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Proceedings of the London Mathematical Society 1999 78(2):369-400; doi:10.1112/S0024611599001677
© 1999 by London Mathematical Society
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© London Mathematical Society

Ideal Spaces of Banach Algebras

D. W. B. Somerset

Department of Mathematical Sciences, University of Aberdeen Aberdeen, AB24 3UE, U.K. E-mail: ds{at}maths.abdn.ac.uk

Received 21 March 1997. Revision received 13 November 1997.

The ideal space Id(A) of a Banach algebra A is studied as a bitopological space Id(A), {tau}u, {tau}n, where {tau}u is the weakest topology for which all the norm functions I -> || a + I|| (with a isin A and I isin Id(A)) are upper semi-continuous, and {tau}n is the de Groot dual of {tau}u. When A is separable, {tau}n{vee}{tau}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.


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