  We prove that a locally compact group has the approximation property (AP),
introduced by Haagerup-Kraus, if and only if a non-commutative Fej\'{e}r
theorem holds for the associated $C^*$- or von Neumann crossed products. As
applications, we answer three open problems in the literature. Specifically, we
show that any locally compact group with the AP is exact. This generalizes a
result by Haagerup-Kraus, and answers a problem raised by Li. We also answer a
question of B\'{e}dos-Conti on the Fej\'{e}r property of discrete
$C^*$-dynamical systems, as well as a question by Anoussis-Katavolos-Todorov
for all locally compact groups with the AP. In our approach, which relies on
operator space techniques, we develop a notion of Fubini crossed product for
locally compact groups, and a dynamical version of the AP for actions
associated with $C^*$- or $W^*$-dynamical systems.
