Regularity points and Jensen measures for R(X)
We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in [16]. We show that, even for the natural uniform algebras R(X) (for compact plane sets X), these two types of regularity point ca...
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| Format: | Article |
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Polskiej Akademii Nauk, Instytut Matematyczny
2016
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| Online Access: | https://eprints.nottingham.ac.uk/35157/ |
| _version_ | 1848795016224309248 |
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| author | Feinstein, Joel Yang, H. |
| author_facet | Feinstein, Joel Yang, H. |
| author_sort | Feinstein, Joel |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in [16]. We show that, even for the natural uniform algebras R(X) (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets X such that R(X) is not regular, but such that R(X) has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set X with the property that the set of points of discontinuity for R(X) has positive area. |
| first_indexed | 2025-11-14T19:25:23Z |
| format | Article |
| id | nottingham-35157 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:25:23Z |
| publishDate | 2016 |
| publisher | Polskiej Akademii Nauk, Instytut Matematyczny |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-351572020-05-04T18:08:46Z https://eprints.nottingham.ac.uk/35157/ Regularity points and Jensen measures for R(X) Feinstein, Joel Yang, H. We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in [16]. We show that, even for the natural uniform algebras R(X) (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets X such that R(X) is not regular, but such that R(X) has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set X with the property that the set of points of discontinuity for R(X) has positive area. Polskiej Akademii Nauk, Instytut Matematyczny 2016-09-30 Article PeerReviewed Feinstein, Joel and Yang, H. (2016) Regularity points and Jensen measures for R(X). Studia Mathematica, 235 (1). pp. 1-15. ISSN 1730-6337 swiss cheeses uniform algebras regularity of R(X) https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/studia-mathematica/all/235/1/91779/regularity-points-and-jensen-measures-for-r-x doi:10.4064/sm8351-7-2016 doi:10.4064/sm8351-7-2016 |
| spellingShingle | swiss cheeses uniform algebras regularity of R(X) Feinstein, Joel Yang, H. Regularity points and Jensen measures for R(X) |
| title | Regularity points and Jensen measures for R(X) |
| title_full | Regularity points and Jensen measures for R(X) |
| title_fullStr | Regularity points and Jensen measures for R(X) |
| title_full_unstemmed | Regularity points and Jensen measures for R(X) |
| title_short | Regularity points and Jensen measures for R(X) |
| title_sort | regularity points and jensen measures for r(x) |
| topic | swiss cheeses uniform algebras regularity of R(X) |
| url | https://eprints.nottingham.ac.uk/35157/ https://eprints.nottingham.ac.uk/35157/ https://eprints.nottingham.ac.uk/35157/ |