CCAP for Universal Discrete Quantum Groups

By K. De Commer, Amaury Freslon, Makoto Yamashita. 2013

88 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15ed59293fe5ff078485aeb3155d7f68065ef76b

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Citations in the corpus

Representation Theory for Subfactors, $${\lambda}$$λ-Lattices and C*-Tensor Categories (2014, methodology): “the tensor category, like property (T) and the Haagerup property, are equivalent with their corresponding “central” counterparts for discrete quantum groups studied in [DFY13, A14]. Using the work of [DFY13] and the equivalence between the bimodule category of an A∞ subfactor and the representation ”
Annular representation theory for rigid $C^{*}$-tensor categories (2015, background): “They use results about the corresponding quantum groups SUq(2) obtained in [6].”
The Connes embedding property for quantum group von Neumann algebras (2014, background): “These connections occur at the lev l of quantum symmetries and asymptotic freeness results [3, 6, 10, 17, 18], and also at th e operator algebra level [9, 19, 21, 22, 23, 35].”
Drinfeld Center and Representation Theory for Monoidal Categories (2015, methodology): “setting it is almost immediate that for representation categories of compact quantum groups the completion of the fusion algebra coincides with the one obtained by embedding it in the Drinfeld double [DCFY14]. On the other hand, for general categories it is more diﬃcult to see in our approach that t”