  In this paper, we introduce a new notion of biprojectivity, called
$WAP$-biprojectivity for $F(\mathcal{A})$, the enveloping dual Banach algebra
associated to a Banach algebra $\mathcal{A}$. We find some relations between
Connes biprojectivity, Connes amenability and this new notion. We show that,
for a given dual Banach algebra $\mathcal{A}$, if $F(\mathcal{A})$ is Connes
amenable, then $\mathcal{A}$ is Connes amenable.
  For an infinite commutative compact group $G$, we show that the convolution
Banach algebra $F(L^2(G))$ is not $WAP$-biprojective. Finally, we provide some
examples of the enveloping dual Banach algebras and we study their
$WAP$-biprojectivity and Connes amenability.
