\(\sigma\)-Biflat Banach Algebras; the Older and the New Notions

  • Sanaz Haddad Sabzevar
  • Amin MahmoodiEmail author
Research Paper
Part of the following topical collections:
  1. Mathematics


We introduce a new concept of \(\sigma\)-biflatness for Banach algebras. We show difference between this new concept and the classical one. A characterization of \(\sigma\)-biflatness is also given. As we shall see, \(\sigma\)-biflatness is closely related to the notions of \(\sigma\)-amenability and \(\sigma\)-biprojectivity. We compare our definition with the existing notion of \(\sigma\)-biflat Banach algebras in the sense of Ghorbani and Bami (Bull Iran Math Soc 39(3):507–515, 2013). In addition, \(\sigma\)-biflatness of tensor product of Banach algebras is discussed.


\(\sigma\)-Amenable \(\sigma\)-Biflat \(\sigma\)-Virtual diagonal \(\sigma\)-Biprojective \(\sigma\)-Derivation 

Mathematics Subject Classification

Primary 46H25 Secondary 16E40 43A20 



We would like to thank the anonymous reviewer(s) for much helpful advice that improved the presentation of the paper, especially for directing us to the reference Mirzavaziri and Moslehian (2011) and pointing out Theorems 4.7 and 5.2.


  1. Bonsall FF, Duncan J (1973) Complete normed algebras. Springer, BerlinCrossRefGoogle Scholar
  2. Dales HG (2000) Banach algebras and automatic continuity. London Mathematical Society Monographs 24. Clarendon Press, OxfordzbMATHGoogle Scholar
  3. Ghorbani Z, Bami ML (2013) \(\varphi\)-amenable and \(\varphi\)-biflat Banach algebras. Bull Iran Math Soc 39(3):507–515MathSciNetzbMATHGoogle Scholar
  4. Gronbaek N (1988) Amenability of weighted discrete convolution algebras. Proc R Soc Edinb Sect A 110:351–360MathSciNetCrossRefGoogle Scholar
  5. Helemskii AY (1989) The Homology of Banach and topological algebras. Kluwer, DordrechtCrossRefGoogle Scholar
  6. Johnson BE (1972) Cohomology in Banach Algebras. Mem Am Math Soc 127:1–96MathSciNetGoogle Scholar
  7. Kaniuth E, Lau AT, Pym J (2008) On \(\varphi\)-amenability of Banach algebras. Math Proc Camb Philos Soc 144:85–96MathSciNetCrossRefGoogle Scholar
  8. Mirzavaziri M, Moslehian MS (2006) \(\sigma\)-derivation in Banach algebras. Bull Iran Math Soc 32(1):65–78MathSciNetzbMATHGoogle Scholar
  9. Mirzavaziri M, Moslehian MS (2009) \(\sigma\)-amenability of Banach algebras. Southeast Asian Bull Math 33:89–99MathSciNetzbMATHGoogle Scholar
  10. Mirzavaziri M, Moslehian MS (2011) \((\sigma, \tau )\)-amenability of C\(^*\)-algebras. Georgian Math J 18(1):137–145MathSciNetzbMATHGoogle Scholar
  11. Moslehian MS, Motlagh AN (2008) Some notes on (\(\sigma,\tau\))-amenable of Banach algebras. Stud Univ Babeş-Bolyai Math 53(3):57–68MathSciNetzbMATHGoogle Scholar
  12. Ramsden P (2009) Biflatness of semigroup algebras. Semigroup Forum 79:515–530MathSciNetCrossRefGoogle Scholar
  13. Runde V (2002) Lectures on amenability, vol 1774. Lecture Notes in Mathematics. Springer, BerlinCrossRefGoogle Scholar
  14. Sahami A, Pourabbas A (2013) On \(\phi\)-biflat and \(\phi\)-biprojective Banach algebras. Bull Belg Math Soc 20(5):789–801MathSciNetzbMATHGoogle Scholar
  15. Yazdanpanah T, Najafi H (2010) \(\sigma\)-contractible and \(\sigma\)-biprojective Banach algebras. Quaest Math 33:485–495MathSciNetCrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Central Tehran BranchIslamic Azad UniversityTehranIran

Personalised recommendations