Primitive ideals and derivations on non-commutative Banach algebras

The Singer-Wermer Conjecture states that if D is a (possibly un-bounded) derivation on a commutative Banach algebra then the range of D is contained in the (Jacobson) radical of the algebra. This conjecture is now known to be true. However, it is still not currently known whether or not the Singer-Wermer Conjecture on derivations extends to non-commutative Banach algebras in the following sense: if D is a (possibly unbounded) derivation then is D(P) C P for all primitive ideals P of the algebra? This has become known as the non-commutative version of the Singer-Wermer Conjecture. We first correct an automatic continuity result in the literature concern-ing which (and how many) primitive ideals can fail to be invariant. Using this result together with some representation theory we prove a theorem about derivations whose second iteration annihilates some element (specifically, D2a = 0 implies that Da is quasinilpotent). This theorem does not require commutativity of the algebra and it i