ON A BREŠAR–ŠEMRL CONJECTURE AND DERIVATIONS OF BANACH ALGEBRAS

In this paper, we answer a question on derivations of dense algebras of linear operators posed by Brešar and šemrl. Our theorem implies the following result: let ℬ be a complex Banach algebra, and let d and g be continuous derivations of ℬ. If dg(x) is quasi-nilpotent for every x ∈ ℬ, then dg(x)³ lies in the radical of ℬ for every x ∈ ℬ. This result was proved by Brešar and Šemrl with the additional assumption gd = dg.