Finitely-Generated Left Ideals in Banach Algebras on Groups and\n Semigroups

2016/12/18 by Jared T. White, White, Jared T · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.1612.05915

openalex publication_date 2016/12/18 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let G be a locally compact group. We prove that the augmentation ideal in\nL¹(G) is (algebraically) finitely-generated as a left ideal if and only if\nG is finite. We then investigate weighted versions of this result, as well as\na version for semigroup algebras. Weighted measure algebras are also\nconsidered. We are motivated by a recent conjecture of Dales and .Zelazko,\nwhich states that a unital Banach algebra in which every maximal left ideal is\nfinitely-generated is necessarily finite-dimensional. We prove that this\nconjecture holds for many of the algebras considered. Finally, we use the\ntheory that we have developed to construct some examples of commutative Banach\nalgebras that relate to a theorem of Gleason.\n

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