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Báo cáo toán học: "Weak operator and weak* topologies on singly generated algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhà điều hành * yếu và yếu cấu trúc liên kết trên đại số đơn lẻ tạo ra. | J. OPERATOR THEORY 15 1986 267-280 Copyright by INCREST 1986 WEAK OPERATOR AND WEAK TOPOLOGIES ON SINGLY GENERATED ALGEBRAS D. WESTWOOD 0. INTRODUCTION Let. be a separable infinite dimensional complex Hilbert space and let F denote the algebra of all bounded linear operators on As is well known ỈẾựC can be identified as the dual of the trace class re via the pairing i tr A A G G re and thus carries a weak topology. For T G S G F let denote the smallest unital weak closed subalgebra of T JT containing T and let iKT denote the smallest unital subalgebra of 3 which is closed in the weak operator topology WOT . Let 9 r w respectively 9lr wot wot denote the topological space 9Ir respectively 9Ir 7 equipped with the weak topology respectively WOT WOT . In 3 it is shown that for a large class of operators 9Ir w 7 z r wot . Furthermore for certain operators T agreement of the weak operator and weak topologies on 9Ir implies via 3 Theorem 6.13 among other things that such T have large invariant subspace lattices and are reflexive. The questions then arise whether or not agreement of these topologies is a general phenomenon and if there exist operators T for which 9Ir WT. The answer to this second question is still unknown even in the case that the weak and relative weak operator topologies on 91T are the same. However below we show that there exist operators Tfor which 9Ir w 9Ir wot . In his survey article on shifts 9 Question 5 Allen Shields raises the question of what is the relationship between the weak and weak operator topologies on iKT in the case T is an injective shift. It is easy to show that 9Ir 1 TT in this case. The example given in Section 3 serves to show that even in this special case the two topologies can be different. The contents of this paper formed a part of the author s Ph. D. thesis and the support and guidance of Professor Carl Pearcy is here gratefully acknowledged. 268 D. WESTWOOD 1. PRELIMINARIES Let j-2Ĩt denote the preannihilator of UIT in tc .