Document Type : Research Paper


1 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran


Let $\mathcal{A}$ be a Banach algebra and $X$ be a Banach $\mathcal{A}-$bimodule. We study the notion of approximate $n-$ideal amenability for module extension Banach algebras $\mathcal{A}\oplus X$. First, we describe the structure of ideals of this kind of algebras and we present the necessary and sufficient conditions for a module extension Banach algebra to be approximately n-ideally amenable.


[1] W. G. Bade, P. G. Curtis, H. G. Dales, Amenability and weak amenability for Beurling and Lipschits algebra, Proc. London Math. Soc. 55 (1987), 359-377.
[2] A. Bodaghi, F. Anousheh, S. Etemad, Generalized notion of character amenability, J. Linear. Topological. Algebra. 2 (4) (2013), 185-194.
[3] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc, Monographs, 24, Clarenden Press, Oxford, 2000.
[4] H. G. Dales, F. Ghahramani, N. Gronbaek, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), 19-54.
[5] F. Ghahramani, R. J. Loy, Generalized notions of amenability, J. Func. Anal. 208 (2004), 229-260.
[6] R. Gholami, H. Rahimi, On character amenability and approximate character amenability of Banach algebras, Int. J. Math. Anal. 10 (5) (2019), 54-64.
[7] M. E. Gordji, A. Jabbari, Approximate ideal amenability of Banach algebras, U. P. B. Sci. Bull. 74 (2012), 57-64.
[8] M. E. Gordji, T. Yazdanpanah, Derivations into duals of ideals of Banach algebras, Proc. Indian Acad. Sci. 114 (4) (2004), 399-408.
[9] S. A. R. Hosseinioun, A. Valadkhani, The relations between φ-amenability and some special ideals, Afr. J. Pure Appl. Math. 2 (2017), 1-6.
[10] B. E. Johnson, Cohomology in Banach algebras, Amer. Math. Soc., 1972.
[11] M. S. Monfared, Character amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008), 697-706.
[12] H. Rahimi, M. Amini, Group congruences and amenability of semi-group algebras, Acta Math. Hungarica., in press.
[13] H. Rahimi, E. Tahmasebi, Amenability and contractibility modulo an ideal of Banach algebras, Abstr. Appl. Anal. 2014, 2014:514761.
[14] H. Rahimi, E. Tahmasebi, A note on amenability modulo an ideal of unitial Banach algebras, J. Math. Ext. 9 (1) (2015), 13-21.
[15] M. Shadab, G. H. Esslamzadeh, Approximate n−ideal amenability of Banach algebras, Int. Math. Forum. 13 (5) (2010), 775-779.
[16] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Soc. 354 (2002) 4131-4151.