  In this paper, we countinue our work in \cite{11}.
  We show that $L^{1}(G,w)$ is $\phi_{0}$-biprojective if and only if $G$ is
compact, where $\phi_{0}$ is the augmentation character. We introduce the
notions of character Johnson amenability and character Johnson contractibility
for Banach algebras. We show that $\ell^{1}(S)$ is pseudo-amenable if and only
if $\ell^{1}(S)$ is character Johnson-amenable, provided that $S$ is a
uniformly locally finite band semigroup.
  We give some conditions whether $\phi$-biprojectivity ($\phi$-biflatness) of
$\ell^{1}(S)$ implies the finiteness (amenability) of $S$, respectively.
