Normed algebras of differentiable functions on compact plane sets

Dales, H.G.; Feinstein, Joel

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 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.

Citation

Dales, H., & Feinstein, J. (2010). Normed algebras of differentiable functions on compact plane sets. Indian Journal of Pure and Applied Mathematics, 41(1), https://doi.org/10.1007/s13226-010-0005-1

• Journal Article Type: Article
• Publication Date: Feb 1, 2010
• Deposit Date: Jan 28, 2015
• Publicly Available Date: Jan 28, 2015
• Journal: Indian Journal of Pure and Applied Mathematics
• Print ISSN: 0019-5588
• Electronic ISSN: 0019-5588
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 41
• Issue: 1
• DOI: https://doi.org/10.1007/s13226-010-0005-1
• Keywords: Normed algebra, differentiable functions, Banach function algebra, completions, pointwise regularity of compact plane sets
• Public URL: https://nottingham-repository.worktribe.com/output/1012172
• Publisher URL: http://link.springer.com/article/10.1007%2Fs13226-010-0005-1
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s13226-010-0005-1