In an earlier paper, 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, for an approximately regular, commutative, unital Banach algebra with character space X, we prove that the topological stable rank of A is no less than that of C(X).

This is joint work with H.G. Dales.