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Archiv der Mathematik

, Volume 112, Issue 6, pp 623–631 | Cite as

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

  • Sergio A. PérezEmail author
  • Michael A. Rincón-Villamizar
Article
  • 44 Downloads

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.

Keywords

Embeddings of \(c_{0}(\Gamma )\) spaces \(l_{\infty }(\Gamma )\) spaces Spaces of bounded linear operators 

Mathematics Subject Classification

Primary 46B03 46E15 Secondary 46E40 46B25 

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Notes

Acknowledgements

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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Escuela de Matemáticas y EstadísticaUniversidad Pedagógica y Tecnológica de ColombiaTunjaColombia
  2. 2.Escuela de MatemáticasUniversidad Industrial de SantanderBucaramangaColombia

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