The Singer-Wermer Conjecture
states that if D is a (possibly unbounded) 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) ⊆ 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
concerning 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 is easily seen to imply the Singer-Wermer
Conjecture. The proof itself is done by contradiction in which the remaining case
leads to a new derivation on a commutative subalgebra, and this case can
be contradicted by the arguments used in the proof of the Singer-Wermer
Conjecture.
