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 all a, 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 for some a0 ? A then ? is symmetric, i.e. ?(a, b) = ?(b, a) and there exists a unital homomorphism ? : A ? B such that ?(a, b) = ?(ab)?(a0) ?1.