This property was originally formulated by Uffe Haagerup for locally compact groups (see Haagerup property). It was generalized to the context of locally compact quantum groups [DFSW13]. There is also a definition of a Haagerup property for von Neumann algebras [Jol02]. Here, we focus on the case of discrete quantum groups (note that every compact quantum groups has the Haagerup property).
Let be a discrete quantum group. The following equivalent statements provide a definition of having a Haagerup property [DFSW13]
A von Neumann algebra equipped with a faithful normal trace is said to have the Haagerup property if there exists a net of -preserving unital completely positive maps on such that each extends to a compact operator on and in the point-ultraweak topology.
An unimodular discrete quantum group has the Haagerup property if and only if has the Haagerup property.
If has the Haagerup property, then
Discrete quantum group has the Haagerup property if
Discrete quantum group has the Haagerup property and property (T) if and only if is finite [DFSW13].