Left $\varphi$-biprojectivity of some Banach algebras

In this paper, we introduce a homological notion of left $\varphi$-biprojectivity for Banach algebras, where $\varphi$ is a non-zero multiplicative linear functional. We show that for a locally compact group $G$, the Segal algebra $S(G)$ is left $\varphi$-contractible if and only if $G$ is compact. Also, we prove that the Fourier algebra $A(G)$ is left $\varphi$-biprojective if and only if $G$ is discrete. Finally, we study left $\varphi$-biprojectivity of certain semigroup algebras. We give some examples which show the differences between our new notion and the classical ones.