On approximately biprojective and approximately biflat Banach algebras

In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ?-amenability and ? -contractibility, where ? is a character on A. Among other things, we show that ?-Lau product algebra L1(G) ?? A(G) is approximately biprojective if and only if G is finite, where L1(G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.