A characterization of the disk algebra

Brian J. Cole, Nazim Sadik, Evgeny Alexander Poletsky

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)533-539
Number of pages7
JournalIllinois Journal of Mathematics
Volume46
Issue number2
StatePublished - Jun 2002

Fingerprint

Algebra
Uniform Algebra
Isomorphic
Subalgebra
Closed
Cantor set
Unital
Unit Disk
Interior
Generator
If and only if
Polynomial

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Illinois Journal of Mathematics, Vol. 46, No. 2, 06.2002, p. 533-539.

Research output: Contribution to journalArticle

Cole, BJ, Sadik, N & Poletsky, EA 2002, 'A characterization of the disk algebra', Illinois Journal of Mathematics, vol. 46, no. 2, pp. 533-539.
Cole, Brian J. ; Sadik, Nazim ; Poletsky, Evgeny Alexander. / A characterization of the disk algebra. In: Illinois Journal of Mathematics. 2002 ; Vol. 46, No. 2. pp. 533-539.
@article{754dd717d9214837b4789757f3bdd6c9,
title = "A characterization of the disk algebra",
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.",
author = "Cole, {Brian J.} and Nazim Sadik and Poletsky, {Evgeny Alexander}",
year = "2002",
month = "6",
language = "English (US)",
volume = "46",
pages = "533--539",
journal = "Illinois Journal of Mathematics",
issn = "0019-2082",
publisher = "University of Illinois at Urbana-Champaign",
number = "2",

}

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 -