Operator Algebras, Jordan Mappings And Lie Ideals

Recently,many people have investigated the structure of Jordan maps and Lie ideals in operator algebras vastly.When it refers to triangular algebra,a special and clearly structured operator algebra.The additivity of Jordan maps is valuable to be considered too. Let T be a triangular algebra,B be an associative algebra over the field Q of rational numbers and r is a rational number.The first part of this paper aims to investigate the Jordan triple maps from T to B.It shows that if mapφfrom T onto B that is bijective and is Jordan triples maps,thenφis additive.Both the matrix structure of triangular algebra and purely algebric method called Pierce decomposition are used in the proof.The theory of operator algebras' Lie structure is one of the wealthiest fields of operator algebras from 1950's.Many people have been studying the Lie structure(Lie ideal,Lie derivations,Lie isomorphism) because it is very important to reveal the structure of various operator algebras.In many instances,the Lie ideals can be exactly determined,or there are close connections between the Lie ideal structure and the associative structure of algebras.This connection has been investigated for some special algebras in recent years,and get a plentiful harvest.In the case of non-self-adjoint operator algebras,the weakly closed Lie ideals in Nest algebras,the norm-closed Lie ideals in TUHF algebras and TAF algebras have been fulfilled.However,because of the complexity of von Neumann algebras' structure(lack of enough rank one operator).The result of Lie ideals in Nest algebras in von Neumann algebras and normal Hilbert spaces may be similar.But there are striking differences in the process of proof.Making use of the special structure and matrix algebric ways,the second part of this paper shows the structure of maximal n—nilpotent ideal of atomic nest algebras in factors.