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Maximal left ideals of the Banach algebra of bounded operators on a Banach space

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

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Maximal left ideals of the Banach algebra of bounded operators on a Banach space. / Dales, H.G.; Kania, Tomasz; Kochanek, Tomasz et al.
In: Studia Mathematica, Vol. 218, No. 3, 2013, p. 245-286.

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

Harvard

Dales, HG, Kania, T, Kochanek, T, Koszmider, P & Laustsen, N 2013, 'Maximal left ideals of the Banach algebra of bounded operators on a Banach space', Studia Mathematica, vol. 218, no. 3, pp. 245-286. https://doi.org/10.4064/sm218-3-3

APA

Dales, H. G., Kania, T., Kochanek, T., Koszmider, P., & Laustsen, N. (2013). Maximal left ideals of the Banach algebra of bounded operators on a Banach space. Studia Mathematica, 218(3), 245-286. https://doi.org/10.4064/sm218-3-3

Vancouver

Dales HG, Kania T, Kochanek T, Koszmider P, Laustsen N. Maximal left ideals of the Banach algebra of bounded operators on a Banach space. Studia Mathematica. 2013;218(3):245-286. doi: 10.4064/sm218-3-3

Author

Dales, H.G. ; Kania, Tomasz ; Kochanek, Tomasz et al. / Maximal left ideals of the Banach algebra of bounded operators on a Banach space. In: Studia Mathematica. 2013 ; Vol. 218, No. 3. pp. 245-286.

Bibtex

@article{a425fcc5049b46c99e289b57b2c3a36a,
title = "Maximal left ideals of the Banach algebra of bounded operators on a Banach space",
abstract = "We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.",
keywords = "Finitely-generated, maximal left ideal, Banach algebra, bounded operator, inessential operator, Banach space, Argyros-Haydon space, Sinclair-Tullo theorem",
author = "H.G. Dales and Tomasz Kania and Tomasz Kochanek and Piotr Koszmider and Niels Laustsen",
year = "2013",
doi = "10.4064/sm218-3-3",
language = "English",
volume = "218",
pages = "245--286",
journal = "Studia Mathematica",
issn = "0039-3223",
publisher = "Instytut Matematyczny",
number = "3",

}

RIS

TY - JOUR

T1 - Maximal left ideals of the Banach algebra of bounded operators on a Banach space

AU - Dales, H.G.

AU - Kania, Tomasz

AU - Kochanek, Tomasz

AU - Koszmider, Piotr

AU - Laustsen, Niels

PY - 2013

Y1 - 2013

N2 - We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.

AB - We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some non-zero x in E? Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.

KW - Finitely-generated, maximal left ideal

KW - Banach algebra

KW - bounded operator

KW - inessential operator

KW - Banach space

KW - Argyros-Haydon space

KW - Sinclair-Tullo theorem

U2 - 10.4064/sm218-3-3

DO - 10.4064/sm218-3-3

M3 - Journal article

VL - 218

SP - 245

EP - 286

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 3

ER -