A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

Abstract

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set 𝐾 and holomorphic in the interior 𝐾∘ of 𝐾. In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on 𝐾.