Skip to main content
SpringerLink
Log in

Two Applications of Boolean Valued Analysis

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

The paper contains two main results that are obtained by using Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices onto the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of ALp and c0 (Γ) for the class of the so-called \(\mathbb{B}\)-cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.

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. Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [Russian], Nauka, Moscow (2005).

    MATH  Google Scholar 

  2. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis: Selected Topics (Trends Sci. South Russia), Vladikavkaz Sci. Center Press, Vladikavkaz (2014).

    Google Scholar 

  3. Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht (2001).

    MATH  Google Scholar 

  4. Meyer-Nieberg P., Banach Lattices, Springer, Berlin etc. (1991).

    Book  Google Scholar 

  5. Abramovich Y. A. and Kitover A. K., Inverses of Disjointness Preserving Operators, Amer. Math. Soc., Providence (2000) (Mem. Amer. Math. Soc.; No. 679).

    Book  Google Scholar 

  6. Lindenstrauss J. and Tzafriri L., Classical Banach Spaces. Vol. 2. Function Spaces, Springer-Verlag, Berlin etc. (1979).

    Book  Google Scholar 

  7. Gordon E. I., “Real numbers in Boolean-valued models of set theory and K-spaces,” Dokl. Akad. Nauk SSSR, vol. 237, no. 4, 773–775 (1977).

    MathSciNet  Google Scholar 

  8. Gutman A. E., Kusraev A. G., and Kutateladze S. S., “The Wickstead problem,” Sib. Èlektron. Mat. Izv., vol. 5, 293–333 (2008).

    MathSciNet  MATH  Google Scholar 

  9. Gutman A. E., “Locally one-dimensional K-spaces and σ-distributive Boolean algebras,” Siberian Adv. Math., vol. 5, no. 2, 99–121 (1995).

    MathSciNet  Google Scholar 

  10. Harmand P., Werner D., and Wener W., M-Ideals in Banach Spaces and Banach Algebras, Springer-Verlag, Berlin etc. (1993) (Lecture Notes Math.; V. 1547).

    Book  Google Scholar 

  11. Kusraev A. G., “The Boolean transfer principle for injective Banach lattices,” Sib. Math. J., vol. 25, no. 1, 888–900 (2015).

    Article  MathSciNet  Google Scholar 

  12. Takeuti G., “Von Neumann algebras and Boolean valued analysis,” J. Math. Soc. Japan., vol. 35, no. 1, 1–21 (1983).

    Article  MathSciNet  Google Scholar 

  13. Takeuti G., “C*-algebras and Boolean valued analysis,” Japan J. Math., vol. 9, no. 2, 207–246 (1983).

    Article  MathSciNet  Google Scholar 

  14. Ozawa M., “Boolean valued interpretation of Banach space theory and module structure of von Neumann algebras,” Nagoya Math. J., vol. 117, 1–36 (1990).

    Article  MathSciNet  Google Scholar 

  15. Coppel W. A., Foundations of Convex Geometry, Cambridge University Press, Cambridge (1988).

    MATH  Google Scholar 

  16. Haydon R., “Injective Banach lattices,” Math. Z., Bd 156, 19–47 (1977).

    Article  MathSciNet  Google Scholar 

  17. Gordon E. I. and Lyubetsky V. A., “Some applications of nonstandard analysis in the theory of Boolean valued measures,” Dokl. Akad. Nauk SSSR, vol. 256, no. 5, 1037–1041 (1981).

    MathSciNet  Google Scholar 

  18. Gordon E. I. and Lyubetsky V. A., “Boolean completion of uniformities,” in: Studies on Nonclassical Logics and Formal Systems [Russian], Nauka, Moscow, 1983, 82–153.

    Google Scholar 

  19. Ando T., “DeBanachverbande und positive Projektionen,” Math. Z., Bd 109, 121–130 (1969).

    Article  MathSciNet  Google Scholar 

  20. Kusraev A. G., “Atomicity in injective Banach lattices,” Vladikavkaz. Mat. Zh., vol. 17, no. 3, 34–42 (2015).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciences, North Ossetian State University, Vladikavkaz, Russia

    A. G. Kusraev

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. S. Kutateladze

Authors
  1. A. G. Kusraev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. S. Kutateladze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. G. Kusraev or S. S. Kutateladze.

Additional information

To Yu. G. Reshetnyak on the occasion of his 90th birthday.

Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 5, pp. 1153–1164.

The authors were supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant No. I.1.2, Project 0314-2019-0005).

The authors express their gratitude to the referee for removing numerous typos and inaccuracies.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kusraev, A.G., Kutateladze, S.S. Two Applications of Boolean Valued Analysis. Sib Math J 60, 902–910 (2019). https://doi.org/10.1134/S0037446619050124

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446619050124

Keywords

Search

Navigation