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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**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*, Proc. Amer. Math. Soc.**28**(1971), 147-154. MR**43**#5300. MR**0279578 (43:5300)****[7]**-,*Sets in**analytically equivalent to the open ball*. Duke Math. J.**39**(1972), 711-717. MR**0352981 (50:5467)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46J99

Retrieve articles in all journals with MSC: 46J99

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