The three space problem for locally pseudoconvex algebras

  • Mati Abel
  • Reyna María Pérez-TiscareñoEmail author


The three space problem for locally pseudoconvex algebras is considered in this paper. It is shown that a locally pseudoconvex algebra E, with jointly continuous multiplication, is locally m-pseudoconvex, if E contains a two-sided ideal I such that I (in the subset topology) and E / I (in the quotient topology) are locally m-pseudoconvex algebras.


Topological algebras Locally pseudoconvex algebras Locally m-pseudoconvex algebras Three space problem 

Mathematics Subject Classification

Primary 46H05 Secondary 46H20 



Research is supported by the institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.


  1. 1.
    Beckenstein, E., Narici, L., Warner, S.: Three-space problems in commutative algebras. Arch. Math. 41(5), 447–453 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bruguera, M., Tkachenko, M.: The three space problem in topological groups. Topol. Appl. 153, 2278–2302 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Castillo, J.M.F., González, M.: Three-Space Problems in Banach Space Theory. Springer, Berlin (2007)zbMATHGoogle Scholar
  4. 4.
    Dierolf, S., Heintz, Th.: The three-space-problem for locally-m-convex algebras. Rev. R. Acad. Cien. Serie A. Mat. 97(2), 223–227 (2003)Google Scholar
  5. 5.
    Enflo, P., Lindenstrauss, J., Pisier, G.: On the ”three space problem”. Math. Scand. 36, 199–210 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fragolopoulou, M.: Topological Algebras with Involution. North-Holland Math. Studies, vol. 200. North Holland Publ. Co., Amsterdam (2005)CrossRefGoogle Scholar
  7. 7.
    Kalton, N.J.: The three space problem for locally bounded \(F\)-spaces. Compos. Math. 37(3), 243–276 (1978)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Köthe, G.: Topological Vector Spaces I. Springer, Berlin (1969)zbMATHGoogle Scholar
  9. 9.
    Roelcke, W., Dierolf, S.: On the three-space-problem for topological vector spaces. Collect. Math. 32(1), 13–35 (1981)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Sawoń, Z.: Some remarks about the three spaces problem in Banach algebras. Czechoslovak Math. J. 28(103), 56–58 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Turpin, M. P.: Sur une classe d’algebres topologiques generalisant les algebres localement bornees, Ph.D. thesis Faculty of sciences, University of Grenoble, (1966)Google Scholar
  12. 12.
    Waelbroeck, L.: Topological vector spaces and algebras. Lecture Notes in Math, vol. 230. Springer-Verlag, Berlin, New York (1971)Google Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsUniversity of TartuTartuEstonia

Personalised recommendations