CCAP for universal discrete quantum groups

Abstract

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SUq(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SUq(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.

29 pages; minor modifications, and a new appendix by S. Vaes

References in corpus (6)

• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
• Orientation of quantum Cayley trees and applications
• Amalgamated free product type III factors with at most one Cartan subalgebra
• Haagerup property for quantum reflection groups
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• Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories
• Drinfeld center and representation theory for monoidal categories
• The Connes embedding property for quantum group von Neumann algebras
• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups
• Lacunary Fourier series for compact quantum groups
• Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
• Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups
• C*-tensor categories and subfactors for totally disconnected groups
• A few remarks on the tube algebra of a monoidal category
• The Fourier algebra of a rigid $C^{\ast }$-tensor category
• Haagerup approximation property and positive cones associated with a von Neumann algebra
• Categorically Morita equivalent compact quantum groups
• Complete metric approximation property for $q$-Araki-Woods algebras
• Character density in central subalgebras of compact quantum groups
• $L_p$-Representations of Discrete Quantum Groups
• Quantum groups and generalized circular elements
• Haagerup approximation property for arbitrary von Neumann algebras
• Property RD and hypercontractivity for orthogonal free quantum groups
• Pointwise convergence of noncommutative Fourier series
• Howe-Moore type theorems for quantum groups and rigid C*-tensor categories
• The approximation property for locally compact quantum groups
• On bi-exactness of discrete quantum groups
• Riesz transforms on compact quantum groups and strong solidity
• Strong 1-Boundedness of Unimodular Free Orthogonal Quantum Groups
• Actions of measured quantum groupoids on a finite basis
• Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras