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

Abstract

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.