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Ideal structure of the algebra of bounded operators acting on a Banach space

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Ideal structure of the algebra of bounded operators acting on a Banach space. / Kania, Tomasz; Laustsen, Niels Jakob.
In: Indiana University Mathematics Journal, Vol. 66, No. 3, 2017, p. 1019-1043.

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

Harvard

Kania, T & Laustsen, NJ 2017, 'Ideal structure of the algebra of bounded operators acting on a Banach space', Indiana University Mathematics Journal, vol. 66, no. 3, pp. 1019-1043. https://doi.org/10.1512/iumj.2017.66.6037

APA

Kania, T., & Laustsen, N. J. (2017). Ideal structure of the algebra of bounded operators acting on a Banach space. Indiana University Mathematics Journal, 66(3), 1019-1043. https://doi.org/10.1512/iumj.2017.66.6037

Vancouver

Kania T, Laustsen NJ. Ideal structure of the algebra of bounded operators acting on a Banach space. Indiana University Mathematics Journal. 2017;66(3):1019-1043. doi: 10.1512/iumj.2017.66.6037

Author

Kania, Tomasz ; Laustsen, Niels Jakob. / Ideal structure of the algebra of bounded operators acting on a Banach space. In: Indiana University Mathematics Journal. 2017 ; Vol. 66, No. 3. pp. 1019-1043.

Bibtex

@article{aa0fefb8645d46e396cf35637a8794a7,
title = "Ideal structure of the algebra of bounded operators acting on a Banach space",
abstract = "We construct a Banach space Z such that the Banach algebra B(Z) of bounded operators on Z contains exactly four non-zero, proper closed ideals, including two maximal ideals.We then determine which kinds of approximate identities (bounded/left/right), if any, each of these four ideals contains, and we show that one of the two maximal ideals is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (Studia Math. 2013). In contrast, the other maximal ideal is not finitely generated as a left ideal.The Banach space Z is the direct sum of Argyros and Haydon's Banach space XAH which has very few operators and a certain subspace Y of XAH. The key property of Y is that every bounded operator from Y into XAH is the sum of ascalar multiple of the inclusion map and a compact operator.non-zer",
keywords = "Banach algebra, lattice of closed ideals, bounded) approximate identity, finitely-generated, maximal left ideal, bounded operator, Banach space, Argyros-Haydon space, Bourgain-Delbaen construction, script L infinity space",
author = "Tomasz Kania and Laustsen, {Niels Jakob}",
year = "2017",
doi = "10.1512/iumj.2017.66.6037",
language = "English",
volume = "66",
pages = "1019--1043",
journal = "Indiana University Mathematics Journal",
issn = "0022-2518",
publisher = "Indiana University",
number = "3",

}

RIS

TY - JOUR

T1 - Ideal structure of the algebra of bounded operators acting on a Banach space

AU - Kania, Tomasz

AU - Laustsen, Niels Jakob

PY - 2017

Y1 - 2017

N2 - We construct a Banach space Z such that the Banach algebra B(Z) of bounded operators on Z contains exactly four non-zero, proper closed ideals, including two maximal ideals.We then determine which kinds of approximate identities (bounded/left/right), if any, each of these four ideals contains, and we show that one of the two maximal ideals is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (Studia Math. 2013). In contrast, the other maximal ideal is not finitely generated as a left ideal.The Banach space Z is the direct sum of Argyros and Haydon's Banach space XAH which has very few operators and a certain subspace Y of XAH. The key property of Y is that every bounded operator from Y into XAH is the sum of ascalar multiple of the inclusion map and a compact operator.non-zer

AB - We construct a Banach space Z such that the Banach algebra B(Z) of bounded operators on Z contains exactly four non-zero, proper closed ideals, including two maximal ideals.We then determine which kinds of approximate identities (bounded/left/right), if any, each of these four ideals contains, and we show that one of the two maximal ideals is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (Studia Math. 2013). In contrast, the other maximal ideal is not finitely generated as a left ideal.The Banach space Z is the direct sum of Argyros and Haydon's Banach space XAH which has very few operators and a certain subspace Y of XAH. The key property of Y is that every bounded operator from Y into XAH is the sum of ascalar multiple of the inclusion map and a compact operator.non-zer

KW - Banach algebra

KW - lattice of closed ideals

KW - bounded) approximate identity

KW - finitely-generated, maximal left ideal

KW - bounded operator

KW - Banach space

KW - Argyros-Haydon space

KW - Bourgain-Delbaen construction

KW - script L infinity space

U2 - 10.1512/iumj.2017.66.6037

DO - 10.1512/iumj.2017.66.6037

M3 - Journal article

VL - 66

SP - 1019

EP - 1043

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -