The bidual of a Banach algebra associated with a bilinear map

  • Sedigheh BarootkoobEmail author
  • AminAllah Khosravi
  • Hamid Reza Ebrahimi Vishki


We associate a Banach algebra \(\mathbf A _\Phi \) to each bounded bilinear map \(\Phi : \mathrm X\times \mathrm Y\rightarrow \mathrm {Z}\) on Banach spaces, and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras. For example, the construction of \(\mathbf A _\Phi \) enables us to provide many simple examples of Banach algebras with different topological centers, which are neither Arens regular nor either left or right strongly Arens irregular. It also gives examples of Banach algebras which are not n-weakly amenable for each natural number n. We also find out that the dual of \(\mathbf A _\Phi \) does not enjoy the factorization property of any level.


Bidual Arens product Bilinear map Topological center Weak amenability 

Mathematics Subject Classification

Primary 46H20 Secondary 46H25 



  1. 1.
    Arens, R.: The adjoint of a bilinear operation. Proc. Am. Math. Soc. 2, 839–848 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arikan, N.: Arens regularity and reflexivity. Quart. J. Math. Oxford Ser. (2) 32(2), 383–388 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dales, H.G., Ghahramani, F., Grønbæk, N.: Derivations into iterated duals of Banach algebras. Studia Math. 128(1), 19–54 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dales, H.G., Lau, A.T.M.: The second duals of Beurling algebras. Mem. Am. Math. Soc. 177(836), vi+191 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Eshaghi Gordgi, M., Filali, M.: Arens regularity of module actions. Studia Math. 181, 237–254 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Johnson, B.E.: Cohomology in Banach algebras. Mem. Am. Math. Soc. 127 (1972)Google Scholar
  7. 7.
    Khadem-Maboudi, A.A., Ebrahimi Vishki, H.R.: Strong Arens irregularity of bilinear mappings and reflexivity. Banach J. Math. Anal. 6, 155–160 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lau, A.T.-M., Losert, V.: On the second conjugate algebra of \(L^1({\cal{G}})\) of a locally compact group. J. Lond. Math. Soc. 37, 464–470 (1988)CrossRefzbMATHGoogle Scholar
  9. 9.
    Losert, V., Neufang, M., Pachl, J., Steprns, J.: Proof of the Ghahramani–Lau conjecture. Adv. Math. 290, 709–738 (2016)Google Scholar
  10. 10.
    Mohammadzadeh, S., Ebrahimi Vishki, H.R.: Arens regularity of module actions and the second adjoint of a derivation. Bull. Aust. Math. Soc. 77, 465–476 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Neufang, M.: On a conjecture by Ghahramani-Lau and related problems concerning topological centers. J. Funct. Anal. 224, 217–229 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Raisi Tousi, Y.R., Kamyabi-Gol, R.A., Ebrahimi Vishki, H.R.: On the topological centre of \(L^1(G/H)^{**}\). Bull. Aust. Math. Soc. 86, 119–125 (2012)Google Scholar
  13. 13.
    Runde, V.: Lectures On Amenability. Lecture Notes in Mathematics, vol. 1774. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Young, N.J.: The irregularity of multiplication in group algebras. Quart. J. Math. Oxford 24, 59–62 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang, Y.: \(2m\)-weak amenability of group algebras. J. Math. Anal. Appl. 396, 412–416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Basic SciencesUniversity of BojnordBojnordIran
  2. 2.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran
  3. 3.Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

Personalised recommendations