  In this paper, for a Banach algebra A, we introduced the new notions of
approximately left $\phi$-biprojective and approximately left character
biprojective, where $\phi$ is a non-zero multiplicative linear functional on A.
We show that for SIN group G, Segal algebra S(G) is approximately left
$\phi_1$- biprojective if and only if G is amenable, where $\phi_1$ 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.
