In this paper, we countinue our work in [16]. We show that L1 (G, w) is ϕ0-biprojective if and only if G is compact, where ϕ0 is the augmentation character. We introduce the notions of character Johnson amenability and character Johnson contractibility for Banach algebras. We show that ℓ 1 (S) is pseudo-amenable if and only if ℓ 1 (S) is character Johnson-amenable, provided that S is a uniformly locally finite band semigroup. We give some conditions whether ϕ-biprojectivity (ϕ-biflatness) of ℓ 1 (S) implies the finiteness (amenability) of S, respectively
