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

Research output: Contribution to journalJournal articlepeer-review

E-pub ahead of print
<mark>Journal publication date</mark>18/04/2017
<mark>Journal</mark>Studia Mathematica
Volume239
Number of pages33
Pages (from-to)67-99
Publication StatusE-pub ahead of print
Early online date18/04/17
<mark>Original language</mark>English

Abstract

Let G be a locally compact group. We prove that the augmentation ideal in L1(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.