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 if?U?U is a nonlinear map satisfying?([[A,B],C])]=[[?(A),B],C]+[[A,?(B)],C]+[[A,B],?(C)]for any A,B,C?U with ABC=0,then 6(A)=d(A)+?(A)I for any A?U,d:U?U is an additive derivation and ?:U?CI is a nonlinear map such that?([[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??:T2=0}.We prove that if?:T T is a nonlinear map satisfying?([A,B])=[?(A),B]+[A,?(B)]for all A,B? ? with AB ?Q,then ?(A)=d(A)+?(A)for all A??,where d:??? is an additive derivation and ? is a nonlinear map from ? into its centre Z(?)such that ?([A,B])=0 for all A,B?? with AB?Q.