Let A be a Banach algebra and let be a non-zero multiplicative linear functional on A: We introduce a new notion of approximate left -biprojectivity for Banach algebra A. We show that for a SIN group G, the Segal algebra S(G) is approximate left 1-biprojective if and only if G is amenable, where 1 is the augmentation character on S(G). Also we show that the measure algebra M(G) is approximate left character biprojective if and only if G is discrete and amenable. For a Cli ord semigroup S, we show that `1(S) is approximate left character biprojective if and only if `1(S) is pseudo-amenable. We study the hereditary property of these notions. Finally we give some examples to show the di erences of these notions and the classical ones.
