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Finitely-generated left ideals in Banach algebras on groups and semigroups

Volume 239 / 2017

Jared T. White Studia Mathematica 239 (2017), 67-99 MSC: 46H10, 43A10, 43A20. DOI: 10.4064/sm8743-1-2017 Published online: 18 April 2017

Abstract

Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason.

Authors

  • Jared T. WhiteDepartment of Mathematics and Statistics
    University of Lancaster
    Lancaster LA1 4YF, United Kingdom
    e-mail

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