On I-biflat and I-biprojective Banach algebras

Document Type : Research Paper


1 Department of Mathematics Faculty of Basic Science, Ilam University, P.O. Box 69315-516 Ilam, Iran.

2 Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914 Tehran, Iran.

3 Department of Basic Sciences, Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran, Post Code:47148-71167.



In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$  is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra  $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.


[1] H.G. Dales and R.J. Loy, Approximate amenability of semigroup algebras and Segal algebras, Diss. Math., 474 (2010), 1-58.
[2] J. Duncan and A.L.T. Paterson, Amenability for discrete convolution semigroup algebras, Math. Scand., 66 (1990), 141-146.
[3] F. Ghahramani and A.T. Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal., 132 (1995), 170-191.
[4] F. Ghahramani, R.J. Loy and G.A. Willis, Amenability and weak amenability of second conjugate Banach algebras, Proc. Am. Math. Soc., 124 (1996), 1489-1497.
[5] A.Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Univ.Press, Oxford, 1993.
[6] A.Ya. Helemskii, The Homology of Banach and Topological Algebras, Kluwer, Academic Press, Dordrecht, 1989.
[7] B.E. Johnson, Cohomology in Banach Algebras, Memoirs of the American Mathematical Society, 1972.
[8] A.R. Medghalchi and M.H. Sattari, Biflatness and biprojectivity of triangular Banach algebras, Bull. Iran. Math. Soc., 34 (2008), 115-120.
[9] R. Nasr Isfahani and S. Soltani Renani, Character contractibility of Banach algebras and homological properties of Banach modules, Stud. Math., 202(3) (2011), 205-225.
[10] V. Runde, Lectures on Amenability, Springer, New York, 2002.
[11] A. Sahami and A. Pourabbas, On $phi$-biflat and $phi$-biprojective  Banach algebras, Bull. Belg. Math. Soc. Simon Stevin, 20(5) (2013), 789-801.