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Abstract Swiss Cheese Space and the Classicalisation of Swiss Cheeses | | J. F. Feinstein
; S. Morley
; H. Yang
; | | Date: |
12 Mar 2015 | | 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 a paper of
Feinstein and Heath from 2010) with various desired properties.
We first give a new proof of the Feinstein-Heath classicalisation theorem. 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 (1979). | | Source: | arXiv, 1503.3785 | | Services: | Forum | Review | PDF | Favorites | |
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