it is shown that if Ể8 is any norm closed algebra close to a reflexive algebra sđ containing a maximal ơ-complete Boolean algebra of idempotents then there is a complete lattice isomorphism 0 of Lat onto Lat Ổ which is close to the identity. This leads to a number of applications including a new proof of a slight strengthening of a theorem of Lance on perturbations of nest algebras and its analogue for nest subalgebras of von Neumann algebras. The study of perturbations of operator algebras started with Kadison and Kastler 12 . Then Christensen 5 showed that for von Neumann algebras of type I closeness implies unitary equivalence via a unitary near the identity. This will be one of our main tools together with a theorem of Wermer 19 for orthogonalizing Boolean algebras of idempotents. Again it is crucial to our purposes that Boolean algebras which are already almost orthogonal can be straightened out by a small change. 1. PRELIMINARIES In this paper yf will denote a separable Hilbert space. The bounded operators on f are denoted by .i .xf . Given a subset of . MjC Lat sal denotes the complete lattice of subspaces left invariant by each element of sẩ. Dually if J is a lattice of subspaces then Alg ỈẾ denotes the weakly closed algebra of all operators which leave each element of invariant. An algebra sđ is reflexive if XỈ Alg Lat sđ. The word projection will always mean orthogonal projection in distinction from idempotents. If L is a closed

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