In this paper, we study approximate biprojectivity and approximate biflatness of a Banach algebra A and find some relations between theses concepts with φ-amenability and φ-contractibility, where φ is a character on A. Among other things, we show that θ-Lau product algebra L 1 (G) ×θ A(G) is approximately biprojective if and only if G is finite, where L 1 (G) and A(G) are the group algebra and the Fourier algebra of the locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.
