Geodesic
Problems concerning $n$-weak amenability of a Banach algebra
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 863-876.

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In this paper we extend the notion of $n$-weak amenability of a Banach algebra $\mathcal A$ when $n\in \mathbb{N}$. Technical calculations show that when $\mathcal A$ is Arens regular or an ideal in $\mathcal A^{**}$, then $\mathcal A^*$ is an $\mathcal A^{(2n)}$-module and this idea leads to a number of interesting results on Banach algebras. We then extend the concept of $n$-weak amenability to $n \in \mathbb{Z}$.
Classification : 46H20, 46H40
Keywords: Banach algebra; weakly amenable; Arens regular; $n$-weakly amenable 
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     author = {Medghalchi, Alireza and Yazdanpanah, Taher},
     title = {Problems concerning $n$-weak amenability of a {Banach} algebra},
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     zbl = {1081.46031},
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Medghalchi, Alireza; Yazdanpanah, Taher. Problems concerning $n$-weak amenability of a Banach algebra. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 863-876. http://geodesic.mathdoc.fr/item/CMJ_2005__55_4_a3/