  In this work we introduce and study a new notion of amenability for actions
of locally compact groups on $C^*$-algebras. Our definition extends the
definition of amenability for actions of discrete groups due to Claire
Anantharaman-Delaroche. We show that our definition has several
characterizations and permanence properties analogous to those known in the
discrete case. For example, for actions on commutative $C^*$-algebras, we show
that our notion of amenability is equivalent to measurewise amenability.
Combined with a recent result of Alex Bearden and Jason Crann, this also
settles a long standing open problem about the equivalence of topological
amenability and measurewise amenability for a second countable $G$-space $X$.
We use our new notion of amenability to study when the maximal and reduced
crossed products agree. One of our main results generalizes a theorem of
Matsumura: we show that for an action of an exact locally compact group $G$ on
a locally compact space $X$ the full and reduced crossed products
$C_0(X)\rtimes_\max G$ and $C_0(X)\rtimes_{\operatorname{red}} G$ coincide if
and only if the action of $G$ on $X$ is amenable. We also show that the
analogue of this theorem does not hold for actions on noncommutative
$C^*$-algebras. Finally, we study amenability as it relates to more detailed
structure in the case of $C^*$-algebras that fibre over an appropriate
$G$-space $X$, and the interaction of amenability with various regularity
properties such as nuclearity, exactness, and the (L)LP, and the equivariant
versions of injectivity and the WEP.
