Approximation properties for free orthogonal and free unitary quantum groups

In this paper, we study the structure of the reduced C*-algebras and von Neumann algebras associated to the free orthogonal and free unitary quantum groups. We show that the reduced von Neumann algebras of these quantum groups always have the Haagerup approximation property. Combining this result with a Haagerup-type inequality due to Vergnioux, we also show that the reduced C*-algebras always have the metric approximation property.