H.G. Dales
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 | 0975-7465 |
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 |
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