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
