The(m,n)-triple Derivatives On A Non-self-adjoint Operator Algebra

Functional analysis is one of the important research areas of modern mathematics.Operator algebra theory is a research direction developed in the 1930 s.It has become an extremely important branch of functional analysis and is widely used in other mathematical fields.It is also an important mathematical tool which can be used in studying many subjects,such as,quantum theory,mathematical physics,communication network and information analysis,engineering technology and probability theory.Operator algebra is divided into self-adjoint operator algebras and non-self-adjoint operator algebras.Compared with the self-adjoint operator algebras,the non-self-adjoint operator algebras have a more abundant phenomenon in mathematics,and it affects each other with various mathematical filds,becoming a very active research direction of operator algebra.Nest algebra and CDC algebra are two very important types of non-self-adjoint operator algebras.At present,domestic and foreign mathematicians have studied the derivation,triple derivation,(m,n)-derivation,Jordan derivation,Lie-derivation and higher-order derivable mappings on operator algebras,and it provides a powerful tools for further studying the properties and structure of operator algebras.This paper introduces the concept of(m,n)-triple derivation in operator algebras,and uses the method of algebraic decomposition to study the(m,n)-triple derivation and(m,n)-Lie centralizer on operator algebra,and gives their characterization.In the first part,the research background,current research status at home and abroad,the main research content and related basic knowledge are given.In the second part,the(m,n)-triple derivatives on the nested algebra are studied.It is proved that the(m,n)-triple derivations are triple derivations on the nested algebra.In the third part,the(m,n)-triple derivatives on a factor von Neumann algebra are studied.And it is proved that the(m,n)-triple derivations are triple derivations on the factor von Neumann algebra.In the fourth part,the characterization of the CDC(m,n)-Lie centralizers is given.