The Haagerup property for groups and for tracial von Neumann algebras in terms of invariant and mixing states

The aim of the article is to provide characterizations of the Haage-rup property for locally compact, second countable groups in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital $C^{\ast }$-algebras one the one hand, and for pairs of tracial von Neumann algebras by mixing binormal states on the other hand.