The Canonical Test Case for the Non-commutative Singer-wermer Conjecture

It is a famous conjecture that every derivation on each Banach algebra leaves every primitive ideal of the algebra invariant. This conjecture is known to be true if, in addition, the derivation is assumed to be continuous. It is also known to be true if the algebra is commutative, in which case the derivation necessarily maps into the (Jacobson) radical. Because I. M. Singer and J. Wermer originally raised the question in 1955 for the case of commutative Banach algebras, the conjecture is now usually referred to as the non-commutative Singer-Wermer conjecture (the non-commutative situation being the unresolved case). In a previous paper we demonstrated that if the conjecture fails for some non-commutative Banach algebra with discontinuous derivation, then it fails for at most finitely many primitive ideals, and each of these primitive ideals must be of finite codimension. In this paper we first show that one can make an additional reduction of any counter-example to the simplest case of a non-commutative radical Banach algebra with identity adjoined and discontinuous derivation D such that D does not leave the (Jacobson) radical (which is of codimension one) invariant. Second, we show that this radical Banach algebra with identity adjoined has a formal power series quotient of the form A0[[t]] based at an element t in the radical which is mapped to an invertible element by the discontinuous derivation. Finally, we specialize to the case of a separable Banach algebra and show that the pre-image of the algebra A0 is a unital subalgebra which is not an analytic set. In particular, this shows that A0 cannot be countably generated.