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Transactions of the American Mathematical Society

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Analytic equivalence in the disk algebra


Author: Hugh E. Warren
Journal: Trans. Amer. Math. Soc. 192 (1974), 219-226
MSC: Primary 46J99
DOI: https://doi.org/10.1090/S0002-9947-1974-0333742-6
MathSciNet review: 0333742
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Abstract: The notion of analytically equivalent domains can be extended from the complex plane to commutative Banach algebras with identity. In $ C(X)$ a domain equivalent to the unit ball must have a boundary that is in a certain sense continuous. This paper shows that in the disk algebra ``continuous'' must be replaced with ``analytic.'' These results set limits in the classical Riemann mapping theorem on how smoothly the mapping can respond to changes in the domain being mapped.


References [Enhancements On Off] (What's this?)

  • [1] E. Bishop, Foundations of constructive analysis, McGraw-Hill, New York, 1967, p. 112. MR 36 #4930. MR 0221878 (36:4930)
  • [2] B. W. Glickfeld, On the inverse function theorem in commutative Banach algebras, Illinois J. Math. 15 (1971), 212-221. MR 42 #8287. MR 0273408 (42:8287)
  • [3] G. M. Goluzin, Geometric theory of functions of a complex variable, GITTL, Moscow, 1952; English transl., Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, R.I., 1969. MR 15, 112; MR 40 #308. MR 0247039 (40:308)
  • [4] E. Hille, Analytic function theory. Vol. II, Introductions to Higher Math., Ginn, Boston, Mass., 1962. MR 34 #1490. MR 0201608 (34:1490)
  • [5] E. R. Lorch, The theory of analytic functions in normed abelian vector rings, Trans. Amer. Math. Soc. 54 (1943), 414-425. MR 5, 100. MR 0009090 (5:100a)
  • [6] H. E. Warren, A Riemann mapping theorem for $ C(X)$, Proc. Amer. Math. Soc. 28 (1971), 147-154. MR 43 #5300. MR 0279578 (43:5300)
  • [7] -, Sets in $ C(X)$ analytically equivalent to the open ball. Duke Math. J. 39 (1972), 711-717. MR 0352981 (50:5467)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0333742-6
Keywords: Disk algebra, Lorch analytic function, analytic equivalence
Article copyright: © Copyright 1974 American Mathematical Society

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