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 ∈ 𝒜 with AB = P), then d is a derivation or d(A) = τ(A) + AN for some additive derivation τ : 𝒜 → ℳ and some N ∈ 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.