MathDBDales–Zelazko conjecture on finitely generated maximal left ideals
Dales–Zelazko conjecture on finitely generated maximal left ideals
Let be an infinite-dimensional unital Banach algebra. Dales–Zelazko conjecture. Then has a maximal left ideal which is not finitely generated. The conjecture concerns the existence of non-finitely generated maximal left ideals in infinite-dimensional unital Banach algebras. It is proved in the source for the Beurling algebras considered there when the underlying group is infinite and either virtually soluble or virtually free, but remains open in general.
Equivalent formulations 1
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Dales–Želazko conjecture on finitely generated maximal left ideals
Let be a unital Banach algebra, and suppose that every maximal left ideal of is finitely-generated. Dales–Želazko conjecture. Then is finite-dimensional. This conjecture strengthens the theorem of Sinclair and Tullo that every left Noetherian Banach algebra is finite-dimensional. It remains open for an arbitrary Banach space .
source: Jared T White, “Finitely-Generated Left Ideals in Banach Algebras on Groups and Semigroups”, arXiv:1612.05915 (2016).
Sources & referencesView supporting material
Primary source
Jared T. White, “On the Dales-Zelazko conjecture for Beurling algebras on discrete groups”, arXiv:2206.13989 (2023).
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