A note on ideal spaces of Banach Algebras

Details A note on ideal spaces of Banach Algebras A note on ideal spaces of Banach Algebras

1998-09-16

arxiv.org/abs/math/9809085v1

Mathematics

Functional Analysis

9 pages plain tex

Scientific paper

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it is shown that this topology is Hausdorff if A is the algebra of once continuously differentiable functions on an interval, but that if A is a uniform algebra then this topology 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.

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