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

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.