Chains of functions in C(K)-spaces

Kania, Tomasz and Smith, Richard J. (2015) Chains of functions in C(K)-spaces. Journal of the Australian Mathematical Society, 99 (3). pp. 350-363. ISSN 1446-7887

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Abstract

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.

Item Type:
Journal Article
Journal or Publication Title:
Journal of the Australian Mathematical Society
Additional Information:
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.
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
75405
Deposited By:
Deposited On:
09 Sep 2015 06:32
Refereed?:
Yes
Published?:
Published
Last Modified:
01 Dec 2020 02:51