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  • 1211.2867

    Rights statement: This is the pre-print version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, 401, 1, 2013 DOI: 10.1016/j.jmaa.2012.12.017

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A reflexive Banach space whose algebra of operators is not a Grothendieck space

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  • Tomasz Kania
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<mark>Journal publication date</mark>1/05/2013
<mark>Journal</mark>Journal of Mathematical Analysis and Applications
Issue number1
Volume401
Number of pages2
Pages (from-to)242-243
Publication StatusPublished
Early online date13/12/12
<mark>Original language</mark>English

Abstract

By a result of Johnson, the Banach space $F=(\bigoplus_{n=1}^\infty
\ell_1^n)_{\ell_\infty}$ contains a complemented copy of $\ell_1$. We identify
$F$ with a complemented subspace of the space of (bounded, linear) operators on
the reflexive space $(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_p}$ ($p\in
(1,\infty))$, thus giving a negative answer to the problem posed in the
monograph of Diestel and Uhl which asks whether the space of operators on a
reflexive Banach space is Grothendieck.

Bibliographic note

This is the pre-print version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, 401, 1, 2013 DOI: 10.1016/j.jmaa.2012.12.017