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Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support

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Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support. / Johnson, William B. ; Kania, Tomasz; Schechtman, Gideon.
In: Proceedings of the American Mathematical Society, Vol. 144, No. 10, 10.2016, p. 4471-4485.

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

Harvard

Johnson, WB, Kania, T & Schechtman, G 2016, 'Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support', Proceedings of the American Mathematical Society, vol. 144, no. 10, pp. 4471-4485. https://doi.org/10.1090/proc/13084

APA

Johnson, W. B., Kania, T., & Schechtman, G. (2016). Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support. Proceedings of the American Mathematical Society, 144(10), 4471-4485. https://doi.org/10.1090/proc/13084

Vancouver

Johnson WB, Kania T, Schechtman G. Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support. Proceedings of the American Mathematical Society. 2016 Oct;144(10):4471-4485. Epub 2016 Apr 25. doi: 10.1090/proc/13084

Author

Johnson, William B. ; Kania, Tomasz ; Schechtman, Gideon. / Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support. In: Proceedings of the American Mathematical Society. 2016 ; Vol. 144, No. 10. pp. 4471-4485.

Bibtex

@article{996f1cdb458a4292a360c88ab5d31ff0,
title = "Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support",
abstract = "Let λ be an infinite cardinal number and let ℓc ∞(λ) denote the subspace of ℓ∞(λ) consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of ℓc ∞(λ), proving that they are isomorphic to ℓc ∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on ℓc ∞(λ) or ℓ∞(λ) has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws{\textquoteright} approach description of the lattice of all closed ideals of B(X), where X = c0(λ) or X = ℓp(λ) for some p ∈ [1,∞), and we classify the closed ideals of B(ℓc ∞(λ)) that contains the ideal of weakly compact operators. {\textcopyright} 2016 American Mathematical Society.",
author = "Johnson, {William B.} and Tomasz Kania and Gideon Schechtman",
note = "Author no longer at Lancaster",
year = "2016",
month = oct,
doi = "10.1090/proc/13084",
language = "English",
volume = "144",
pages = "4471--4485",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "10",

}

RIS

TY - JOUR

T1 - Closed ideals of operators on and complemented subspaces of banach spaces of functions with countable support

AU - Johnson, William B.

AU - Kania, Tomasz

AU - Schechtman, Gideon

N1 - Author no longer at Lancaster

PY - 2016/10

Y1 - 2016/10

N2 - Let λ be an infinite cardinal number and let ℓc ∞(λ) denote the subspace of ℓ∞(λ) consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of ℓc ∞(λ), proving that they are isomorphic to ℓc ∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on ℓc ∞(λ) or ℓ∞(λ) has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws’ approach description of the lattice of all closed ideals of B(X), where X = c0(λ) or X = ℓp(λ) for some p ∈ [1,∞), and we classify the closed ideals of B(ℓc ∞(λ)) that contains the ideal of weakly compact operators. © 2016 American Mathematical Society.

AB - Let λ be an infinite cardinal number and let ℓc ∞(λ) denote the subspace of ℓ∞(λ) consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of ℓc ∞(λ), proving that they are isomorphic to ℓc ∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on ℓc ∞(λ) or ℓ∞(λ) has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws’ approach description of the lattice of all closed ideals of B(X), where X = c0(λ) or X = ℓp(λ) for some p ∈ [1,∞), and we classify the closed ideals of B(ℓc ∞(λ)) that contains the ideal of weakly compact operators. © 2016 American Mathematical Society.

U2 - 10.1090/proc/13084

DO - 10.1090/proc/13084

M3 - Journal article

VL - 144

SP - 4471

EP - 4485

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 10

ER -