Normed algebras of differentiable functions on compact plane sets

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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Correspondence to H. G. Dales.

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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