Normed Algebras of Differentiable Functions on Compact Plane Sets

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 F-derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D (1) F (X) 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 (1) F (X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘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.