  Let $A$ be a dual Banach algebra with predual $A_\ast$ and consider the
following assertions: (A) $A$ is Connes-amenable; (B) $A$ has a normal, virtual
diagonal; (C) $A_\ast$ is an injective $A$-bimodule. For general $A$, all that
is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B),
and (C) are equivalent. We show that (C) always implies (B) whereas the
converse is false for $A = M(G)$ where $G$ is an infinite, locally compact
group. Furthermore, we present partial solutions towards a characterization of
(A) and (B) for $B(G)$ in terms of $G$: For amenable, discrete $G$ as well as
for certain compact $G$, they are equivalent to $G$ having an abelian subgroup
of finite index. The question of whether or not (A) and (B) are always
equivalent remains open. However, we introduce a modified definition of a
normal, virtual diagonal and, using this modified definition, characterize the
Connes-amenable, dual Banach algebras through the existence of an appropriate
notion of virtual diagonal.
