  We investigate the notion of Connes-amenability for dual Banach algebras, as
introduced by Runde, for bidual algebras and weighted semigroup algebras. We
provide some simplifications to the notion of a $\sigma WC$-virtual diagonal,
as introduced by Runde, especially in the case of the bidual of an Arens
regular Banach algebra. We apply these results to discrete, weighted, weakly
cancellative semigroup algebras, showing that these behave in the same way as
C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also
show that for each one of these cancellative semigroup algebras
$l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect
to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable,
which is in turn equivalent to $S$ being an amenable group. This latter point
was first shown by Gr{\"o}nb\ae k, but we provide a unified proof. Finally, we
consider the homological notion of injectivity, and show that here, weighted
semigroup algebras do not behave like C$^*$-algebras.
