Non-self-with The Ideal Of The Algebra, The Lie Ideal

This thesis is devoted to the investigation of some problems about reflexive operator algebras. It consists of four chapers.Chapter 1: We introduce some prepared knowledge ,which is lattices, lattices corresponding operator algebras , usual several operators ,and such like. Finally,we summarize the background.Chapter 2 : We discuss CSL algebras and CDCSL algebras. Then we obtain the foiling results.One : Let A be a. CDCSL algebra; LatA included at least a non-trivial comparable element. (1)If I⁰ is trace-zero part of I which is a wcakly-covcrcd ideal of A,we get:(2)Let L be a weakly-covered subspace of A If there exists a weakly-covered ideal G of Asuch thatthen L is a conjugation-invariant subspace of A(3)Let L be a weakly-covered Lie ideal of A.There exist an atomic-diagonal disjointideal I of A and a subset A₁ of the atom A_I adjacent to I such thatTwo: Let T be an atomic-diagonal disjoint weakly-covered ideal of A A_I are atoms adjacent to I. Chapter 3: We show the relations between Lie ideals and simitary subspaces.Let N be a subspace nest acting on Hilbert Space H whose atoms have order type isomorphic to a subset of the integers.Let L₀ be a lattice generated by the interval of N again.If L is a, strongiy closed Lie subspace of AlgL₀ = A, L is invariant under similarities from AChapter 4: We discuss the relations between trace-class operators and rank-one operators in I ,which is an ideal of AlgL.