Jordan Derivations Of The Stability. Operator Algebra And A Great N-nilpotent Ideal

Recently,many people have investigated the stability problem of functional equations and ideals of operator algebras vastly.In 1940,Ulam posed the first stability problem cencerning the stability of group homomorphism.Under what conditions there exists additive mapping near an approximately additive mapping? After that,this result was generalized by a number of mathematicians. And all of these results are known today as Hyers-Ulam-Rassias stability of functional equations.Jordan derivation is a type of important map in Banach algebras.So the generalized Hyers-Ulam-Rassias stability of Jordan derivations is valuable to be considered.Assume that A and B are Banach algebras and the linear mapping L:Aâ†'B is a Jordan derivation.The first part of this paper aims to prove the generalized Hyers-Ulam-Rassias stability of Jordan derivations form a normed algebra onto a Banach algebra by associated to a generalized Jensen's equation f((x+y)/K)=(f(x)+f(y))/K.Nest algebras,as a kind of important non-self-adjoint operator algebra, have a very rich ideal structure and that this should be a focus of interest.In fact,many authors have studied the ideals of nest algebras.The most important and first studied ideal of nest algebras is Jacoboson radical.In his foundational paper"On some algebras of operators",Ringrose gave an elegant characterization. In general algebra theory,the research of n-nilpotent ideal is an important project too.So the n-nilpotent ideal of nest algebras is valuable to be considered too.The second part of this paper investigates the maximal n-nilpotent ideal of nest algebra in a factor.Using the theory of the comparison of projections in a vN algebra,a necessary and sufficient condition that an ideal of nest algebra in a factor is maximal n-nilpotent is given.This result generalizes the result due to Lu Fangyan.