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Journal of Function Spaces and Applications
Volume 2012, Article ID 905650, 15 pages
http://dx.doi.org/10.1155/2012/905650
Research Article

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

1Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, 57045 Metz, France
2Fakultät Allgemeinwissenschaften, Georg-Simon-Ohm-Hochschule Nürnberg, Keßlerplatz 12, 90489 Nürnberg, Germany

Received 13 March 2010; Accepted 19 October 2010

Academic Editor: N. Kalton

Copyright © 2012 Raymond Mortini and Rudolf Rupp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 .