Analytic equivalence in the disk algebra

Warren, Hugh E.

doi:10.1090/S0002-9947-1974-0333742-6

Analytic equivalence in the disk algebra
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Trans. Amer. Math. Soc. 192 (1974), 219-226 Request permission

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.

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