Normed algebras of differentiable functions on compact plane sets
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...
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| Format: | Article |
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Springer
2010
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| Online Access: | https://eprints.nottingham.ac.uk/28114/ |
| _version_ | 1848793509909233664 |
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| author | Dales, H.G. Feinstein, Joel |
| author_facet | Dales, H.G. Feinstein, Joel |
| author_sort | Dales, H.G. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:01:26Z |
| format | Article |
| id | nottingham-28114 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:01:26Z |
| publishDate | 2010 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-281142020-05-04T20:25:09Z https://eprints.nottingham.ac.uk/28114/ Normed algebras of differentiable functions on compact plane sets Dales, H.G. Feinstein, Joel 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. Springer 2010-02 Article PeerReviewed Dales, H.G. and Feinstein, Joel (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 Normed algebra differentiable functions Banach function algebra completions pointwise regularity of compact plane sets http://link.springer.com/article/10.1007%2Fs13226-010-0005-1 doi:10.1007/s13226-010-0005-1 doi:10.1007/s13226-010-0005-1 |
| spellingShingle | Normed algebra differentiable functions Banach function algebra completions pointwise regularity of compact plane sets Dales, H.G. Feinstein, Joel Normed algebras of differentiable functions on compact plane sets |
| title | Normed algebras of differentiable functions on compact plane sets |
| title_full | Normed algebras of differentiable functions on compact plane sets |
| title_fullStr | Normed algebras of differentiable functions on compact plane sets |
| title_full_unstemmed | Normed algebras of differentiable functions on compact plane sets |
| title_short | Normed algebras of differentiable functions on compact plane sets |
| title_sort | normed algebras of differentiable functions on compact plane sets |
| topic | Normed algebra differentiable functions Banach function algebra completions pointwise regularity of compact plane sets |
| url | https://eprints.nottingham.ac.uk/28114/ https://eprints.nottingham.ac.uk/28114/ https://eprints.nottingham.ac.uk/28114/ |