A Non-commutative Fejér Theorem for Crossed Products, the Approximation Property, and Applications

Abstract We prove that a locally compact group has the approximation property (AP), introduced by Haagerup–Kraus [ 21], if and only if a non-commutative Fejér theorem holds for its 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 [ 21] and answers a problem raised by Li [ 27]. We also answer a question of Bédos–Conti [ 4] on the Fejér property of discrete $C^*$-dynamical systems, as well as a question by Anoussis–Katavolos–Todorov [ 3] for all locally compact groups with the AP. In our approach, we develop a notion of Fubini crossed product for locally compact groups and a dynamical version of the slice map property.