This paper continues the investigation initially begun for left -biflatness and left -biprojectivity. We show that left -biflatness and left -biprojectivity are closely related to the notions of left -amenability and -inner amenability. We characterize left -biprojectivity and left -biflatness of certain semigroup algebras and some algebras related to a locally compact group. We discuss non left -biflatness of some specified triangular Banach algebras
