Normed algebras of differentiable functions on compact plane sets

TL;DR

This paper studies the properties of normed algebras of differentiable functions on compact plane sets, focusing on their completeness, structure, and the relationship between different classes of such algebras.

Contribution

It constructs examples of non-complete algebras, characterizes when these algebras are complete, and explores the relationship between various classes of differentiable function algebras.

Findings

Constructed a compact plane set where $D^{(1)}(X)$ is not complete.

Established conditions under which $D^{(1)}(X)$ is complete, such as pointwise regularity.

Showed that the completion of $D^{(1)}(X)$ is semisimple when certain density conditions are met.

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.