on left ϕ-connes biprojectivity of dual banach algebras

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 ∈ δw∗ (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 ∈ δw∗ (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.