Resumen de Automatic continuity for Banach algebras with finite-dimensional radical

Hungq Le Pham

• The paper \cite{dalesloy1997} proved a necessary algebraic condition for a Banach algebra $A$ with finite-dimensional radical $R$ 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 $A$ is separable and $A/R$ is commutative, but is shown to fail in general. Similar questions for derivations are discussed.