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CCAP for Universal Discrete Quantum Groups

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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 SU q (2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SU q (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.

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Authors and Affiliations

  1. Département de mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise, 95000, Cergy-Pontoise, France

    Kenny De Commer

  2. Sorbonne Paris Cité, UMR 7586, Université Paris Diderot, 8 place FM/13, 75013, Paris, France

    Amaury Freslon

  3. Institut for Matematiske Fag, Københavns Universitet, Universitetsparken 5, 2100, Copenhagen, Denmark

    Makoto Yamashita

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  1. Kenny De Commer
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  2. Amaury Freslon
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  3. Makoto Yamashita
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Correspondence to Makoto Yamashita.

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Communicated by Y. Kawahigashi

Makoto Yamashita: on leave from Ochanomizu University.

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De Commer, K., Freslon, A. & Yamashita, M. CCAP for Universal Discrete Quantum Groups. Commun. Math. Phys. 331, 677–701 (2014). https://doi.org/10.1007/s00220-014-2052-7

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