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Reiter’s properties for the actions of locally compact quantum groups on von Neumann algebras. (English) Zbl 1251.43002

Daws and Runde introduced the Reiter’s conditions \(P_1\) and \(P_2\) for a general locally compact quantum group. The authors present a slightly different approach to study these conditions for the action of a locally compact quantum group on a von Neumann algebra. In particular they show that the action has Reiter’s condition \(P_1\) if and only if it is left amenable and Reiter’s condition \(P_2\) if and only if the corresponding unitary corepresentation (which implements the action) has the weak containment property.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
22F05 General theory of group and pseudogroup actions
22D10 Unitary representations of locally compact groups
46L07 Operator spaces and completely bounded maps
46L65 Quantizations, deformations for selfadjoint operator algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
46L55 Noncommutative dynamical systems
11Y50 Computer solution of Diophantine equations
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