Abstract
At the present paper, we study the notions of \(\varphi \)-biprojectivity, \(\varphi \)-Johnson contractibility, and \(\varphi \)-contractibility of Banach algebras, where \(\varphi \) is a nonzero character. We introduce the condition (Q) which is weaker than \(\varphi \)-biprojectivity. For classes of Banach algebras with a left and right approximate identity, we obtain some relations between these notions. Moreover, we apply these results for the hypergroup algebra \(L^{1}(K)\) and some Segal algebras with respect to the \(L^{1}(K)\). As a main result, for a hypergroup K, we prove that the hypergroup algebra \(L^{1}(K)\) is \(\varphi \)-biprojective (left \(\varphi \)-contractible) if and only if K is compact.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azimifard, A.: On the amenability of compact and discrete hypergroup algebras (2009). ArXiv:0908.1590v2 [Math.FA]
Bloom, W.R., Heyer, H.: Harmonic Analysis of probability measures on hypergroups. Walter de Gruyter, Berlin (1995)
Dales, H.G.: Banach Algebras and Automatic Continuity. Oxford University Press, New York (2000)
Dunkl, C.F.: The measure algebra of a locally compact hypergroup. Trans. Am. Math. Soc. 179, 331–348 (1973)
Essmaili, M., Rostami, M., Amini, M.: A characterization of biflatness of Segal algebras based on a character. Glasnik Math. 51, 45–58 (2016)
Helemskii, A.Ya.: The Homology of Banach and Topological Algebras. Kluwer Academic Publishers Group, Dordrecht (1989)
Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Springer, New York (1970)
Hu, Z., Monfared, M.S., Traynor, T.: On character amenable Banach algebras. Stud. Math. 193, 53–78 (2009)
Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)
Kaniuth, E., Lau, A.T.M., Pym, J.: On \(\varphi \)-amenability of Banach algebras. Math. Proc. Camb. Philos. Soc. 144, 85–96 (2008)
Kaniuth, E., Lau, A.T., Pym, J.S.: On character amenability of Banach algebras. J. Math. Anal. Appl. 344, 942–955 (2008)
Lasser, R.: Various amenability properties of the \(l^1\)-algebra of polynomial hypergroups and applications. J. Comput. Appl. Math. 233, 786–792 (2009)
Nasr-Isfahani, R., Soltani Renani, S.: Character contractibility of Banach algebras and homological properties of Banach modules. Stud. Math. 202, 205–225 (2011)
Pourmahmood-Aghababa, H., Shi, L.Y., Wu, Y.J.: Generalized notions of character amenability. Acta Math. Sin. Engl. Ser. 29, 1329–1350 (2013)
Runde, V.: Lectures on amenability, Lecture Note in Mathematics 1774. Springer, Berlin (2002)
Sahami, A., Pourabbas, A.: On \(\phi \)-biflat and \(\phi \)-biprojective Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 20, 789–801 (2013)
Sahami, A., Pourabbas, A.: On character biprojectivity of Banach algebras. UPB Sci. Bull. Ser. A Appl. Math. Phys. 78, 163–174 (2016)
Skantharajah, M.: Amenable hypergroups. Illinois J. Math. 36, 15–46 (1992)
Vrem, R.C.: Harmonic analysis on compact hypergroups. Pac. J. Math. 85, 239–251 (1979)
Acknowledgements
The first author was partially supported by a grant from IPM (no. 94470069).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hamid Reza Ebrahimi Vishki.
Rights and permissions
About this article
Cite this article
Essmaili, M., Medghalchi, A.R. & Ramezani, R. \(\varphi \)-Biprojectivity of Banach Algebras with Applications to Hypergroup Algebras. Bull. Iran. Math. Soc. 45, 359–376 (2019). https://doi.org/10.1007/s41980-018-0137-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-018-0137-3
Keywords
- Hypergroup algebras
- \(\varphi \)-Biprojectivity
- \(\varphi \)-Contractibility
- \(\varphi \)-Johnson contractibility
- Abstract Segal algebras