  The Haagerup approximation property (HAP) is defined for finite von Neumann
algebras in such a way that the group von Neumann algebra of a discrete group
has the HAP if and only if the group itself has the Haagerup property. The HAP
has been studied extensively for finite von Neumann algebras and it is recently
generalized for arbitrary von Neumann algebras by Caspers-Skalski and
Okayasu-Tomatsu. One of the motivations behind the generalization is the fact
that quantum group von Neumann algebras are often infinite even though the
Haagerup property has been defined successfully for locally compact quantum
groups by Daws-Fima-Skalski-White. In this paper, we partly fill this gap by
proving that the von Neumann algebra of a locally compact quantum group with
the Haagerup property has the HAP. This is new even for genuine locally compact
groups.
