In this paper, we continue our work [13] in the study of ????-biprojective Banach algebra A, with respect to a character ???? of A. For a locally compact group G, we show that a Segal algebra S(G) is ????-biprojective if and only if G is compact. We also introduce and study character biprojectivity of Banach algebras. We show that the measure algebra M(G) is character biprojective if and only if G is a finite group. For a commutative character biprojective Banach algebra A, we prove that the character space Δ(????) is discrete. Finally we show that some triangular Banach algebras are never ????-biprojective.
