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  • 1211.2770

    Rights statement: This article has been accepted for publication in Quarterly Journal of Mathematics Published by Oxford University Press.

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A chain condition for operators from C(K)-spaces

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  • Klaas Pieter Hart
  • Tomasz Kania
  • Tomasz Kochanek
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<mark>Journal publication date</mark>26/06/2014
<mark>Journal</mark>The Quarterly Journal of Mathematics
Issue number2
Volume65
Number of pages13
Pages (from-to)703-715
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We introduce a chain condition (bishop), defined for operators acting on C(K)-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pe{\l}czy\'nski's characterisation of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies (bishop) if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (bishop), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying (bishop) forms a closed left ideal of B(C(K)).

Bibliographic note

This article has been accepted for publication in Quarterly Journal of Mathematics Published by Oxford University Press.