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 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.

**[1]**Errett Bishop,*Foundations of constructive analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR**0221878****[2]**Barnett W. Glickfeld,*On the inverse function theorem in commutative Banach algebras*, Illinois J. Math.**15**(1971), 212–221. MR**0273408****[3]**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****[4]**Einar Hille,*Analytic function theory. Vol. II*, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR**0201608****[5]**Edgar R. Lorch,*The theory of analytic functions in normed Abelian vector rings*, Trans. Amer. Math. Soc.**54**(1943), 414–425. MR**0009090**, https://doi.org/10.1090/S0002-9947-1943-0009090-0**[6]**Hugh E. Warren,*A Riemann mapping theorem for 𝐶(𝑋)*, Proc. Amer. Math. Soc.**28**(1971), 147–154. MR**0279578**, https://doi.org/10.1090/S0002-9939-1971-0279578-9**[7]**H. E. Warren,*Sets in 𝐶(𝑋) analytically equivalent to the open ball*, Duke Math. J.**39**(1972), 711–717. MR**0352981**

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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