Skip to main content
SpringerLink
Log in

Erased Kantorovich Spaces

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

An erased Kantorovich space is the lattice ordered additive group of a Kantorovich space. We study the special role of erased Kantorovich spaces in extending positive, dominated, and lattice homomorphisms and also the existence of unbounded polar-preserving group homomorphisms. Our method of study is Boolean-valued analysis.

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

Notes

  1. As Veksler reported to the authors in 2005, a similar question was earlier raised by Lozanovskii at Leningrad mathematical seminars.

References

  1. Kutateladze S. S., “Modules admitting convex analysis,” Soviet Math. Dokl., vol. 21, no. 3, 820–823 (1980).

    MATH  Google Scholar 

  2. Kutateladze S. S., “Convex analysis in modules,” Sib. Math. J., vol. 22, no. 4, 575–583 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bigard A., “Modules ordonnes injectifs,” Mathematica, vol. 15, no. 1, 15–24 (1973).

    MathSciNet  MATH  Google Scholar 

  4. Breckner W. W. and Scheiber E., “A Hahn–Banach extension theorem for linear mappings into ordered modules,” Mathematica (Cluj), vol. 19, no. 1, 13–27 (1977).

    MathSciNet  MATH  Google Scholar 

  5. Ghika A., “The extension of general linear functionals in semi-normed modules,” Acad. Repub. Pop. Romane Bul. Sti. Ser. Mat. Fiz. Chim., vol. 2, 399–405 (1950).

    MathSciNet  Google Scholar 

  6. Ohron M., “On the Hahn–Banach theorem for modules over \( C(S) \),” J. London Math. Soc. (2), vol. 1, 363–368 (1969).

    MathSciNet  Google Scholar 

  7. Vincent-Smith G., “The Hahn–Banach theorem for modules,” Proc. London Math. Soc. (3), vol. 17, 72–90 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  8. Vuza D., “The Hahn–Banach extension theorem for modules over ordered rings,” Rev. Roumaine Math. Pures Appl., vol. 27, no. 9, 989–995 (1982).

    MathSciNet  MATH  Google Scholar 

  9. Hager A. and Madden J. J., “Majorizing-injectivity in abelian lattice-ordered groups,” Rend. Sem. Mat. Univ. Padova, vol. 69, no. 9, 181–194 (1983).

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Aliprantis C. D. and Burkinshaw O., Positive Operators, Academic, London (1985).

    MATH  Google Scholar 

  12. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis: Selected Topics, SMI VSC RAS, Vladikavkaz (2014) (Trends in Science: The South of Russia. A Math. Monogr. 6).

    MATH  Google Scholar 

  13. Birkhoff G., Lattice Theory, Amer. Math. Soc., Providence (1967).

    MATH  Google Scholar 

  14. Fuchs L., Partially Ordered Algebraic Systems, Pergamon, Oxford etc. (1963).

    MATH  Google Scholar 

  15. Bigard A., Keimel K., and Wolfenstein S., Groupes et Anneaux Réticulés, Springer, Berlin, Heidelberg, and New York (1977) (Lecture Notes Math.; Vol. 608).

    Book  MATH  Google Scholar 

  16. Hall M., Jr., The Theory of Groups, Amer. Math. Soc., Providence (1998).

    MATH  Google Scholar 

  17. Veksler A. I. and Geiler V. A., “Order and disjoint completeness of linear partially ordered spaces,” Sib. Math. J., vol. 13, no. 1, 30–35 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  18. Bernau S. J., “Lateral and Dedekind completion of archimedean lattice groups,” J. London Math. Soc., vol. 12, no. 2, 320–322 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  19. Bernau S. J., “Lateral completion for arbitrary lattice groups,” Bull. Amer. Math. Soc., vol. 80, no. 2, 334–336 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  20. Conrad P. F., “Epi-archimedean groups,” Czechoslovak Math. J., vol. 24, no. 2, 1–29 (1974).

