Derivable And Lie Derivable Maps On Von Neumann Algebra

Derivable maps,Lie derivable maps,Jordan multiplicative derivations and Jordan multi-plicative maps are very important in operator algebras and operator theory,and have received a fair amount of attention.In this thesis,we first characterize a linear bounded map on a von Neumann algebra,which is derivable at an arbitrary but fixed operator,and show that an arbitrary but fixed operator is generalized full-derivable point.Next we investigate the Jordan semi-triple multiplicative derivations,Jordan semi-triple and Jordan semi-triple-*multiplica-tive maps on von Neumann algebra with no central abelian projections.Finally,we investigate Lie derivable maps on von Neumann algebras.The structure of this thesis are as follows:Part Ⅰ,we characterize derivable maps on von Neumann algebras,and we prove that its arbitrary but fixed operator is a generalized full-derivable point.Let A be a von Neumann algebra and Ω∈A be an arbitrary but fixed operator.Then a linear bounded map δ:A→A is derivable at Ω,that is δ(AB)=δ(A)B+Aδ(B),(?)A,B ∈R,AB=Ω if and only if there is a derivation τ:A→A such that δ(A)=τ(A)+δ(I)A,(?)A ∈A,where 6(I)∈ Z(A)and δ(I)Ω=0.Part Ⅱ,we investigate the multiplicative derivations and multiplicative isomorphism on von Neumann algebras with no central abelian projections.Let A be a von Neumann algebra with no central abelian projections,then the following statements are true.1.If δ:A→A is a Jordan semi-triple multiplicative derivation,that is,δ(ABA)=δ(A)BA+Aδ(B)A+ABδ(A)for all A,B ∈ A,then δ is an additive derivation.2.If φ:A→A is a Jordan semi-triple multiplicative bijective map,that is,φ(ABA)=φ(A)φ(B)φ(A)for all A,B ∈ A,then φ is an additive Jordan ring isomorphism;3.If φ:A→A is a Jordan semi-triple-*multiplicative bijective map,that is,砂(AB*A)=φ(A)φ(B)*φ(A)for all A,B ∈ A,then 0(I)*φ is an additive Jordan-*ring isomorphism.Part Ⅲ,we investigate Lie derivable maps on von Neumann algebras.Let A be a von Neumann algebra,Ω∈A and P∈A be the range projection of Ωsuch that (?)=0,(?)=I.Then additive map δ:A→A is Lie derivable at Ω,that isδ([A,B])=[δ(A),B]+[A,δ(B)],(?)A,B∈A,AB=Ω,if and only if there is a derivationτ:A→A and additive map f:A→Z(A)such that δ(A)=τ(A)+f(A),(?)A ∈A.