An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem

Mason, J.

doi:10.1090/S0002-9939-2010-10447-1

An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem
HTML articles powered by AMS MathViewer

by J. W. D. Mason

Proc. Amer. Math. Soc. 138 (2010), 4423-4432

DOI: https://doi.org/10.1090/S0002-9939-2010-10447-1

Published electronically: June 16, 2010

PDF | Request permission

Abstract:

A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn’s lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.

References

Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
J. F. Feinstein, A counterexample to a conjecture of S. E. Morris, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2389–2397. MR 2052417, DOI 10.1090/S0002-9939-04-07382-4
J. F. Feinstein and M. J. Heath, Regularity and amenability conditions for uniform algebras, Function spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 159–169. MR 2359426, DOI 10.1090/conm/435/08374
J. F. Feinstein and M. J. Heath, Swiss cheeses, rational approximation and universal plane curves, Studia Math. 196 (2010), no. 3, 289–306. MR 2587300, DOI 10.4064/sm196-3-5
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
M. J. Heath, Bounded derivations from Banach algebras, Ph.D. thesis, University of Nottingham, February 2008.
Alice Roth, Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), no. 1, 77–125 (German). MR 1509593, DOI 10.1007/BF01199693
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
Guan Hou Zhang, Theory of entire and meromorphic functions, Translations of Mathematical Monographs, vol. 122, American Mathematical Society, Providence, RI, 1993. Deficient and asymptotic values and singular directions; Translated from the Chinese by Chung-chun Yang. MR 1222870, DOI 10.1090/mmono/122