In this paper, we introduce a new notion of biprojectivity, called WAP-biprojectivity for F(A) , the enveloping dual Banach algebra associated to a Banach algebra A . We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dual Banach algebra A , if F(A) is Connes amenable, then A is Connes amenable. For an infinite commutative compact group G, we show that the convolution Banach algebra F(L2(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.
