The Haagerup property and actions on von Neumann algebras

Abstract: 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 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.