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Abasalt Bodaghi, Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran, e-mail: abasalt.bodaghi@gmail.com; Behrouz Shojaee, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran, e-mail: shoujaei@kiau.ac.ir
Abstract: In the current work, a new notion of $n$-weak amenability of Banach algebras using homomorphisms, namely $(\varphi,\psi)$-$n$-weak amenability is introduced. Among many other things, some relations between $(\varphi,\psi)$-$n$-weak amenability of a Banach algebra $\mathcal{A}$ and $M_m(\mathcal{A})$, the Banach algebra of $m\times m$ matrices with entries from $\mathcal{A}$, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra $L^1(G)$ is ($\varphi,\psi$)-$n$-weakly amenable for any bounded homomorphisms $\varphi$ and $\psi$ on $L^1(G)$.
Keywords: Banach algebra; continuous homomorphism; $(\varphi,\psi)$-derivation; $n$-weak amenability
Classification (MSC 2010): 46H25, 43A20, 22D15
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