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Uniqueness of the Norm Topology for Banach Algebras with Finite‐Dimensional Radical
Author(s) -
Dales HG,
Loy RJ
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s002461159700021x
Subject(s) - mathematics , uniqueness , separable space , norm (philosophy) , commutative property , conjecture , pure mathematics , algebra over a field , mathematical analysis , political science , law
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.