  Amenable actions of locally compact groups on von Neumann algebras are
investigated by exploiting the natural module structure of the crossed product
over the Fourier algebra of the acting group. The resulting characterisation of
injectivity for crossed products generalises a result of Anantharaman-Delaroche
on discrete groups. Amenable actions of locally compact groups on
$C^*$-algebras are investigated in the same way, and amenability of the action
is related to nuclearity of the corresponding crossed product. A survey is
given to show that this notion of amenable action for $C^*$-algebras satisfies
a number of expected properties. A notion of inner amenability for actions of
locally compact groups is introduced, and a number of applications are given in
the form of averaging arguments, relating approximation properties of crossed
product von Neumann algebras to properties of the components of the underlying
$w^*$-dynamical system. We use these results to answer a recent question of
Buss-Echterhoff-Willett.
