J. Aust. Math. Soc.  81 (2006), 279-295
Automatic continuity for Banach algebras with finite-dimensional radical

 Hung Le Pham   Department of Pure Mathematics   University of Leeds   Leeds LS2 9JT   UK  hung@maths.leeds.ac.uk

Abstract
The paper \cite{dalesloy1997} proved a necessary algebraic condition for a Banach algebra with finite-dimensional radical to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where is separable and is commutative, but is shown to fail in general. Similar questions for derivations are discussed.