  We shall develop two notions of pointwise amenability, namely pointwise
Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual
Banach algebras which take the $w^*$-topology into account. We shall study
these concepts for the Banach sequence algebras $\ell^1(\omega)$ and the
weighted semigroup algebras $ \ell^{1}(\mathbb{N}_{\wedge},\omega)$. For a
weight $\omega$ on a discrete semigroup $S$, we shall investigate pointwise
amenability/Connes amenability of $\ell^1(S,\omega)$ in terms of diagonals.
