Derivable maps at commutative products on Banach algebras.

Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and w ∈ A . In this paper, we characterize those continuous linear maps δ : A → M that satisfy one of the following conditions: δ (a b) = δ (a) b + a δ (b) , 2 δ (w) = δ (a) b + a δ (b) , δ (a b) = δ (a) b + a δ (b) - a δ (e) b , for any a , b ∈ A with a b = b a = w , where w is either a separating point with w ∈ Z (A) or an idempotent. [ABSTRACT FROM AUTHOR]