: In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra A is studied, where ϕ is a weak∗ -continuous multiplicative linear functional on A. Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated.
