Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions
We continue the study of the completeness and completions of normed algebras of differentiable functions D(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(K) and results concerning the semisimplicity of the completion of D(K). In particular, we prove that semi-rectifiability is necessary for the completion of D(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.