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

Published online by Cambridge University Press:  01 May 1997

HG Dales  and
RJ Loy
HG Dales
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK. [email protected]
RJ Loy
Affiliation:
Department of Mathematics, Australian National University, ACT 0200, Australia. [email protected]
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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.

1991 Mathematics Subject Classification: 46H20, 46H40.

Keywords

Banach algebrasuniqueness of normradicalanalytic space
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 74 , Issue 3 , May 1997 , pp. 633 - 661
Copyright
© London Mathematical Society 1997

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