Maximal left ideals in Banach algebras

García, Miguel Cabrera; Dales, H. G.; Palacios, Ángel Rodríguez

doi:10.1112/blms.12290

Cited by 3 publications

(1 citation statement)

References 11 publications

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“…Indeed, we have seen that an arbitrary finite-codimension left ideal I is equal to , for some finite-index subgroup H and some , and since is closed, I must be as well. Not every unital Banach algebra has the property that all of its finite-codimension left ideals are closed: for intance, add a unit to any of the many interesting examples in [4]. This property can be rephrased as an automatic continuity property of , namely that every module map from to a finite-dimensional Banach left module is automatically continuous.…”

Section: The Proofsmentioning

confidence: 99%

On the Dales–Żelazko conjecture for Beurling algebras on discrete groups

White¹

2023

Proceedings of the Edinburgh Mathematical Society

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Let G be a group that is either virtually soluble or virtually free, and let ω be a weight on G. We prove that if G is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$ , which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance, we describe a Beurling algebra on an infinite group in which every closed left ideal of finite codimension is finitely generated and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Żelazko’s conjecture.

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Section: The Proofsmentioning

confidence: 99%

On the Dales–Żelazko conjecture for Beurling algebras on discrete groups

White¹

2023

Proceedings of the Edinburgh Mathematical Society

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Prime ideals and Noetherian properties in vector lattices

Kandić

Roelands

2022

Positivity

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In this paper we study the set of prime ideals in vector lattices and how the properties of the prime ideals structure the vector lattice in question. The different properties that will be considered are firstly, that all or none of the prime ideals are order dense, secondly, that there are only finitely many prime ideals, thirdly, that every prime ideal is principal, and lastly, that every ascending chain of prime ideals is stationary (a property that we refer to as prime Noetherian). We also completely characterize the prime ideals in vector lattices of piecewise polynomials, which turns out to be an interesting class of vector lattices for studying principal prime ideals and ascending chains of prime ideals.Keywords Vector lattices • Prime ideals • Principal prime ideals • Ascending chains of prime ideals

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Regularity results for classes of Hilbert C*-modules with respect to special bounded modular functionals

Frank

2024

Ann. Funct. Anal.

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Considering the deeper reasons of the appearance of a remarkable counterexample by Kaad and Skeide (J Operat Theory 89(2):343–348, 2023) we consider situations in which two Hilbert C*-modules $$M \subset N$$ M ⊂ N with $$M^\bot = \{ 0 \}$$ M ⊥ = { 0 } over a fixed C*-algebra A of coefficients cannot be separated by a non-trivial bounded A-linear functional $$r_0: N \rightarrow A$$ r 0 : N → A vanishing on M. In other words, the uniqueness of extensions of the zero functional from M to N is focussed. We show this uniqueness of extension for any such pairs of Hilbert C*-modules over W*-algebras, over monotone complete C*-algebras and over compact C*-algebras. Moreover, uniqueness of extension takes place also for any one-sided maximal modular ideal of any C*-algebra. Such a non-zero separating bounded A-linear functional $$r_0$$ r 0 exist for a given pair of full Hilbert C*-modules $$M \subseteq N$$ M ⊆ N over a given C*-algebra A iff there exists a bounded A-linear non-adjointable operator $$T_0: N \rightarrow N$$ T 0 : N → N , such that the kernel of $$T_0$$ T 0 is not biorthogonally closed w.r.t. N and contains M. This is a new perspective on properties of bounded modular operators that might appear in Hilbert C*-module theory. By the way, we find a correct proof of Lemma 2.4 of Frank (Int J Math 13:1–19, 2002) in the case of monotone complete and compact C*-algebras, but find it not valid in certain particular cases.

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Maximal left ideals in Banach algebras

Cited by 3 publications

References 11 publications

On the Dales–Żelazko conjecture for Beurling algebras on discrete groups

On the Dales–Żelazko conjecture for Beurling algebras on discrete groups

Prime ideals and Noetherian properties in vector lattices

Regularity results for classes of Hilbert C*-modules with respect to special bounded modular functionals

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