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    Rights statement: http://journals.cambridge.org/action/displayJournal?jid=JAZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 99 (3), pp 350-363 2015, © 2015 Cambridge University Press.

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Chains of functions in C(K)-spaces

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Chains of functions in C(K)-spaces. / Kania, Tomasz; Smith, Richard J.

In: Journal of the Australian Mathematical Society, Vol. 99, No. 3, 12.2015, p. 350-363.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Kania, T & Smith, RJ 2015, 'Chains of functions in C(K)-spaces', Journal of the Australian Mathematical Society, vol. 99, no. 3, pp. 350-363. https://doi.org/10.1017/S1446788715000245

APA

Kania, T., & Smith, R. J. (2015). Chains of functions in C(K)-spaces. Journal of the Australian Mathematical Society, 99(3), 350-363. https://doi.org/10.1017/S1446788715000245

Vancouver

Kania T, Smith RJ. Chains of functions in C(K)-spaces. Journal of the Australian Mathematical Society. 2015 Dec;99(3):350-363. https://doi.org/10.1017/S1446788715000245

Author

Kania, Tomasz ; Smith, Richard J. / Chains of functions in C(K)-spaces. In: Journal of the Australian Mathematical Society. 2015 ; Vol. 99, No. 3. pp. 350-363.

Bibtex

@article{d2b896393d2b41cc9e62c71872a01317,
title = "Chains of functions in C(K)-spaces",
abstract = "The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pe{\l}czy{\'n}ski's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\{"}of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.",
keywords = "CHAIN, Bishop property, Banach spaces, Compact space",
author = "Tomasz Kania and Smith, {Richard J.}",
note = "http://journals.cambridge.org/action/displayJournal?jid=JAZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 99 (3), pp 350-363 2015, {\textcopyright} 2015 Cambridge University Press.",
year = "2015",
month = dec,
doi = "10.1017/S1446788715000245",
language = "English",
volume = "99",
pages = "350--363",
journal = "Journal of the Australian Mathematical Society",
issn = "1446-7887",
publisher = "Cambridge University Press",
number = "3",

}

RIS

TY - JOUR

T1 - Chains of functions in C(K)-spaces

AU - Kania, Tomasz

AU - Smith, Richard J.

N1 - http://journals.cambridge.org/action/displayJournal?jid=JAZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 99 (3), pp 350-363 2015, © 2015 Cambridge University Press.

PY - 2015/12

Y1 - 2015/12

N2 - The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

AB - The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

KW - CHAIN

KW - Bishop property

KW - Banach spaces

KW - Compact space

U2 - 10.1017/S1446788715000245

DO - 10.1017/S1446788715000245

M3 - Journal article

VL - 99

SP - 350

EP - 363

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

SN - 1446-7887

IS - 3

ER -