Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 243–255 | Cite as

Character Connes amenability of dual Banach algebras



We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.


dual Banach algebra Connes amenability character amenability locally compact group 

MSC 2010

46H20 46H25 43A07 22D15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. F. Bonsall, J. Duncan: Complete Normed Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete 80, Springer, Berlin, 1973.Google Scholar
  2. [2]
    M. Daws: Connes-amenability of bidual and weighted semigroup algebras. Math. Scand. 99 (2006), 217–246.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    E. G. Effros: Amenability and virtual diagonals for von Neumann algebras. J. Funct. Anal. 78 (1988), 137–153.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    P. Eymard: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. Fr. 92 (1964), 181–236. (In French.)CrossRefMATHGoogle Scholar
  5. [5]
    G. B. Folland: A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1995.Google Scholar
  6. [6]
    G. B. Folland: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley & Sons, New York, 1999.Google Scholar
  7. [7]
    B. Hayati, M. Amini: Connes-amenability of multiplier Banach algebras. Kyoto J. Math. 50 (2010), 41–50.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    A. Ya. Helemskiĭ: The homological essence of Connes amenability: Injectivity of the predual bimodule. Math. USSR, Sb. 68 (1990), 555–566. (In English. Russian original.); translation from Mat. Sb. 180 (1989), 1680–1690.MATHGoogle Scholar
  9. [9]
    Z. Hu, M. S. Monfared, T. Traynor: On character amenable Banach algebras. Stud. Math. 193 (2009), 53–78.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    B. E. Johnson: Cohomology in Banach Algebras. Memoirs of the American Mathematical Society 127, American Mathematical Society, Providence, 1972.Google Scholar
  11. [11]
    B. E. Johnson, R. V. Kadison, J. R. Ringrose: Cohomology of operator algebras. III: Reduction to normal cohomology. Bull. Soc. Math. Fr. 100 (1972), 73–96.MathSciNetMATHGoogle Scholar
  12. [12]
    E. Kaniuth, A. T. Lau, J. Pym: On character amenability of Banach algebras. J. Math. Anal. Appl. 344 (2008), 942–955.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    E. Kaniuth, A. T. Lau, J. Pym: On φ-amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144 (2008), 85–96.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. T. Lau: Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fundam. Math. 118 (1983), 161–175.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Mahmoodi: On φ-Connes amenability of dual Banach algebras. J. Linear Topol. Algebra 3 (2014), 211–217.Google Scholar
  16. [16]
    M. S. Monfared: On certain products of Banach algebras with applications to harmonic analysis. Stud. Math. 178 (2007), 277–294.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    M. S. Monfared: Character amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144 (2008), 697–706.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    R. Nasr-Isfahani, S. S. Renani: Character contractibility of Banach algebras and homological properties of Banach modules. Stud. Math. 202 (2011), 205–225.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    V. Runde: Amenability for dual Banach algebras. Stud. Math. 148 (2001), 47–66.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    V. Runde: Lectures on Amenability. Lecture Notes in Mathematics 1774, Springer, Berlin, 2002.Google Scholar
  21. [21]
    V. Runde: Connes-amenability and normal, virtual diagonals for measure alebras I. J. Lond. Math. Soc., II. Ser. 67 (2003), 643–656.CrossRefMATHGoogle Scholar
  22. [22]
    V. Runde: Connes-amenability and normal, virtual diagonals for measure algebras II. Bull Aust. Math. Soc. 68 (2003), 325–328.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    V. Runde: Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule. Math. Scand. 95 (2004), 124–144.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    V. Runde, F. Uygul: Connes-amenability of Fourier-Stieltjes algebras. Bull. Lond. Math. Soc. 47 (2015), 555–564.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Y. Zhang: Weak amenability of module extensions of Banach algebras. Trans. Am. Math. Soc. 354 (2002), 4131–4151.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceDamghan UniversityDamghanIran

Personalised recommendations