Actions, Quotients and Lattices of Locally Compact Quantum Groups
Michael BrannanDepartment of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
Alexandru ChirvasituDepartment of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
Ami ViselterDepartment of Mathematics, University of Haifa, 31905 Haifa, Israel
We prove a number of property (T) permanence results for locally compact quantum groups under exact sequences and the presence of invariant states, analogous to their classical versions. Along the way we characterize the existence of invariant weights on quantum homogeneous spaces of quotient type, and relate invariant states for LCQG actions on von Neumann algebras to invariant vectors in canonical unitary implementations, providing an application to amenability. Finally, we introduce a notion of lattice in a locally compact quantum group, noting examples provided by Drinfeld doubles of compact quantum groups. We show that property (T) lifts from a lattice to the ambient LCQG, just as it does classically, thus obtaining new examples of non-classical, non-compact, non-discrete LCQGs with property (T).
Cite this article
Michael Brannan, Alexandru Chirvasitu, Ami Viselter, Actions, Quotients and Lattices of Locally Compact Quantum Groups. Doc. Math. 25 (2020), pp. 2553–2582DOI 10.4171/DM/807