Skip to main content
SpringerLink
Log in

On Positive Injective Tensor Products Being Grothendieck Spaces

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. In this paper, we show that the positive injective tensor product \(\lambda {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over \otimes } _{\left| \varepsilon \right|}}X\) is a Grothendieck space if and only if X is a Grothendieck space and every positive linear operator from λ* to X** is compact.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Q. Bu and G. Buskes, Schauder decompositions and the Fremlin projective tensor product of Banach lattices, J. Math. Anal. Appl., 355 (2009), 335–351.

    Article  MathSciNet  Google Scholar 

  2. Q. Bu and N. C. Wong, Some geometric properties inherited by the positive tensor products of atomic Banach lattices, Indag. Math. (N.S.), 23 (2012), 199–213.

    Article  MathSciNet  Google Scholar 

  3. J. Diestel, Grothendieck spaces and vector measures, In: Vector and operator valued measures and applications (Proc. Sympos., Snowbird Resort, Alta, Utah, 1972), Academic Press, New York, 1973, pp. 97–108.

    Chapter  Google Scholar 

  4. D. H. Fremlin, Tensor products of Archimedean vector lattices, Amer. J. Math., 94 (1972), 778–798.

    Article  MathSciNet  Google Scholar 

  5. D. H. Fremlin, Tensor products of Banach lattices, Math. Ann., 211 (1974), 87–106.

    Article  MathSciNet  Google Scholar 

  6. A. Grothendieck, Sur les applications lineaires faiblement compactes d’espaces du type C(K), Canad. J. Math., 5 (1953), 129–173.

    Article  MathSciNet  Google Scholar 

  7. Y. Li and Q. Bu, New examples of non-reflexive Grothendieck spaces, Houston J. Math., 43 (2017), 569–575.

    MathSciNet  MATH  Google Scholar 

  8. P. Meyer-Nieberg, Banach lattices, Springer-Verlag, 1991.

Download references

Acknowledgement

This work was supported by the National Natural Science Foundation of China (11971493).

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, P. R. China

    Shaoyong Zhang

  2. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, P. R. China

    Zhaohui Gu

  3. Department of Mathematics, Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Yongjin Li

Authors
  1. Shaoyong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaohui Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongjin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Shaoyong Zhang, Zhaohui Gu or Yongjin Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, S., Gu, Z. & Li, Y. On Positive Injective Tensor Products Being Grothendieck Spaces. Indian J Pure Appl Math 51, 1239–1246 (2020). https://doi.org/10.1007/s13226-020-0461-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-020-0461-1

Key words

2010 Mathematics Subject Classification

Search

Navigation