Nonlinear Derivable Mapping On Factor Von Neumann Algebra

Recently,the map without additivity and linearity assumping has been an im-portant issues.In this paper,we introduce and mainly study the characterization of nonlinear mixed Lie triple derivable mappings and nonlinear*?-Lie derivable mappings on factors.In Chapter 1,we give some common definitions of factor von Neumann algebras,nonlinear mixed Lie triple derivable mappings,nonlinear*?-Lie derivable mappings and so on.In Chapter 2,we mainly discuss nonlinear mixed Lie triple derivable mappings on factors.Let A be a factor von Neumann with dimension greater than 1.We prove that every mapping ?:A? A satisfies ?([[A,B]*,C])=[[?(A),B]*,C]+[[A,?(B)]*,C]+[[A,B]*,?(C)]for any A,B,C ?A,then is an additivve-derivation.In Chapter 3,we mainly characterize nonlinear*?-Lie derivable mappings on factors.Let A be a factor with dim(A)>1 and ? ? 1.We prove that every mapping?:A? A satisfies ?([A*,B]?)?[?(A)*,B]? +[A*,?(B)]? for any A,B ?A,then ?is an additive*-derivation and ?(?A):??(A).