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CHAINS OF FUNCTIONS IN $C(K)$-SPACES

Part of: Complex dynamical systems Normed linear spaces and Banach spaces; Banach lattices Special properties

Published online by Cambridge University Press:  02 September 2015

TOMASZ KANIA  and
RICHARD J. SMITH
TOMASZ KANIA*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, UK Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-056 Warszawa, Poland email tomasz.marcin.kania@gmail.com
RICHARD J. SMITH
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland email richard.smith@maths.ucd.ie
*
✉tomasz.marcin.kania@gmail.com
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Abstract

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The Bishop property (♗), 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 (♗): 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 (♗) 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 nonmetrizable linearly ordered space, then every member of $\mathscr{D}$ has (♗). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf 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 (♗) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

Keywords

compact spacelinear orderinguncountable chain

MSC classification

Secondary: 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 37F20: Combinatorics and topology 46B26: Nonseparable Banach spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 3 , December 2015 , pp. 350 - 363
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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