banach homology and its recent notions with respect to the maximal ideal space

In this paper, for a banach algebra a and a non-zero multiplicative linear functional ϕ : a → c, we define and study the notion of left ϕ-biflatness and approximate left ϕ-biprojectivity. we show that the segal algebra s(g) is left ϕ-biflat if and only if the locally compact group g is amenable.we study approximate left ϕ-biprojectivity of some algebras related to locally compact groups and some matrix algebras.