| Derivations,ξ-Lie derivations and identity mapping are very important maps on opera-tor algebras and operator theory,and have received a fair amount of attention.In this paper,we characterize additive maps on prime rings,B(X)and von Neumann algebras which are derivable at general elements,and explore the problem that under what conditions,an addi-tive map becomes a derivation,and then get the new equivalent characterization of derivations on prime rings,B(X)and von Neumann algebras.For ξ≠4 0,1,The aim of the paper is to discuss additive maps on rings which are derivable at commutative zero product.For a given positive integer k ≥ 1,we characterize the strong k-skew commutativity preserving and k-skew Jordan product preserving nonlinear maps on C*-algebras which contain the unit I.The structure of this paper is as follows:In the first chapter,some preliminaries and main conclusions are mentioned,we provide some fundamental definitions which will be used in this paper and introduce the background and notations of the discussed problem briefly.In the second chapter,we characterize derivable maps on prime rings,B(X)and von Neumann algebras.Main result is as follows:Let R be a prime ring containing a nontrivial idempotent P.Suppose C ∈ TZ satisfies C = PC.It is shown that an additive map △:R→R is derivable at C,that is,△(AB)=△(A)B + A△(B)for every A,B ∈R with AB = C if and only if there exists a derivationδ:R→R such that △(A)= δ(A)+ △(I)A for all A ∈R.Similar results are obtained for von Neumann algebras with no central abelian projections.As its application,we obtain that,if nonzero operator C E 3(X)such that ran(C)or ker(C)is complementary in X,then an additive map △:B(X)→B(X)is derivable at C if and only if it is a derivation.In particular,we show that an additive map from a factor von Neumann algebra into itself is derivable at an arbitrary but fixed nonzero operator if and only if it is a derivation.In the third chapter,we characterize additive maps on rings which are derivable at commutative zero product.Let R be a,unital 2-torsion free.Let R contains a nontrivial idempotent P such that(1)for A ∈ R,AR(I-P)= {0} implies A = 0;(2)for A ∈ R,PRA = {0} implies A = 0;If an additive map δ:R →R satisfiesδ(AB-ξBA)= δ(A)B + Aδ(B)-ξBδ(A)-ξδ(B)A,VA,B ∈R,AB = BA = 0 then there exists an additive Jordan derivation φ:R→R and a cental element C ∈ Z(R)such that δ(A)= φ(A)+ CA,(?)A ∈R.In the fourth chapter,we main characterize the strong k-skew commutativity preserving and k-skew Jordan product preserving surjection on C*-algebras which contain the unit I.Main results are as follows:1.Let A be a C*-algebra having the unit I,Φ:A → A is a,surjection.Then Φ is strong k-skew commutativity preserving,that is,*[Φ(A),Φ(B)]k,(?)A,B ∈ A if and only if Φ(A)= Φ(I)A,Φ(I)∈ Z(A),Φ(I)*= Φ(I),Φ(I)k+1 = I,(?)A ∈ A.2.Let A be a C*-algebra having the unit I,Φ:A → A is a,surjection.Then Φ is k-skew Jordan product preserving,that is,*(Φ(A)◇Φ(B))k=*(A◇B)k,VA,B ∈ A if and only if Φ(A)= Φ(I)A,Φ(I)E Z(A),Φ(I)*= Φ(I),Φ(I)k+1 = I,(?)A ∈ A. |
