Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space

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CRLaustsenSkillicorn
Rights statement: This is the author’s version of a work that was accepted for publication in Comptes Rendus Mathematique. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Comptes Rendus Mathematique, 354, 5, 2016 DOI: 10.1016/j.crma.2015.12.020
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bounded, linear operator, Banach space, Banach algebra, short-exact sequence, algebraic splitting, strong splitting, singular extension, admissible extension, pullback, second continuous Hochschild cohomology group, homological bidimension

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Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space. / Laustsen, Niels Jakob ; Skillicorn, Richard.
In: Comptes Rendus Mathématique, Vol. 354, No. 5, 05.2016, p. 459-463.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Bibtex

@article{d0817e268ab544baa65f7d02b131bf16,

title = "Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space",

abstract = "We show that there exists a Banach space E such that: - the Banach algebra B(E) of bounded, linear operators on E has a singular extension which splits algebraically, but it does not split strongly; - the homological bidimension of B(E) is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space E that we use was originally introduced by Read (J. London Math. Soc. 1989).",

keywords = "bounded, linear operator, Banach space, Banach algebra, short-exact sequence, algebraic splitting, strong splitting, singular extension, admissible extension, pullback, second continuous Hochschild cohomology group, homological bidimension",

author = "Laustsen, {Niels Jakob} and Richard Skillicorn",

note = "This is the author{\textquoteright}s version of a work that was accepted for publication in Comptes Rendus Mathematique. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Comptes Rendus Mathematique, 354, 5, 2016 DOI: 10.1016/j.crma.2015.12.020",

year = "2016",

month = may,

doi = "10.1016/j.crma.2015.12.020",

language = "English",

volume = "354",

pages = "459--463",

journal = "Comptes Rendus Math{\'e}matique",

issn = "1631-073X",

publisher = "Elsevier Masson",

number = "5",

}

RIS

TY - JOUR

T1 - Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space

AU - Laustsen, Niels Jakob

AU - Skillicorn, Richard

N1 - This is the author’s version of a work that was accepted for publication in Comptes Rendus Mathematique. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Comptes Rendus Mathematique, 354, 5, 2016 DOI: 10.1016/j.crma.2015.12.020

PY - 2016/5

Y1 - 2016/5

N2 - We show that there exists a Banach space E such that: - the Banach algebra B(E) of bounded, linear operators on E has a singular extension which splits algebraically, but it does not split strongly; - the homological bidimension of B(E) is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space E that we use was originally introduced by Read (J. London Math. Soc. 1989).

AB - We show that there exists a Banach space E such that: - the Banach algebra B(E) of bounded, linear operators on E has a singular extension which splits algebraically, but it does not split strongly; - the homological bidimension of B(E) is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space E that we use was originally introduced by Read (J. London Math. Soc. 1989).

KW - bounded, linear operator

KW - Banach space

KW - Banach algebra

KW - short-exact sequence

KW - algebraic splitting

KW - strong splitting

KW - singular extension

KW - admissible extension

KW - pullback

KW - second continuous Hochschild cohomology group

KW - homological bidimension

U2 - 10.1016/j.crma.2015.12.020

DO - 10.1016/j.crma.2015.12.020

M3 - Journal article

VL - 354

SP - 459

EP - 463

JO - Comptes Rendus Mathématique

JF - Comptes Rendus Mathématique

SN - 1631-073X

IS - 5

ER -

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