Let A be a Banach algebra. We introduce the notions of approximate left φ-biprojective and approximate left character biprojective Banach algebras, where φ is a non-zero multiplicative linear functional on A. We show that for a SIN group G, the Segal algebra S(G) is approximate left φ1-biprojective if and only if G is amenable, where φ1 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 Clifford semigroup S, we show that l1(S) is approximate left character biprojective if and only if l1(S) is pseudo-amenable. We study the hereditary property of these notions. Finally we give some examples to show the differences of these notions and the classical ones.
