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 $\mathbb{C}$. 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 $D_{\mathcal{F}}^{(1)}(X)$ corresponding to the normed algebras $D^{(1)}(X)$. In the present paper, we obtain stronger results concerning the questions when $D^{(1)}(X)$ and $D_{\mathcal{F}}^{(1)}(X)$ 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.