Advertisement

Positivity

, Volume 20, Issue 2, pp 399–412 | Cite as

On natural density, orthomodular lattices, measure algebras and non-distributive \(L^p\) spaces

  • Jarno TalponenEmail author
Article
  • 104 Downloads

Abstract

In this note we first show, roughly speaking, that if \(\mathcal {B}\) is a Boolean algebra included in the natural way in the collection \(\mathcal {D}/_\sim \) of all equivalence classes of natural density sets of the natural numbers, modulo null density, then \(\mathcal {B}\) extends to a \(\sigma \)-algebra \(\Sigma \subset \mathcal {D}/_\sim \) and the natural density is \(\sigma \)-additive on \(\Sigma \). We prove the main tool employed in the argument in a more general setting, involving a kind of quantum state function, more precisely, a group-valued submeasure on an orthomodular lattice. At the end we discuss the construction of ‘non-distributive \(L^p\) spaces’ by means of submeasures on lattices.

Keywords

Orthomodular lattice Boolean algebra Measure algebra Group-valued submeasure Natural density Density set Content extension Banach space Function space \(L^p\) space 

Mathematics Subject Classification

06C15 28A60 11B05 46E99 

Notes

Acknowledgments

This research was financially supported by the Academy of Finland Project #268009, the Finnish Cultural Foundation and Väisälä Foundation.

References

  1. 1.
    Austin, D.G.: An isomorphism theorem for finitely additive measures. Proc. Am. Math. Soc. 6, 205–208 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Behrends, E. et al.: \(L^p\)-Structure in Real Banach Spaces. In Lecture Notes in Mathematics 613, Springer, Berlin (1977)Google Scholar
  3. 3.
    Beran, L.: Orthomodular Lattices. Algebraic Approach. Mathematics and Its Applications. D. Reidel Publishing Co., Dordrecht (1985)zbMATHGoogle Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publications, 25. American Mathematical Society, Providence (1979)Google Scholar
  5. 5.
    Blass, A., Frankiewicz, R., Plebanek, G., Ryll-Nardzewski, C.: A note on extensions of asymptotic density. Proc. Am. Math. Soc. 129, 3313–3320 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buck, C.R.: The measure theoretic approach to density. Am. J. Math. 68, 560–580 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buck, E.F., Buck, C.R.: A note on finitely-additive measures. Am. J. Math. 69, 413–420 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    de Lucia, P., Morales, P.: A non-commutative version of a theorem of Marczewski for submeasures. Studia Math 101, 123–138 (1992)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Di Nasso, M.: Fine asymptotic densities for sets of natural numbers. Proc. Am. Math. Soc. 138, 2657–2665 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dow, A., Hart, K.: The measure algebra does not always embed. Fund. Math. 163, 163–176 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fabian, M., Habala, P., Hájek, P., Montesinos Santaluca, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, 8th edn. Springer, New York (2001)CrossRefGoogle Scholar
  12. 12.
    Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fremlin, D.H.: Measure Theory. Torres Fremlin, England (2000)zbMATHGoogle Scholar
  14. 14.
    Pisier, G., Xu, Q.: Non-commutative Lp-Spaces. Handbook of the Geometry of Banach Spaces. North-Holland, Amsterdam (2003)Google Scholar
  15. 15.
    Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27, 199–245 (1975)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wittstock, G.: Ordered Normed Tensor Products, Lecture Notes in Physics, 25, Springer, Berlin (1972)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

Personalised recommendations