Normed algebras of differentiable functions on compact plane sets

Dales, H.G. and Feinstein, J. F. (2010) Normed algebras of differentiable functions on compact plane sets. Indian Journal of Pure and Applied Mathematics, 41 (1). pp. 153-187. ISSN 0019-5588

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 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.