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Ideal Spaces of Banach Algebras
Author(s) -
Somerset D. W. B.
Publication year - 1999
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/s0024611599001677
Subject(s) - mathematics , metrization theorem , hausdorff space , separable space , ideal (ethics) , banach space , pure mathematics , banach manifold , norm (philosophy) , banach algebra , regular space , maximal ideal , discrete mathematics , countable set , topology (electrical circuits) , lp space , combinatorics , mathematical analysis , philosophy , epistemology , political science , law
The ideal space Id( A ) of a Banach algebra A is studied as a bitopological space Id( A ), τ u , τ n , where τ u is the weakest topology for which all the norm functions I → ‖ a + I ‖ (with a ∈ A and I ∈ Id( A )) are upper semi‐continuous, and τ n is the de Groot dual of τ u . When A is separable, τ n ∨ τ u is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF‐algebras are shown to exhibit the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI‐Banach algebras, are given. 1991 Mathematics Subject Classification : 46H10, 46J20.