Amenable Köthe co‐echelon algebras

Piszczek, Krzysztof

doi:10.1002/mana.201700429

Cited by 5 publications

(8 citation statements)

References 23 publications

(60 reference statements)

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Section: Introductionsupporting

confidence: 70%

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Topological amenability and Köthe co-echelon algebras

Pirkovskii¹,

Piszczek²

2020

Preprint

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We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). By using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

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Section: Introductionsupporting

confidence: 70%

“…A comprehensive study of Köthe co-echelon spaces may be found in [1]. Köthe co-echelon algebras appear as a main object of investigation in [3] and [23,24].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Topological amenability and Köthe co-echelon algebras

Pirkovskii¹,

Piszczek²

2020

Preprint

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“…We define topologically amenable algebras in terms of topologically flat modules, and we show that topological amenability for complete barrelled DF-algebras is equivalent to amenability in Johnson's sense. We also obtain a topological amenability criterion for Köthe co-echelon algebras, complementing thereby recent results of the second author [23,24]. Note that such algebras need not be complete, so that our definition of topological amenability is made in terms of the completion (see Definition 3.21).…”

Section: Introductionmentioning

confidence: 59%

“…(iii) To get the nuclearity of k p (V) we repeat exactly the proof of [23, Theorem 5.1]. We can indeed do so, since π −1 is a topological isomorphism not only in the case of amenability (which was the assumption in [23]) but also under the weaker assumption of topological amenability. ◻ Theorem 4.5 Let 1 ≤ p < ∞ , and let k p (V) be a Köthe co-echelon algebra.…”

Section: Define Now a Linear Mapmentioning

confidence: 81%

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Topological amenability and Köthe co-echelon algebras

Pirkovskii

Piszczek

2022

Banach J. Math. Anal.

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We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). Using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii–Sheinberg’s criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

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General approach to Köthe echelon algebras

Ciaś

Piszczek

2021

Mathematische Nachrichten

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We provide a study of Köthe sequence algebras. These are Fréchet sequence algebras which can be viewed as abstract analogoues of algebras of smooth or holomorphic functions. Of particular treatment are the following properties: unitality,-convexity,-property and variants of amenability. These properties are then checked against the topological (DN)-(Ω) type conditions of Vogt-Zaharjuta. Description of characters and closed ideals in Köthe echelon algebras is provided. We show that these algebras are functionally continuous and we characterize completely which of them factor. Moreover, we prove that closed subalgebras of Montel-convex Köthe echelon algebras are Köthe echelon algebras as well and we give a version of Stone-Weiestrass theorem for these algebras. We also emphasize connections with the algebra of smooth operators.

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Amenable Köthe co‐echelon algebras

Abstract: In this note we investigate amenability properties in the class of the so‐called DF‐algebras. We also characterize, in terms of the defining sequence of weights, amenable Köthe co‐echelon algebras.

Cited by 5 publications

References 23 publications

Topological amenability and Köthe co-echelon algebras

Topological amenability and Köthe co-echelon algebras

Topological amenability and Köthe co-echelon algebras

General approach to Köthe echelon algebras

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