| As is well known, higher derivations and Lie derivations are very important maps in both theory and applications, and have received a fair amount of attention. In this paper, we will continue to study the higher derivations and Lie derivation on operator algebras.The structure of this paper are as follows:In the first chapter, we introduce the background of the discussed problem of this thesis briefly and the main results of this thesis. Some notation and basic results are also given in this section.In the second chapter, we give a necessary and sufficient condition for a family of linear maps on CSL algebra AlgL to be higher derivable at Ω. where Ω=0or Ω∈AlgL is a left (or right) separating point in AlgL or PΩP=Ω and Ω|ran(p) is a left (or right) separating point in PAlgLP for some nontrivial projection P∈L. As its applications, we show that a family of linear maps from an irreducible CDCSL algebra or a nest algebra into itself, which is higher derivable at such Ω, is a higher derivation.In the third chapter, a necessary and sufficient condition for a family of linear maps on CSL algebra AlgL to be higher derivable at arbitrary but fixed point Ω∈AlgL is given. Moreover, we show that if there is a faithful projection P in L such that PΩP and (I-P)Ω(I-P) are a left or right separating point in PAlgLP and (I-P)AlgL(I-P) respectively, then a family of linear maps on AlgL is higher derivable at Ω if and only if it is a higher derivation. In particular, if AlgL is an irreducible CDCSL algebra or a nest algebra, then a family of linear maps on AlgL is higher derivable at Ω≠0if and only if it is a higher derivation. By these results, we give a necessary and sufficient condition for a linear map on CSL algebra to be derivable at arbitrary but fixed point, and a full characterization of a linear map which is derivable at Ω≠0on a nest algebra if and only if it is a derivation.In the fourth chapter, we show that a necessary and sufficient condition for an additive map on any ring with a nontrivial idempotent P to be Lie derivable at Ω. Ω=ΩP=PΩ. As its application, we characterize Lie derivable maps on von Neumann algebras with no abelian summands and nest algebras. In particular, we obtain that Lie derivations on B(X) can be determined by additive maps Lie derivable at any finite rank operator. |
