Journal of Mathematical Sciences

, Volume 237, Issue 3, pp 445–459 | Cite as

The Geometry of Projective, Injective, and Flat Banach Modules

  • N. T. NemeshEmail author


In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology and the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an Open image in new window - or Open image in new window -space all its homologically trivial modules possess the Dunford–Pettis property.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Grad. Texts Math., Vol. 233, Springer (2006).Google Scholar
  2. 2.
    D. P. Blecher and N. Ozawa, “Real positivity and approximate identities in Banach algebras,” Pacific J. Math., 277, No. 1, 1–59 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. P. Blecher and C. J. Read, “Operator algebras with contractive approximate identities,” J. Funct. Anal., 261, No. 1, 188–217 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bourgain, New Classes of Open image in new window -Spaces, Springer (1981).Google Scholar
  5. 5.
    J. Bourgain, “On the Dunford–Pettis property,” Proc. Am. Math. Soc., 81, No. 2, 265–272 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. G. Dales and M. E. Polyakov, “Homological properties of modules over group algebras,” Proc. London Math. Soc., 89, No. 2, 390–426 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud., Vol. 176, Elsevier (1992).Google Scholar
  8. 8.
    M. Fabian and P. Habala, Banach Space Theory, Springer (2011).Google Scholar
  9. 9.
    M. González and J. Gutiérrez, “The Dunford–Pettis property on tensor products,” Math. Proc. Cambridge Philos. Soc., 131, No. 1, 185–192 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. W. M. Graven, “Injective and projective Banach modules,” Indag. Math., 82, No. 1, 253–272 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Grothendieck, “Une caractérisation vectorielle-métrique des espaces L 1,” Can. J. Math., 7, 552–561 (1955).CrossRefzbMATHGoogle Scholar
  12. 12.
    A. Ya. Helemskii, Banach and Polynormed Algebras: General Theory, Representations, Homology [in Russian], Nauka, Moscow (1989).Google Scholar
  13. 13.
    A. Y. Helemskii, The Homology of Banach and Topological Algebras, Math. Its Appl., Vol. 41, Springer (1989)Google Scholar
  14. 14.
    A. Ya. Helemskii, “Metric version of flatness and Hahn–Banach type theorems for normed modules over sequence algebras,” Stud. Math., 206, No. 2, 135–160 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Ya. Helemskii, “Metric freeness and projectivity for classical and quantum normed modules,” Sb. Math., 204, No. 7, 1056–1083 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, Transl. Math. Monogr., Vol. 233, Amer. Math. Soc., 2006.Google Scholar
  17. 17.
    W. B. Johnson and J. Lindenstrauss, Handbook of the Geometry of Banach Spaces, Vol. 2, Elsevier (2001).Google Scholar
  18. 18.
    G. Köthe, “Hebbare lokalkonvexe Räume,” Math. Ann., 165, No. 3, 181–195 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer (1974).Google Scholar
  20. 20.
    J. Lindenstrauss and A. Pelczynski, “Absolutely summing operators in Open image in new window-spaces and their applications,” Stud. Math., 29, No. 3, 275–326 (1968).Google Scholar
  21. 21.
    N. T. Nemesh, “Metrically and topologically projective ideals of Banach algebras,” Math. Notes, 99, No. 4, 524–533 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J.-P. Pier, Amenable Locally Compact Groups, Wiley–Interscience (1984).Google Scholar
  23. 23.
    G. Racher, “Injective modules and amenable groups,” Comment. Math. Helv., 88, No. 4, 1023–1031 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Ramsden, Homological Properties of Semigroup Algebras, thesis, University of Leeds (2009).Google Scholar
  25. 25.
    S. M. Shteiner, “Topological freedom for classical and quantum normed modules,” Vestn. SamGU. Estestvennonauchn. ser., No. 9/1 (110), 49–57 (2013).Google Scholar
  26. 26.
    C. P. Stegall and J. R. Retherford, “Fully nuclear and completely nuclear operators with applications to Open image in new window- and Open image in new window-spaces,” Trans. Am. Math. Soc., 163, 457–492 (1972).Google Scholar
  27. 27.
    M. C. White, “Injective modules for uniform algebras,” Proc. London Math. Soc., 3, No. 1, 155–184 (1996).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations