t In this paper, we introduce the notions of ϕ-WAP-biprojectivity and ϕ-WAP-virtual diagonal for the enveloping dual Banach algebras F(A) and we study the relation between these notions. We then show that F(A) is ϕ-WAP-biprojective if A is ϕ-biprojective or ϕ-Johnson contractible. Examples are provided to demonstrate that ϕ-WAP-biprojectivity is distinct from ϕ-biprojectivity and from WAP-biprojectivity. Finally, we define the notion of ϕ-Connes biprojectivity of dual Banach algebras and find some relations between the new notions of ϕ-WAP-biprojectivity, ϕ-Connes biprojectivity and some concepts already known.
