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

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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.

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References

  1. Albiac, F., Kalton, N.J.: Topics in Banach space theory. In: Graduate Texts in Mathematics, vol. 233, Springer, New York (2006)

  2. Banach, S.: Théorie des opérations linéaires. Monografie Matematyczne, Warsaw (1933)

    MATH  Google Scholar 

  3. Bessaga, C., Pełczyński, A.: On bases and unconditional convergence of series in Banach spaces. Studia Math. 17, 151–164 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  4. Feder, M.: On subspaces of spaces with unconditional basis and spaces of operators. Ill. J. Math. 34, 196–205 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ferrando, J.C., Amigó, J.M.: On copies of \(c_0\) in the bounded linear operator space. Czechoslov. Math. J. 50, 651–656 (2000)

    Article  MATH  Google Scholar 

  6. Ferrando, J.C.: On copies of \(c_0\) and \(l_\infty \) in \(L_{w^*}(X^*, Y)\). Bull. Belg. Math. Soc. Simon Stevin 9(2), 259–264 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Ferrando, J.C.: Complemented copies of \(c_0\) in spaces of operators. Acta Math. Hungar. 99(1–2), 57–61 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Josefson, B.: Weak sequential convergence in the dual of a Banach space does not imply norm convergence. Ark. Mat. 13, 79–89 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kalton, N.J.: Spaces of compact operators. Math. Ann. 208, 267–278 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lacey, H.E.: The Isometrical Theory of Classical Banach Spaces. Springer, Berlin (1974)

    Book  Google Scholar 

  11. Lewis, P.: Spaces of operators and \(c_0\). Studia Math. 145, 213–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)

    Book  MATH  Google Scholar 

  13. Nissenzweig, A.: \(\text{ Weak }^{*}\) sequential convergence. Isr. J. Math. 22, 266–272 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rosenthal, H.P.: On injective Banach spaces and the spaces \(L^\infty (\mu )\) for finite measures \(\mu \). Acta Math. 124, 205–247 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rosenthal, H.P.: On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37, 13–36 (1970)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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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). https://doi.org/10.1007/s00013-018-01296-0

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