Some Achievements on Two Variable σ-Derivations

Let b and a be two banach algebras andmbe a banach bbimodule.suppose that ? : a ? b is a linear mapping. a bilinear map? : a×a ?mis called a two variable ?-derivation whenever ?(ab, c) =?(a, c)?(b) + ?(a)?(b, c) and ?(a, bc) = ?(a, b)?(c) + ?(b)?(a, c) for alla, b, c ? a. in this paper, we prove that if a and b are unital and? : a×a?b is a two variable ?-derivation such that ?(1, a0) = 1 forsome a0 ? a then ? is symmetric, i.e. ?(a, b) = ?(b, a) and there existsa unital homomorphism ? : a ? b such that ?(a, b) = ?(ab)?(a0)?1.