, Volume 10, Issue 3, pp 591–606 | Cite as

Mackey Topologies and Mixed Topologies in Riesz Spaces

  • Jurie Conradie


The possibility of characterizing the Mackey topology of a dual pair of vector spaces as a generalized inductive limit (or mixed) topology is investigated. Positive answers are given for a wide range of dual pairs of Riesz spaces (vector lattices) and non-commutative Banach function spaces (or symmetric operator spaces).

Mathematics Subject Classification (2000)

46A70 46A40 46L52 


Generalized inductive limit topology Mackey topology pre–Lebesgue topology Riesz space symmetric operator space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C.D. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York (1988).Google Scholar
  2. 2.
    C.D. Aliprantis, O. Burkinshaw, M. Duhoux, Compactness properties of abstract kernel operators, Pacific J. Math. 100 (1982), 1–22Google Scholar
  3. 3.
    J.B. Cooper, Saks spaces and applications to functional analysis, North Holland, Amsterdam (1978).Google Scholar
  4. 4.
    J.J. Conradie, Generalized precompactness and mixed topologies, Collect. Math., 44 (1993), 164–172.Google Scholar
  5. 5.
    J.J. Conradie, Duality results for order precompact sets in locally solid Riesz spaces, Indag. Mathem. N.S., 2 (1991), 19–28.Google Scholar
  6. 6.
    J.J. Conradie, The coarsest Hausdorff Lebesgue topology, Quaestiones Math. 28 (2005), 287–304.Google Scholar
  7. 7.
    H.S. Collins, Strict, weighted and mixed topologies and applications, Adv. Math., 19 (1976), 207–237.Google Scholar
  8. 8.
    J. Diestel, Sequences and series in Banach space, Graduate texts in mathematics 92 Springer, Berlin–Heidelberg–New York (1984).Google Scholar
  9. 9.
    P.G. Dodds, T.K. Dodds, B. de Pagter, Weakly compact subsets of symmetric operator spaces, Math. Proc. Camb. Phil. Soc., 111 (1992), 355–368.Google Scholar
  10. 10.
    P.G. Dodds, T.K. Dodds, B. de Pagter, Non-commutative Köthe duality, Trans. Amer. Math. Soc., 339 (1993), 717–750.Google Scholar
  11. 11.
    J. Dazourd, M. Jourlin, Une topologie mixte sur l'espace L, Publ. Dp. Math. Lyon, 11 (1974), fasc.2, 1–18.Google Scholar
  12. 12.
    M. Duhoux, Order precompactness in topological Riesz spaces, J. London Math. Soc., (2) 23 (1981), 509–522.Google Scholar
  13. 13.
    D.H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press (1974).Google Scholar
  14. 14.
    D.H. Fremlin, D.J.H. Garling, R. Haydon, Bounded measures on topological spaces, Proc. Lond. Math. Soc., (3) 25 (1972), 115–136.Google Scholar
  15. 15.
    D.J.H. Garling, A generalized form of inductive limit topology for vector spaces, Proc. Lond. Math. Soc., 14 (1964), 1–28.Google Scholar
  16. 16.
    D. Guido, T. Isola, Singular traces on semifinite von Neumann algebras, J. Funct. Anal., 134 (1995), 451–485.Google Scholar
  17. 17.
    H. Hogbe-Nlend, Théorie des bornologies et applications, Springer, Berlin–Heidelberg–New York (1971).Google Scholar
  18. 18.
    H. Jarchow, Topological vector spaces, Teubner, Stuttgart (1981).Google Scholar
  19. 19.
    I. Kawai, Locally convex lattices, J. Math. Soc. Japan, 29 (1957), 281–314.Google Scholar
  20. 20.
    L.C. Moore, J.C. Reber, Mackey topologies which are locally convex Riesz topologies, Duke Math. J., 39 (1972), 105–119.Google Scholar
  21. 21.
    M. Nowak, On the finest Lebesgue topology on the space of essentially bounded measurable functions, Pacific J. Math., 140 (1989), 155–161.Google Scholar
  22. 22.
    H.H. Schaefer, Banach lattices and positive operators, Springer, Berlin–Heidelberg–New York (1974).Google Scholar
  23. 23.
    A. Persson, A generalization of two-norm spaces, Ark. Math., 5 (1963), 27–36.Google Scholar
  24. 24.
    A. Wiweger, Linear spaces with mixed topologies, Studia Math., 20 (1961), 47–68.Google Scholar
  25. 25.
    R.F. Wheeler, The strict topology, separable measures and paracompactness, Pacific J. Math., 47 (1973), 287–302.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

Personalised recommendations