Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions

Heiko Hoffmann¹

¹ Department of Mathematics Institute of Analysis Karlsruhe Institute of Technology (KIT) 76128 Karlsruhe, Germany

Studia Mathematica, Tome 207 (2011) no. 1, pp. 19-45

Résumé

We continue the study of the completeness and completions of normed algebras of differentiable functions $D^n(K)$ (where $K$ is a perfect, compact plane set), initiated by Bland, Dales and Feinstein [Studia Math. 170 (2005) and Indian J. Pure Appl. Math. 41 (2010)]. We prove new characterizations of the completeness of $D^{1}(K)$ and results concerning the semisimplicity of the completion of $D^{1}(K)$. In particular, we prove that semi-rectifiability is necessary for the completion of $D^{1}(K)$ to be semisimple in the case where $K$ lies on a rectifiable, injective curve. Furthermore, we answer a question posed by Dales and Feinstein and show that another question posed by them has an affirmative answer in some special cases. As compared with the approach taken by Bland, Dales and Feinstein, which comes from the theory of function algebras, we move within an operator-theoretic framework by investigating the mapping properties of certain derivation operators.

Détail

DOI : 10.4064/sm207-1-2

Keywords: continue study completeness completions normed algebras differentiable functions where perfect compact plane set initiated bland dales feinstein studia math indian pure appl math prove characterizations completeness results concerning semisimplicity completion particular prove semi rectifiability necessary completion semisimple where lies rectifiable injective curve furthermore answer question posed dales feinstein another question posed has affirmative answer special cases compared approach taken bland dales feinstein which comes theory function algebras move within operator theoretic framework investigating mapping properties certain derivation operators