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Banach function algebras with dense invertible group

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<mark>Journal publication date</mark>2008
<mark>Journal</mark>Proceedings of the American Mathematical Society
Issue number4
Volume136
Number of pages10
Pages (from-to)1295-1304
<mark>State</mark>Published
Early online date21/12/07
<mark>Original language</mark>English

Abstract

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $ \mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))$ whenever $ A$ is approximately regular.