  We generalize the notion of weakly mixing unitary representations to locally
compact quantum groups, introducing suitable extensions of all standard
characterizations of weak mixing to this setting. These results are used to
complement the noncommutative Jacobs-de Leeuw-Glicksberg splitting theorem of
Runde and the author ["Ergodic theory for quantum semigroups", J. Lond. Math.
Soc. (2) 89 (2014) 941-959]. Furthermore, a relation between mixing and weak
mixing of state-preserving actions of discrete quantum groups and the
properties of certain inclusions of von Neumann algebras, which is known for
discrete groups, is demonstrated.
