Generators of maximal left ideals in Banach algebras

Volume 212 / 2012

H. G. Dales, W. Żelazko Studia Mathematica 212 (2012), 173-193 MSC: Primary 46H10; Secondary 46J10. DOI: 10.4064/sm212-2-5

Abstract

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over $\mathbb C$ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of $A$'. We give a shorter proof of this latter result, together with some extensions and related examples.

We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra $A$ are finitely generated. Then $A$ is finite-dimensional.

Authors

  • H. G. DalesDepartment of Mathematics and Statistics
    Fylde College
    University of Lancaster
    Lancaster LA1 4YF, United Kingdom
    e-mail
  • W. ŻelazkoInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    P.O. Box 21
    00-956 Warszawa, Poland
    e-mail

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