Characterization Of Derivations And Isomorphisms On Von Neumann Algebras

As is well known,derivations,Jordan derivations,Lie derivations,multiplicative maps and completely preserving problems are very important maps in operator algebras and op-erator theory,and have received a fair amount of attention.In this paper,we investigate the skew Lie-triple derivations,skew Jordan-triple derivations on von Neumann algebras.We characterize the skew Lie-triple multiplicative maps on von Neumann algebras,and show such maps are*-isomorphism.Also the completely preserving skew Lie zero product maps and completely preserving skew Jordan zero product maps between*-rings with the identity element I are also characterized.The structure of this paper are as follows:In the first chapter,we introduce the background of the discussed problem of this thesis briefly,the main results,notations and basic theorems in this thesis.In the second chapter,we give new equivalent characterization of*-derivation on von Neumann algebras.1.Characterization of nonlinear skew Lie-triple derivations on von Neumann algebras.Let A be a von Neumann algebra with no central abelian projections.Then ?:A?A is a nonlinear skew Lie-triple derivation,that is,?([[A,B]*,C]*)=[[?(A),B]*,C]*+[[A,?(B)]*,C]*+[[A,B]*,?(C)]*,(?)A,B,C?A if and only if ? is an additive*-derivation.In particularly,if A is a factor von Neumann algebra,then ?:A?A is a nonlinear skew Lie-triple derivation if and only if there exists T E A with T*=-T such that?(A)=AT-TA,(?)A ? A.2.Characterization of nonlinear Jordan triple*-derivations on von Neumann algebras.Let A be a von Neumann algebra with no central abelian projections.Then ?:A?A is a Jordan triple*-derivation,that is ?(A(?)B(?)C)= ?(A)(?)B(?)C+A(?)?(B)(?)C + A(?)B(?)?(C),(?)A,B,C?A if and only if ? is an additive*-derivation.In particularly,if A is a factor von Neumann algebra,then ?:A?A is a nonlinear Jordan triple*-derivations if and only if there exists T?A with T*= T such that ?(A)= AT + TA,(?)A?A.In the third chapter,we give new equivalent characterization of*-isomorphism on von Neumann algebras.1.Characterization of*-isomorphism on von Neumann algebras.Let A be a von Neu-mann algebra with no central abelian projections,B be a*-algebra.If ?:A?B is a skew Lie triple multiplicative bijective map,that is ?([[A,B]*,C]*)=[[?(A),?(B)]*,?(C)]*,VA,B,C?A,then ?=?(I)-1?1+?2,where ?1|PAP is a linear*-isomorphism,?2|(I-P)A(I-P)is a conjugate linear*-isomorphism,P is a central projection.In particularly,if A is a factor von Neumann algebra,?:A?A is a skew Lie triple multiplicative bijective map if and only if ? or-? is a linear*-isomorphism or conjugate linear*-isomorphism.In the fourth chapter,we characterize maps completely preserving skew Lie zero product and maps completely preserving skew Jordan zero product on von Neumann algebras.1.Characterization of maps completely preserving skew Lie zero product.Let A be a von Neumann algebra with no central abelian projections.If ?:A?A is a surjective map,then the following statements are equivalent.(1)? is 2-preserving skew Lie zero product in both directions;(2)? is completely preserving the skew Lie zero product in both directions;(3)there exists the central element Z?Z(A)such that ?=Z?= ?1+?2,where ?1|PAP is a linear*-isomorphism,?2|(I-P)A(I-P)is a conjugate linear*-isomorphism.2.Characterization of maps completely preserving skew Jordan zero product.Let A be a von Neumann algebra with no central abelian projections.If ?:A?A is a surjective map,then the following statements are equivalent.(1)? is 2-preserving skew Jordan zero products map in both directions;(2)? is completely preserving the skew Jordan zero product in both directions;(3)there exists the central element Z ?Z(A)such that(?)= Z?=(?)1+(?)2,where(?)1|PAP is a linear*-isomorphism,(?)2|(I-P)A(I-P)is a conjugate linear*-isomorphism.