Derivations preserving quasinilpotent elements

We consider a Banach algebra A with the property that, roughly speaking, sufficiently many irreducible representations of A on non‐trivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large: it includes C ∗ ‐algebras, group algebras on arbitrary locally compact groups, commutative algebras, L ( X ) for any Banach space X , and various other examples. Our main result states that every derivation of A that preserves the set of quasinilpotent elements has its range in the radical of A .