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

Hart, Klaas Pieter and Kania, Tomasz and Kochanek, Tomasz (2014) A chain condition for operators from C(K)-spaces. The Quarterly Journal of Mathematics, 65 (2). pp. 703-715. ISSN 0033-5606

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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)).

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Journal Article
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The Quarterly Journal of Mathematics
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This article has been accepted for publication in Quarterly Journal of Mathematics Published by Oxford University Press.
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18 Jun 2015 05:44
Last Modified:
22 Oct 2023 00:21