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Abstract Swiss cheese space and classicalisation of Swiss cheeses

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Abstract

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].

Details

Original languageEnglish
Pages (from-to)119-141
JournalJournal of Mathematical Analysis and Applications
Volume438
Issue number1
Early online date6 Feb 2016
DOIs
Publication statusPublished - 1 Jun 2016
Peer-reviewedYes
Externally publishedYes

Keywords

    Research areas

  • Swiss cheeses, Rational approximation, Uniform algebras, Bounded point derivations, Regularity of, R(X)

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