Equivalent characterization of derivations on operator algebras

Abstract

Let R be a unital ring with a nontrivial idempotent p, and let M be an R-bimodule. We say that an additive map δ : R → M is derivable at β ∈ R if δ(x y) = δ(x)y + xδ(y) for any x, y ∈ R with x y = β. In this paper, we give a necessary and sufficient condition for an additive map δ : R → M to be derivable at β with β = pβ = βp. Moreover, we show that if R is a prime Banach algebra with the unit 1, then an additive map δ : R → R is derivable at β with β = pβ = βp if and only if there is a derivation τ : R → R such that δ(x) = τ (x) + δ(1)x for all x ∈ R. 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 δ : B(X ) → B(X ) 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.