Characterizations Of Some Derivations On Operator Algebras

In this paper,we give a characterization of the linear mappings on operator algebras.The mappings that we study include(m,n)-Jordan derivations,Jordan*-derivations,Jordan left*-derivations and left*-derivable mapping.The algebras as that we study include C*-algebras,von Neumann algebras,matrix algebras.This paper splits into five chapters.In Chapter one,we introduce the background of this study,review the development and achievements until now,and give some preliminary concepts we need in this paper.In Chapter two,we study the Hyers-Ulam-Rassias stability of(m,n)-Jordan derivations.As applications,we characterize(m,n)-Jordan derivations on C*-algebras and some non self-adjoint operator algebras.In Chapter three,we characterize the Jordan*-derivations on Banach*-algebras.Suppose that A is a real or complex unital Banach*-algebra,M is a unital Banach A-bimodule,and G ? A is a left separating point of M.In this chapter,we investigate whether the additive mapping ?:A?M satisfies the condition AB=G(?)A?(B)+?(A)B*=?(G)characterize Jordan*-derivat-ions.Initially,we prove that if A is a real unital C*-algebra and G=I is the unit element in A,then ?(non-necessarily continuous)is a Jordan*-derivation.In addition,we prove that if A is a real unital C*_algebra and ? is continuous,then ? is a Jordan*-derivation.Finally,we show that if A is a complex factor von Neumann algebra and ? is linear,then ?(non-necessarily continuo-us)is a Jordan*-derivation.In Chapter four,we characterize the additive Jordan left*-derivations on C*-algebras.An additive mapping ? from a*-algebra A into a left A-mod-ule M is called an additive Jordan left*-derivation if ?(A2)=A?(A)+A*?(A)for every A in A.In this chapter,we prove that every additive Jordan left*-derivation from a complex unital C*-algebra to its Banach left-module is equal to zero.An additive mapping ? from a*-algebra A into a left A-module M is called left*-derivable at G in A if ?(AB)=A?(B)+B*?(A)for each A,B in A with ABG.We prove that every continuous additive left*-derivable mapping at I from a complex unital C*-algebra to its Banach left-module is equal to zero.In Chapter five,we give a summarization of the whole paper.