Abstract
Let π be a Banach algebra with unity I containing a non-trivial idempotent P and β³ be a unital π-bimodule. Under several conditions on π, β³ and P, we show that if dβ:βπββββ³ is an additive mapping derivable at P (i.e. d(AB)β=βAd(B)β+βd(A)B for any A,βBβββπ with ABβ=βP), then d is a derivation or d(A)β=βΟ(A)β+βAN for some additive derivation Οβ:βπββββ³ and some Nββββ³, 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.
Acknowledgements
The author like to express his sincere thanks to the referees for this paper.