Functiones et Approximatio Commentarii Mathematici

When does a closed ideal of a commutative unital Banach algebra contain a dense subideal?

Wiesław Żelazko

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Abstract

The question formulated in the title is answered in the case of a separable algebra. The necessary and sufficient condition in this case is that the ideal in question is not finitely (algebraically) generated. We conjecture that this result is true in the general case.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 285-287.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749132

Digital Object Identifier
doi:10.7169/facm/1308749132

Mathematical Reviews number (MathSciNet)
MR2841187

Zentralblatt MATH identifier
1230.46047

Subjects
Primary: 46J20: Ideals, maximal ideals, boundaries

Keywords
Banach algebra dense subideal

Citation

Żelazko, Wiesław. When does a closed ideal of a commutative unital Banach algebra contain a dense subideal?. Funct. Approx. Comment. Math. 44 (2011), no. 2, 285--287. doi:10.7169/facm/1308749132. https://projecteuclid.org/euclid.facm/1308749132


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References

  • R. Choukri and A. El Kinani, Topological algebras with ascending and descending chain conditions, Acta Math. (Basel) 72 (1999), 438–443.
  • R. Choukri, A. El Kinani and M. Oudadess, Algèbres topologiques à idéaux à gauche fermés, Studia Math. 168 (2005), 159–164.
  • A.V. Ferreira and G. Tomassini, Finitness properties of topological algebras, Ann. Scuola Norm. Sup. Pisa 5 (1978), 471–488.
  • H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer, 1971.
  • A.M. Sinclair and A.W. Tullo, Noetherian Banach algebras are finite dimensional, Math. Ann. 211 (1974), 151–153.
  • W. Żelazko, A characterization of commutative Fréchet algebras with all ideals closed, Studia Math. 138 (2000), 293–300.
  • W. Żelazko, A characterization of $F$-algebras with all one-sided ideals closed, Studia Math. 168 (2005), 135–145.