Additive mappings derivable at non-trivial idempotents on Banach algebras

Let A be a Banach algebra with unity I containing a non-trivial idempotent P and M be a unital A-bimodule. Under several conditions on A, M and P, we show that if d : A -> M is an additive mapping derivable at P (i.e. d(AB) = Ad(B) + d(A)B for any A, B is an element of A with AB = P), then d is a derivation or d(A) = tau(A) + AN for some additive derivation tau : A -> M and some N is an element of M, and various examples are given which illustrate limitations on extending some of the theory developed. Also, we describe the additive mappings derivable at P on semiprime Banach algebras and C*-algebras. As applications of the above results, we characterize the additive mappings derivable at P on matrix algebras, Banach space nest algebras, standard operator algebras and nest subalgebras of von Neumann algebras. Moreover, we obtain some results about automatic continuity of linear (additive) mappings derivable at P on various Banach algebras.