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

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.
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