an approximate notion in the homology of the enveloping dual banach algebras

In this paper, we introduce a notion of approximate $wap$-biprojectivity for the enveloping dual banach algebras. we also find some relations between approximate connes amenability, approximate biprojectivity, left $varphi$-contractibility and approximate $wap$-biprojectivity. moreover, we propose a criterion to show that the enveloping dual banach algebras associated to triangular banach algebras are not approximately $wap$-biprojective. finally, we present some examples of the enveloping dual banach algebras associated to $l^1(s)$ and $ell^1(mathbb{n})$ (equipped with a new multiplication) and also study their approximate $wap$-biprojectivity, where $s$ is a unital weakly cancellative semigroup.