On approximately left phi-biprojective Banach algebras

In this paper, for a Banach algebra A, we introduced the new notions of approximately left $\varphi$-biprojective and approximately left character biprojective, where $\varphi$ is a non-zero multiplicative linear functional on A. We show that for SIN group G, Segal algebra S(G) is approximately left ${\varphi }_1$- biprojective if and only if G is amenable, where ${\varphi }_1$ is the augmentation character on S(G). Also we showed that the measure algebra M(G) is approximately left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that 1(S) is approximately left character biprojective if and only if 1(S) is pseudo-amenable. We study the hereditary property of these new notions. Finally we give some examples among semigroup algebras and Triangular Banach algebras to show the differences of these notions and the classical ones.