## Abstract

We investigate the completeness and completions of the normed algebras (*D*
^{(1)}(*X*), ‖ · ‖) for perfect, compact plane sets *X*. In particular, we construct a radially self-absorbing, compact plane set *X* such that the normed algebra (*D*
^{(1)}(*X*), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets *X* for which the completeness of (*D*
^{(1)}(*X*), ‖ · ‖) is equivalent to the pointwise regularity of *X*. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.

In an earlier paper of Bland and Feinstein, the notion of an \( \mathcal{F} \)-derivative of a function was introduced, where \( \mathcal{F} \) is a suitable set of rectifiable paths, and with it a new family of Banach algebras \( D_\mathcal{F}^{\left( 1 \right)} \left( X \right) \) corresponding to the normed algebras *D*
^{(1)}(*X*). In the present paper, we obtain stronger results concerning the questions when *D*
^{(1)}(*X*) and \( D_\mathcal{F}^{\left( 1 \right)} \left( X \right) \) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever *X* is ‘\( \mathcal{F} \)-regular’.

An example of Bishop shows that the completion of (*D*
^{(1)}(*X*), ‖ · ‖) need not be semisimple. We show that the completion of (*D*
^{(1)}(*X*), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in *X* is dense in *X*.

We prove that the character space of *D*
^{(1)}(*X*) is equal to *X* for all perfect, compact plane sets *X*, whether or not (*D*
^{(1)}(*X*), ‖ · ‖) is complete. In particular, characters on the normed algebras (*D*
^{(1)}(*X*), ‖ · ‖) are automatically continuous.

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Dales, H.G., Feinstein, J.F. Normed algebras of differentiable functions on compact plane sets.
*Indian J Pure Appl Math* **41, **153–187 (2010). https://doi.org/10.1007/s13226-010-0005-1

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### Key words

- Normed algebra
- differentiable functions
- Banach function algebra
- completions
- pointwise regularity of compact plane sets