### Abstract

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(Ū), where Ū 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 ∈ A such that f(X) = Ū.

Original language | English (US) |
---|---|

Pages (from-to) | 533-539 |

Number of pages | 7 |

Journal | Illinois Journal of Mathematics |

Volume | 46 |

Issue number | 2 |

State | Published - Jun 2002 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Illinois Journal of Mathematics*,

*46*(2), 533-539.

**A characterization of the disk algebra.** / Cole, Brian J.; Sadik, Nazim; Poletsky, Evgeny Alexander.

Research output: Contribution to journal › Article

*Illinois Journal of Mathematics*, vol. 46, no. 2, pp. 533-539.

}

TY - JOUR

T1 - A characterization of the disk algebra

AU - Cole, Brian J.

AU - Sadik, Nazim

AU - Poletsky, Evgeny Alexander

PY - 2002/6

Y1 - 2002/6

N2 - 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(Ū), where Ū 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 ∈ A such that f(X) = Ū.

AB - 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(Ū), where Ū 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 ∈ A such that f(X) = Ū.

UR - http://www.scopus.com/inward/record.url?scp=0036628272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036628272&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036628272

VL - 46

SP - 533

EP - 539

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 2

ER -