In this paper, we investigate the homological notion of left /-biprojectivity for certain Banach algebras, where / is a nonzero multiplicative linear functional. Our initial result states that this notion is equivalent to left /-contractibility, provided that A has a left approximate identity. As an application, we study left /-biprojectivity of Banach algebras related to locally compact groups. For instance, we show that for a locally compact group G, the Segal algebra SðGÞ is left /- biprojective if and only if G is compact and the Fourier algebra AðGÞ is left /-biprojective if and only if G is discrete. Finally, we give some examples which show the differences between our new notion and the classical ones.
