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Biprojectivity and Biflatness of Bi-amalgamated Banach Algebras

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Abstract

Let A, X be two Banach algebras and let X be an algebraic Banach A-module equipped with a bounded bilinear map \(\Theta :X\times X\rightarrow A\) which is compatible with the A-module operations of X. Then the \(\ell ^1\)-direct sum \(A\times X\) endowed with the multiplication

$$\begin{aligned} (a,x)(b,y)=(ab+\Theta (x,y),ay+xb+xy) \quad (a,b\in A, x, y\in X) \end{aligned}$$

is a Banach algebra, denoted by \(A\boxtimes _\Theta X\) and will be called a bi-amalgamated Banach algebra. Many known Banach algebras such as (generalized) module extension Banach algebras, Lau product Banach algebras, generalized matrix Banach algebras have this general framework. The main aim of this paper is to investigate biprojectivity and biflatness of \(A\boxtimes _\Theta X\). Our results extend several results in the literature and provide simple direct proofs for some known results. In particular, we characterize the biprojectivity and biflatness of certain classes of the module extension Banach algebras and generalized matrix Banach algebras. Some unsolved questions are also included.

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References

  1. Abtahi, F., Ghafarpanah, A., Rejali, A.: Biprojectivity and biflatness of Lau product of banach algebras defined by a Banach algebra morphism. Bull. Aust. Math. Soc. 91, 134–144 (2015)

    Article  MathSciNet  Google Scholar 

  2. Dales, H.G.: Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, vol. 24. The Clarendon Press, Oxford (2000)

    Google Scholar 

  3. Dales, H.G., Lau, A.T.-M.: The second duals of Beurling algebras. Mem. Amer. Math. Soc 177(836) (2005)

  4. Ebrahimi Vishki, H.R., Khoddami, A.R.: $n$-Weak amenability for Lau product of Banach algebras. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 77, 177–184 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Ebrahimi Vishki, H.R., Khoddami, A.R.: Character inner amenability of certain Banach algebras. Colloq. Math. 122, 225–232 (2011)

    Article  MathSciNet  Google Scholar 

  6. Ettefagh, M.: Biprojectivity and biflatness of generalized module extension Banach algebras. Filomat 32, 5895–5905 (2018)

    Article  MathSciNet  Google Scholar 

  7. Helemskii, A.Y.: Flat Banach modules and amenable algebras. Trudy Moskov. Math. Obshch. 47, 199–244 (1985)

    MathSciNet  Google Scholar 

  8. Khadem-Maboudi, A.A., Ebrahimi Vishki, H.R.: Strong Arens irregularity of bilinear mappings and reflexivity. Banach J. Math. Anal 6, 155–160 (2012)

    Article  MathSciNet  Google Scholar 

  9. Khodami, A.R., Ebrahimi Vishki, H.R.: Biflatness and biprojectivity of Lau product of Banach algebras. Bull. Iran. Math. Soc 39, 559–568 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Lakzian, H., Ebrahimi Vishki, H.R., Barootkoob, S.: Biduals and derivations of Morita context Banach algebras (preprint)

  11. Lakzian, H., Ebrahimi Vishki, H.R., Barootkoob, S.: Itrated duals and weak amenability of bi-amalgamated Banach algebras (preprint)

  12. Li, Y., Wei, F.: Semi-centralizing maps of generalized matrix algebras. Linear Algebra Appl. 436, 1122–1153 (2012)

    Article  MathSciNet  Google Scholar 

  13. Javanshiri, H., Nemati, M.: Amalgamated duplication of the Banach algebra $A$ along a $A$-bimodule $A$. J. Algebra Appl. 17, 1–21 (2018)

    Article  MathSciNet  Google Scholar 

  14. Medghalchi, A.R., Sattari, M.H.: Biflatness and biprojectivity of triangular Banach algebras. Bull. Iran. Math. Soc. 34, 115–120 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Pourmahmood Aghababa, H., Shirmohammadi, N.: On amalgamated Banach algebras. Period. Math. Hungar. 75, 1–13 (2017)

    Article  MathSciNet  Google Scholar 

  16. Ramezanpour, M., Barootkoob, S.: Generalized module extension Banach algebras: derivations and weak amenability. Quaest. Math. 40, 451–465 (2017)

    Article  MathSciNet  Google Scholar 

  17. Runde, V.: Lectures on Amenability, Lecture Notes in Mathematics, vol. 1774. Springer, Berlin (2002)

    Book  Google Scholar 

  18. Sahami, A., Pourabbas, A.: On $\phi $-biflat and $\phi $-biprojective Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 20, 789–801 (2013)

    Article  MathSciNet  Google Scholar 

  19. Sands, A.D.: Radicals and Morita contexts. J. Algebra 24, 335–345 (1973)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Y.: Weak amenableility of module extension of Banach algebras. Trans. Am. Math. Soc. 354, 4131–4151 (2002)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, 91775, Iran

    Hosein Lakzian

  2. Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran

    Sedigheh Barootkoob

Authors
  1. Hosein Lakzian
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  2. Sedigheh Barootkoob
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Correspondence to Sedigheh Barootkoob.

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Communicated by Shirin Hejazian.

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Lakzian, H., Barootkoob, S. Biprojectivity and Biflatness of Bi-amalgamated Banach Algebras. Bull. Iran. Math. Soc. 47, 63–74 (2021). https://doi.org/10.1007/s41980-020-00366-w

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  • DOI: https://doi.org/10.1007/s41980-020-00366-w

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