Equivalent characterization of derivations on operator algebras

Abstract

Let be a unital ring with a nontrivial idempotent , and let be an -bimodule. We say that an additive map is derivable at if for any with . In this paper, we give a necessary and sufficient condition for an additive map to be derivable at with . Moreover, we show that if is a prime Banach algebra with the unit , then an additive map is derivable at with if and only if there is a derivation such that for all . As an application, we get a full characterization of derivable maps on some reflexive algebras and von Neumann algebras with no abelian summands. In particular, we show that an additive map is derivable at any nonzero finite rank operator if and only if it is a derivation. New equivalent characterization of derivations on these algebras are obtained.