Dales, H.G. and Loy, Richard J.
(1997)
*Uniqueness of the norm topology for Banach algebras with finite-dimensional radical.*
Proceedings of the London Mathematical Society, 74 (3).
pp. 633-661.
ISSN 0024-6115

## Abstract

Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.