Map On The Stability Of The Banach Algebra Joran Guide The Child And Sets Of Algebraic Stability

Recently, many people have investigated the stability problem of func-tional equations. In 1940. Ulam posed the first stability problem cencerning the stability of group homomorphisms. Under what conditions there exits addi-tive 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 functions equations. Bi-Jordan derivation is a type of important map in Banach algebras, So the generalized Hyers-Ulam-Rassias stability of Jordan derivation is valuable to be considered. Assume that B is a complex associative algebras with unity. And the linear mapping L:Bâ†'B is a Bi-Jordan derivation, the first part of this paper aims to prove the Hyers-Ulam-Rassias stability of Bi-Jordan derivation in Banach algebras.Nest algebras is a type of important non-self-adjoint operator algebra, many people have investigated the nature of derivation in operator algebra. Semrl has proved an important result:H is an infinite dimensional Banach space and A is a standard subalgebra in H.φ:Aâ†'A is a bijective. Givenε> 0,||φ(xy)-φ(x)φ(y)||≤ε, for every x.y∈A, thenφis either linear or conjugate linear. Basing on the result, I investigated the the Hyers-Ulam-Rassias stability of elementary map and multiplicative map in nest algebra in the second part of this paper.