Characterization Of Non-global Non-linear Derivable Mapping

In recent years,the maps without additivity and linearity assumping have at-tracted the attentions of many scholars.In this paper,with the help of algebraic decomposition,we mainly study the characterization of two types of non-global nonlinear derivable mappings on factor von Neumann algebra and on triangular algebras.The main content are as follows:In Chapter 1,we mainly introduce some definitions,symbols(for example,factor von Neumann algebra,triangular algebra,derivable mappings)and so on.In Chapter 2,we mainly characterize a class of non-global nonlinear Lie triple derivable mappings on factor Neumann algebra.Specifically,let U be a factor von Neumann algebra acting on a Hilbert space H with dim H>1.We prove that ifδU→U is a nonlinear map satisfyingδ([[A,B],C])]=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)]for any A,B,C∈U with ABC=0,then 6(A)=d(A)+τ(A)I for any A∈U,d:U→U is an additive derivation and τ:U→CI is a nonlinear map such thatτ([[A,B],C])=0 with ABC=0 for all A,B,C ∈U.In Chapter 3,we mainly study a class of non-global nonlinear Lie derivable mappings on triangular algebras and characterize the concrete structure.Specifi-cally,let T be a triangular algebra and Q={T∈τ:T2=0}.We prove that ifδ:T T is a nonlinear map satisfyingδ([A,B])=[δ(A),B]+[A,δ(B)]for all A,B∈ τ with AB ∈Q,then δ(A)=d(A)+τ(A)for all A∈τ,where d:τ→τ is an additive derivation and τ is a nonlinear map from τ into its centre Z(τ)such that τ([A,B])=0 for all A,B∈τ with AB∈Q.