*-isomorphisms And Lie Derivations On Von Neumann Algebras

Completely preserving problems,derivations,Jordan derivations and Lie derivations are very important concepts in operator algebras and operator theory,and have received a fair amount of attention.In this paper,we characterize the maps completely preserving Jordan?-*-zero product,Lie-derivations and Jordan ?-*-multiplicative derivations on von Neumann algebras.The structure of this paper are as follows:In the first chapter,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 maps completely preserving Jordan ?-*-zero prod-uct and give equivalent characterizations of*-isomorphisms on von Neumann algebras.Let M be a von Neumann algebra,? be a,non-zero scalar.In this paper,we show that,if ?:M?M is a surjective map and completely preserves Jordan ?-*-zero product,then?(I)-1? is a*-ring isomorphism.In particular,if M has no central abelian projections,since ?=±1,then ?(I)-1? is a direct sum of a linear*-isomorphism or a conjugate linear*-isomorphism,since |?|= 1,?(?)R,then ?(I)-1 is a direct sum of a linear*-isomorphism;if M is a factor von Neumann algebra,then ?(I)-1 is a linear*-isomorphism or a conjugate linear*-isomorphism.In the third chapter,we give new equivalent characterizations of Lie derivations on von Neumann algebras.Let M be a von Neumann algebra with no central abelian projections,P ?M be a projection with (?)=0 and (?)=I.An additive map ?:M?M is said to be Lie derivable at??M,if ?([A,B])=[?(A),B]+[A,?(B)]for any A,B?M with AB=?.We show that,if ??M such that P?=?,then ? is Lie derivable at ? if and only if there exist a derivation?:M?M and and additive map f:M? Z(M)vanishing at commutators[A,B]with AB=? such that ?(A)=d(A)+f(A),(?)A ? M.In particular,if M is a factor von Neumann algebra and ??SM such that ker(?)?0 or (?)?H,then ? is Lie derivable at ? if and only if it has the above form.In the fourth chapter,we give equivalent characterization of*-derivation on von Neu-mann algebras.Let M be a von Neumann algebra with no central abelian projections,? be a non-zero scalar.we show ?:M? is a Jordan ?-*-derivation,that is,?(A ?? B)=?(A)?? B+A ????(B),(?)A,B,?M,??±1 if and only if it is an additive*-derivation.