    MathSciNet  MATH  Google Scholar 

  21. Jakubík J., “Strong projectability of lattice ordered groups,” Czechoslovak Math. J., vol. 55, no. 4, 957–973 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  22. Conrad P. F. and Mcalister D., “The completion of a lattice-ordered group,” J. Austral. Math. Soc., vol. 9, 182–208 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  23. Buskes G. and van Rooij A., “The vector lattice cover of certain partially ordered groups,” Austral. Math. Soc. (Ser. A), vol. 54, 352–367 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  24. Jakubík J., “Lateral completion of a projectable lattice ordered group,” Czechoslovak Math. J., vol. 50, no. 2, 431–444 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  25. Conrad P. F., “Minimal vector lattice covers,” Bull. Austral. Math. Soc., vol. 4, 35–39 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  26. Wickstead A. W., “Representation and duality of multiplication operators on Archimedean Riesz space,” Compositio Math., vol. 35, no. 3, 225–238 (1977).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. Aczél J. and Dhombres J., Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Cambridge University, Cambridge (2008) (Encyclopaedia of Mathematics and Its Applications).

    MATH  Google Scholar 

  29. Conrad P. F. and Diem J. E., “The ring of polar preserving endomorphisms of an abelian lattice-ordered group,” Illinois J. Math., vol. 15, 222–240 (1971).

    MathSciNet  MATH  Google Scholar 

  30. Gutman A. E., “Locally one-dimensional \( K \)-spaces and \( \sigma \)-distributive Boolean algebras,” Sib. Elektron. Mat. Izv., vol. 5, no. 1, 42–48 (1995).

    MathSciNet  MATH  Google Scholar 

  31. Hayes A., “Indecomposable positive additive functionals,” J. London Math. Soc., vol. 41, 318–322 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  32. Kusraev A. G. and Kutateladze S. S., Subdifferentials: Theory and Applications, Kluwer, Dordrecht (1995).

    Book  MATH  Google Scholar 

  33. Aron E. R., Hager A. W., and Madden J. J., “Extensions of \( \ell \)-homomorphisms,” Rocky Mount. J. Math., vol. 12, no. 3, 481–490 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  34. Hayes A., “Additive functionals on groups,” Proc. Cambridge Phil. Soc., vol. 58, 196–205 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  35. Lipecki L., “Extension of vector-lattice homomorphisms,” Proc. Amer. Math. Soc., vol. 79, no. 2, 247–248 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  36. Luxemburg W. A. J. and Schep A. R., “An extension theorem for Riesz homomorphisms,” Indag. Math., vol. 41, no. 2, 145–154 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  37. Kutateladze S. S., “The Krein–Mil’man theorem and its converse,” Sib. Math. J., vol. 21, no. 1, 97–103 (1980).

    Article  MATH  Google Scholar 

  38. Buskes G. J. H. M. and van Rooij A. C. M., “Hahn–Banach for Riesz homomorphisms,” Indag. Math. (Proc.), vol. A92, no. 1, 25–34 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  39. Bonnice W. and Silvermann R., “The Hahn–Banach extension and the least upper bound properties are equivalent,” Proc. Amer. Math. Soc., vol. 18, no. 5, 843–850 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  40. Ioffe A. D., “A new proof of the equivalence of the Hahn–Banach extension and the least upper bound properties,” Proc. Amer. Math. Soc., vol. 82, no. 3, 385–389 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  41. Lipecki Z., “Extensions of positive operators and extreme points. III,” Colloquium Math., vol. 46, no. 2, 263–268 (1982).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The research was supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement 075–02–2021–1844) and carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314–2019–0005).

Author information

Authors and Affiliations

  1. Southern Mathematical Institute, North-Caucasian Mathematical Research Center, 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 author

Correspondence to A. G. Kusraev.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 1, pp. 123–144. https://doi.org/10.33048/smzh.2022.63.109

To the blessed memory of Leonid Vital’evich Kantorovich.

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. Erased Kantorovich Spaces. Sib Math J 63, 102–118 (2022). https://doi.org/10.1134/S0037446622010098

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446622010098

Keywords

UDC

Search

Navigation