On the distance of the composition of two derivations to the generalized derivations

A well-known theorem of E. Posner 10 states that if the composition d 1 d 2 of derivations d 1 d 2 of a prime ring A of characteristic not 2 is a derivation, then either d 1 = 0 or d 2 = 0. A number of authors have generalized this theorem in several ways (see e.g. 1, 2, and 5, where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in 3 (resp. 12). Recently, M. Mathieu 8 extended Posner's theorem to arbitrary C * -algebras.