Characterizations Of Some Mappings On Operator Algebras

This dissertation is devoted to characterizing linear mappings,such as(Jordan)derivable mappings,(Jordan)left derivable mappings,anti-derivable mappings and cen-tralized mappings on some operator algebras with the local properties,studying the the necessary and sufficient conditions for Lie n-derivations to be standard,and discussing the relationship between Lie n-derivations and Lie n-higher derivations.It consists of five chapters.In chapter 1,we introduce the background of this study,the definition of related terms,and some preparatory results.In chapter 2,without requiring continuity of mappings,and only with the algebraic method,we discuss linear mappings on a unital Banach algebra A.Suppose that ? is a linear mapping from A into the unital A-bimodule M.Denote that Z(A)is the center of A.Suppose that W ?Z(A)is a separating point of M.The main results are:(i)? is a Jordan left derivation,if and only if ? is left derivable at W,if and only if ? is Jordan left derivable at W,if and only if A?(B)+Bb(A)=b(W)for each A,B?A with AB=BA=W;(ii)if ? is anti-derivable at W,or if ?(B)A+Bb(A)=?(W)for each A,B ? A withAB=BA=W,then ? is a Jordan derivation.In Chapter 3,we discuss continuous linear mappings on Banach algebras with weaker conditions.Suppose that ? is a continuous linear mapping from a unital Banach algebra A with property(B)into the unital Banach A-bimodule M.(For the definition of Banach algebra satisfying property(B),see page 23 of this dissertation.)The main results are:(i)? is a generalized left derivation,if and only if ? is left derivable at zero,if and only if AC?(B)=0 for each A,B,C?A with AB=BC=0;(ii)? is a generalized Jordan left derivation,if and only if ? is Jordan left derivable at zero,if and only if Ab?(B)+BS(A)=0 for each A,B?A with AB=BA=0;(iii)if A?(B)C+C?(B)A=0 for each A,B,C?A with AB=BC=0,then ? is a generalized Jordan derivation;(iv)if C?(B)A=0 for each A,B,C ? A with AB=BC=0,then ? is a generalized anti-derivation:(v)if ? is anti-derivable at zero,then ? is a generalized Jordan derivation.When?(I)=0,? is an anti-derivation;(vi)if ? is right centralized at zero,then ? is a right centralizer;(vii)if A is generated by idempotents,and A O ?(B)+?(A)o B=0 for each A,B?A with AB=BA=0,then ? is a generalized Jordan derivation.In Chapter 4,we discuss continuous linear mappings on another kind of algebras with weaker conditions.Suppose that A is a unital algebra with a nontrivial idempotent e.For each x in A,the Pierce decomposition form satisfiesThe main results are:(i)Suppose that A is 2-and(n-1)-torsion free.Each Lie n-derivation on A can be expressed as the sum of a derivation,a singular Jordan derivation and a central mapping vanishing on all(n-1)-commutators of A,if and only if A satisfies conditions(ii)Suppose that A is 2-torsion free,and and ? is a linear mapping on A.If x o?(y)+?(x)o y=0 for each x,y ? A with xy=yx=0,then ? is a generalized Jordan derivation,and ? can be expressed as the sum of a derivation,a singular Jordan derivation and a left centralizer.In Chapter 5,we characterize Lie n-higher derivations on a unital torsion-free algebra.Suppose that A is a unital torsion-free algebra,RA and RA are two non-empty subsets of An.We prove that:(i)if every Lie n-derivation on A is standard,then every Lie n-higher derivation on A is standard:(ii)if every linear mapping Lie n-derivable at RA is a Lie n-derivation,then every sequence {dm} of linear mappings Lie n-higher derivable at RA is a Lie n-higher derivation;(iii)if every linear mapping Lie n-derivable at RA is a sum of a derivation and a linear mapping vanishing on all(n-1)-th commutators of RA,then every sequence {dm}of linear mappings Lie n-higher derivable at RA is a sum of a higher derivation and a sequence of linear mappings vanishing on all(n-1)-th commutators of RA.