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Copies of \(c_0(\Gamma )\) in the space of bounded linear operators

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The space \({{\mathcal {L}}}(X, Y)\) stands for the Banach space of all bounded linear operators from X to Y endowed with the operator norm. It is shown that \(c_{0}(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) if and only if \(l_{\infty }(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) or \(c_{0}(\Gamma )\) embeds into Y. As a consequence, we extend a classical Kalton theorem by proving that if \(c_{0}(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) and X has the \(|\Gamma |\)-Josefson–Nissenzweig property, then \(l_{\infty }(\Gamma )\) also embeds into \({{\mathcal {L}}}(X, Y)\). We also show that, for certain Banach spaces X and Y, \(c_{0}(\Gamma )\) embeds complementably into \({{\mathcal {L}}}(X, Y)\) if and only if \(c_{0}(\Gamma )\) embeds into Y.

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We thank to Professor E. M. Galego for suggesting us the problems studied in this article. The second author also thanks to Vicerrectoría de Investigación y Extensión (VIE) de la Universidad Industrial de Santander for supporting this work, which is part of the VIE Project C-2018-02.

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Correspondence to Sergio A. Pérez.

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Pérez, S.A., Rincón-Villamizar, M.A. Copies of \(c_0(\Gamma )\) in the space of bounded linear operators. Arch. Math. 112, 623–631 (2019).

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