Weak mixing for locally compact quantum groups

TL;DR

This paper extends the concept of weak mixing to locally compact quantum groups, providing new characterizations and linking mixing properties with von Neumann algebra inclusions, advancing quantum ergodic theory.

Contribution

It introduces a generalized notion of weak mixing for quantum groups and connects it with existing theorems and algebraic properties, broadening the scope of quantum ergodic theory.

Findings

Extended weak mixing to locally compact quantum groups.

Provided characterizations of weak mixing in this setting.

Linked mixing properties with von Neumann algebra inclusions.

Abstract

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.