Geodesic
$\varphi$-approximate biprojective and $\varphi$-approximate amenable Banach algebras
Eurasian mathematical journal, Tome 13 (2022) no. 3, pp. 33-40.

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Let $A$ be a Banach algebra and $\varphi$ be a continuous homomorphism on $A$. We define the notions of a $\varphi$-approximate biprojective and $\varphi$-approximate amenable Banach algebra $A$, and consider relations between them and some their properties. 
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Z. Ghorbani. $\varphi$-approximate biprojective and $\varphi$-approximate amenable Banach algebras. Eurasian mathematical journal, Tome 13 (2022) no. 3, pp. 33-40. http://geodesic.mathdoc.fr/item/EMJ_2022_13_3_a2/

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