Abstract Swiss cheese space and classicalisation of Swiss cheeses
Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call “abstract S...
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| Format: | Article |
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Elsevier
2016
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| Online Access: | https://eprints.nottingham.ac.uk/32267/ |
| _version_ | 1848794373340266496 |
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| author | Feinstein, Joel Morley, S. Yang, H. |
| author_facet | Feinstein, Joel Morley, S. Yang, H. |
| author_sort | Feinstein, Joel |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call “abstract Swiss cheeses”. Working within this topological space, we show how to prove the existence of “classical” Swiss cheese sets (as discussed in [6]) with various desired properties. We first give a new proof of the Feinstein–Heath classicalisation theorem [6]. We then consider when it is possible to “classicalise” a Swiss cheese while leaving disks which lie outside a given region unchanged. We also consider sets obtained by deleting a sequence of open disks from a closed annulus, and we obtain an analogue of the Feinstein–Heath theorem for these sets. We then discuss regularity for certain uniform algebras. We conclude with an application of these techniques to obtain a classical Swiss cheese set which has the same properties as a non-classical example of O’Farrell. |
| first_indexed | 2025-11-14T19:15:10Z |
| format | Article |
| id | nottingham-32267 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:15:10Z |
| publishDate | 2016 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-322672020-05-04T17:48:24Z https://eprints.nottingham.ac.uk/32267/ Abstract Swiss cheese space and classicalisation of Swiss cheeses Feinstein, Joel Morley, S. Yang, H. Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call “abstract Swiss cheeses”. Working within this topological space, we show how to prove the existence of “classical” Swiss cheese sets (as discussed in [6]) with various desired properties. We first give a new proof of the Feinstein–Heath classicalisation theorem [6]. We then consider when it is possible to “classicalise” a Swiss cheese while leaving disks which lie outside a given region unchanged. We also consider sets obtained by deleting a sequence of open disks from a closed annulus, and we obtain an analogue of the Feinstein–Heath theorem for these sets. We then discuss regularity for certain uniform algebras. We conclude with an application of these techniques to obtain a classical Swiss cheese set which has the same properties as a non-classical example of O’Farrell. Elsevier 2016-06-01 Article PeerReviewed Feinstein, Joel, Morley, S. and Yang, H. (2016) Abstract Swiss cheese space and classicalisation of Swiss cheeses. Journal of Mathematical Analysis and Applications, 438 (1). pp. 119-141. ISSN 0022-247X Swiss Cheeses Rational Approximation Uniform Algebras Bounded Point Derivations Regularity of R(X) http://www.sciencedirect.com/science/article/pii/S0022247X16001232 doi:10.1016/j.jmaa.2016.02.004 doi:10.1016/j.jmaa.2016.02.004 |
| spellingShingle | Swiss Cheeses Rational Approximation Uniform Algebras Bounded Point Derivations Regularity of R(X) Feinstein, Joel Morley, S. Yang, H. Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title | Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title_full | Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title_fullStr | Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title_full_unstemmed | Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title_short | Abstract Swiss cheese space and classicalisation of Swiss cheeses |
| title_sort | abstract swiss cheese space and classicalisation of swiss cheeses |
| topic | Swiss Cheeses Rational Approximation Uniform Algebras Bounded Point Derivations Regularity of R(X) |
| url | https://eprints.nottingham.ac.uk/32267/ https://eprints.nottingham.ac.uk/32267/ https://eprints.nottingham.ac.uk/32267/ |