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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}rtheorem holds for the associated $C^*$- or von Neumann crossed products. Asapplications, we answer three open problems in the literature. Specifically, weshow that any locally compact group with the AP is exact. This generalizes aresult by Haagerup-Kraus, and answers a problem raised by Li. We also answer aquestion of B\'{e}dos-Conti on the Fej\'{e}r property of discrete$C^*$-dynamical systems, as well as a question by Anoussis-Katavolos-Todorovfor all locally compact groups with the AP. In our approach, which relies onoperator space techniques, we develop a notion of Fubini crossed product forlocally compact groups, and a dynamical version of the AP for actionsassociated with $C^*$- or $W^*$-dynamical systems.

A non-commutative Fej\'{e}r theorem for crossed products, the approximation property, and applications