Illinois Journal of Mathematics

A characterization of the disk algebra

Brian J. Cole, Evgeny A. Poletsky, and Nazim Sadik

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We prove that a complex unital uniform algebra is isomorphic to the disk algebra if and only if every closed subalgebra with one generator is isomorphic to the whole algebra. Moreover, every such subalgebra of the disk algebra is isometrically isomorphic to the disk algebra. On the way we prove: (1) for a function $f$ in the disk algebra the interior of the polynomial hull of the set $f(\overline U)$, where $\overline U$ is the closed unit disk, is a Jordan domain; (2) if a uniform algebra $A$ on a compact Hausdorff set $X$ containing the Cantor set separates points of $X$, then there is $f\in A$ such that $f(X)=\overline U$.

Article information

Illinois J. Math., Volume 46, Number 2 (2002), 533-539.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
Secondary: 30H05: Bounded analytic functions


Cole, Brian J.; Sadik, Nazim; Poletsky, Evgeny A. A characterization of the disk algebra. Illinois J. Math. 46 (2002), no. 2, 533--539. doi:10.1215/ijm/1258136209.

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