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  • Uniquemaxideal

    Rights statement: The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 262 (11), 2012, © ELSEVIER.

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Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

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Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1]). / Kania, Tomasz; Laustsen, Niels.
In: Journal of Functional Analysis, Vol. 262, 01.06.2012, p. 4831-4850.

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

Harvard

Kania, T & Laustsen, N 2012, 'Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])', Journal of Functional Analysis, vol. 262, pp. 4831-4850. https://doi.org/10.1016/j.jfa.2012.03.011

APA

Kania, T., & Laustsen, N. (2012). Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1]). Journal of Functional Analysis, 262, 4831-4850. https://doi.org/10.1016/j.jfa.2012.03.011

Vancouver

Kania T, Laustsen N. Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1]). Journal of Functional Analysis. 2012 Jun 1;262:4831-4850. doi: 10.1016/j.jfa.2012.03.011

Author

Kania, Tomasz ; Laustsen, Niels. / Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1]). In: Journal of Functional Analysis. 2012 ; Vol. 262. pp. 4831-4850.

Bibtex

@article{eddfe44adc534726aadcf2cd5f7fb9e0,
title = "Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])",
abstract = "Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).",
keywords = "Continuous functions on ordinals, Bounded operators on Banach spaces, Maximal ideal , Loy–Willis ideal",
author = "Tomasz Kania and Niels Laustsen",
note = "The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 262 (11), 2012, {\textcopyright} ELSEVIER.",
year = "2012",
month = jun,
day = "1",
doi = "10.1016/j.jfa.2012.03.011",
language = "English",
volume = "262",
pages = "4831--4850",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",

}

RIS

TY - JOUR

T1 - Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

AU - Kania, Tomasz

AU - Laustsen, Niels

N1 - The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 262 (11), 2012, © ELSEVIER.

PY - 2012/6/1

Y1 - 2012/6/1

N2 - Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).

AB - Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).

KW - Continuous functions on ordinals

KW - Bounded operators on Banach spaces

KW - Maximal ideal

KW - Loy–Willis ideal

U2 - 10.1016/j.jfa.2012.03.011

DO - 10.1016/j.jfa.2012.03.011

M3 - Journal article

VL - 262

SP - 4831

EP - 4850

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

ER -