We introduce the notion of left (right) ϕ-Connes biprojective for a dual Banach algebra A, where ϕ is a non-zero wk∗-continuous multiplicative linear functional on A. We discuss the relationship of left ϕ-Connes biprojectivity with ϕ-Connes amenability and Connes biprojectivity. For a unital weakly cancellative semigroup S, we show that ℓ1(S) is left ϕS-Connes biprojective if and only if S is a finite group, where ϕS ∈ ∆w∗ (ℓ1(S)). We prove that for a non-empty totally ordered set I with the smallest element, the upper triangular I ×I-matrix algebra UP(I,A) is right ψϕ-Connes biprojective if and only if A is right ϕ-Connes biprojective and I is a singleton, provided that A has a right identity and ϕ ∈ ∆w∗ (A). Also for a finite set I, if Z(A) ∩ (A − ker ϕ) = ∅, then the dual Banach algebra UP(I,A) under this new notion forced to have a singleton index.
