- 1211.2770
**Rights statement:**This article has been accepted for publication in Quarterly Journal of Mathematics Published by Oxford University Press.Submitted manuscript, 197 KB, PDF document

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**A chain condition for operators from C(K)-spaces.** / Hart, Klaas Pieter; Kania, Tomasz; Kochanek, Tomasz.

Research output: Contribution to journal › Journal article › peer-review

Hart, KP, Kania, T & Kochanek, T 2014, 'A chain condition for operators from C(K)-spaces', *The Quarterly Journal of Mathematics*, vol. 65, no. 2, pp. 703-715. https://doi.org/10.1093/qmath/hat006

Hart, K. P., Kania, T., & Kochanek, T. (2014). A chain condition for operators from C(K)-spaces. *The Quarterly Journal of Mathematics*, *65*(2), 703-715. https://doi.org/10.1093/qmath/hat006

Hart KP, Kania T, Kochanek T. A chain condition for operators from C(K)-spaces. The Quarterly Journal of Mathematics. 2014 Jun 26;65(2):703-715. https://doi.org/10.1093/qmath/hat006

@article{d7eed40c58184f318199da6d2b92b87c,

title = "A chain condition for operators from C(K)-spaces",

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)). ",

author = "Hart, {Klaas Pieter} and Tomasz Kania and Tomasz Kochanek",

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

year = "2014",

month = jun,

day = "26",

doi = "10.1093/qmath/hat006",

language = "English",

volume = "65",

pages = "703--715",

journal = "The Quarterly Journal of Mathematics",

issn = "0033-5606",

publisher = "Oxford University Press",

number = "2",

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TY - JOUR

T1 - A chain condition for operators from C(K)-spaces

AU - Hart, Klaas Pieter

AU - Kania, Tomasz

AU - Kochanek, Tomasz

N1 - This article has been accepted for publication in Quarterly Journal of Mathematics Published by Oxford University Press.

PY - 2014/6/26

Y1 - 2014/6/26

N2 - 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)).

AB - 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)).

U2 - 10.1093/qmath/hat006

DO - 10.1093/qmath/hat006

M3 - Journal article

VL - 65

SP - 703

EP - 715

JO - The Quarterly Journal of Mathematics

JF - The Quarterly Journal of Mathematics

SN - 0033-5606

IS - 2

ER -