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A Note on Ideal Spaces of Banach Algebras
Author(s) -
Feinstein J. F.,
Somerset D. W. B.
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398004512
Subject(s) - mathematics , metrization theorem , hausdorff space , banach algebra , separable space , ideal (ethics) , countable set , pure mathematics , bounded function , second countable space , maximal ideal , banach space , discrete mathematics , mathematical analysis , philosophy , epistemology
In a previous paper, the second author introduced a compact topology τ r on the space of closed ideals of a unital Banach algebra A . If A is separable, then τ r is either metrizable or else neither Hausdorff nor first countable. Here it is shown that τ r is Hausdorff if A is C 1 [0, 1], but that if A is a uniform algebra, then τ r is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis. 1991 Mathematics Subject Classification 46H10.