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...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2016
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| _version_ | 1826269538404007936 |
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| author | Feinstein, JF Morley, S Yang, H |
| author_facet | Feinstein, JF Morley, S Yang, H |
| author_sort | Feinstein, JF |
| collection | OXFORD |
| 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 [5].
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| first_indexed | 2024-03-06T21:26:38Z |
| format | Journal article |
| id | oxford-uuid:434cdcf5-ea3d-4a0b-a823-25ee93d1cdab |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:26:38Z |
| publishDate | 2016 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:434cdcf5-ea3d-4a0b-a823-25ee93d1cdab2022-03-26T14:54:40ZAbstract Swiss cheese space and classicalisation of Swiss cheesesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:434cdcf5-ea3d-4a0b-a823-25ee93d1cdabEnglishSymplectic ElementsElsevier2016Feinstein, JFMorley, SYang, HSwiss 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 [5]. |
| spellingShingle | Feinstein, JF 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 |
| work_keys_str_mv | AT feinsteinjf abstractswisscheesespaceandclassicalisationofswisscheeses AT morleys abstractswisscheesespaceandclassicalisationofswisscheeses AT yangh abstractswisscheesespaceandclassicalisationofswisscheeses |