SOME ACHIEVEMENTS ON TWO VARIABLES $\sigma$-derivations

Abstract

Let

B and A be two Banach algebras and M be a Banach B

: A ! B

-

: A × A ! M is called a two variables -derivation whenever -(ab, c

-

(a, c)(b)+(a)-(b, c) and -(a, bc) = -(a, b)(c)+(b)-(a, c) for all a, b, c 2 A

A and B are unital and - : A × A ! B

-derivation such that -(1, a0) = 1 for some a0 2 A then

is symmetric, i.e.

-(a, b) = -(b, a

: A ! B such that -(a, b) = (ab)(a0)−1

.

) and there exists a unital homomorphism

-

is atwo variables

.In this paper, we prove that if

) =

is a linear mapping. A bilinear map

bimodule.Suppose that