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Regularity points and Jensen measures for R(X)

Feinstein, Joel; Yang, H.

Regularity points and Jensen measures for R(X) Thumbnail


H. Yang


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 (2000). 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.

Journal Article Type Article
Acceptance Date Jul 1, 2016
Online Publication Date Sep 30, 2016
Deposit Date Jul 19, 2016
Publicly Available Date Sep 30, 2016
Journal Studia Mathematica
Print ISSN 0039-3223
Electronic ISSN 1730-6337
Publisher Polskiej Akademii Nauk, Instytut Matematyczny
Peer Reviewed Peer Reviewed
Volume 235
Issue 1
Keywords swiss cheeses, uniform algebras, regularity of R(X)
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