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Uniqueness of the norm topology for Banach algebras with finite-dimensional radical

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>05/1997
<mark>Journal</mark>Proceedings of the London Mathematical Society
Issue number3
Volume74
Number of pages29
Pages (from-to)633-661
<mark>State</mark>Published
<mark>Original language</mark>English

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.