Jordan Derivations And Lie Derivations On B(X)

As is well known,derivations,Jordan derivations and Lie derivations are very important maps in operator algebras and operator theory,and have received a fair amount of atten-tion.In this paper,we will continue to study Jordan derivations and Lie derivations on B(X)by their local property at a point.We characterize additive maps on B(X)which are Jor-dan derivable at an arbitrary nontrivial idempotent P,and Lie derivable at Z E B(X)with PZ = ZP = Z for some nontrivial idempotent P ? B(X).We characterize Jordan derivable maps on Hilbert space nest algebra,Jordan derivable maps on B(X),Lie derivable maps on Hilbert space nest algebra,and Lie derivable maps on B(X).Main results of this paper are as follows:1.Assume that A? AlgN is a Hilbert space nest algebra,M=B(H),and P?N is a nontrivial projection.Then an additive map ?:A ?M is Jordan derivable at P,that is,?(X)(?)Y+X(?)?(Y)=?(X(?)Y)for any X,Y ?A with X(?)Y=P,if and only if ? is a derivation,here X(?)Y = XY + YX is the usual Jordan product.2.Assume that A =B(X),P ? B(X)is an arbitrary but fixed nontrivial idempotent.Then an additive map ?:A ? A is Jordan derivable at P,that is,?(X)(?)Y + X(?)?(Y)=?(X(?)Y)for any X,Y ? A with X(?)Y = P,if and only if ? is a derivation.3.Assume that A = AlgN is a Hilbert space nest algebra,M = B(H),and P ?N is a nontrivial projection,Z ? AlgN with PZ = ZP = Z.Then an additive map ?:A ?M is Lie derivable at Z,that is,?([X,Y])=[?(X),Y]+[X,?(Y)],(?)X,Y ?A,XY = Z,if and only if there exist a derivation d:AlgN?B(H)and an additive map ?:AlgN?FI vanishing on commutators[X,Y]with XY = Z such that ?(X)= d(X)+ ?(X),X E AlgN,here[X,Y]= XY-YX is the usual Lie product.4.Let A = B(X),P ?B(X)be a nontrivial idempotent,Z ? B(X)with PZ = ZP = Z.Then an additive map ?:A ? A is Lie derivable at Z,that is,?([X,Y])=[?(X),Y]+[X,?(Y)],(?)X,Y?A,XY=Z,if and only if there exist a derivation d:B(X)?B(X)and an additive map ?:B(X)?FI vanishing on commutators[X,Y]with XY = Z such that?(X)= d(X)+?(X),X ?A.