  In Chapter 2 of "Groups with the Haagerup Property", Jolissaint gives on the
one hand a characterization of the Haagerup property in terms of strongly
mixing actions on standard probability spaces; on the other hand he gives a
noncommutative analogue of this result in terms of actions on factors. In the
recent paper "A new characterization of the Haagerup property by actions on
infinite measure spaces", the authors give a characterization of the Haagerup
property but this time dealing with $C_0$-actions on infinite measure spaces.
Following the spirit of this section, we give a noncommutative analogue in
terms of $C_0$-actions on von Neumann algebras. Next we discuss some natural
questions which remained open around $C_0$-dynamical systems. In particular we
give examples of $C_0$- dynamical systems for groups acting properly on trees.
Finally, we give a positive answer to the question of the ergodicity of such
systems for non-periodic groups.
