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Haagerup approximation property for quantum reflection groups


Author: François Lemeux
Journal: Proc. Amer. Math. Soc. 143 (2015), 2017-2031
MSC (2010): Primary 46L54, 16T20; Secondary 46L65, 20G42
DOI: https://doi.org/10.1090/S0002-9939-2015-12402-1
Published electronically: January 21, 2015
MathSciNet review: 3314111
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Abstract: In this paper we prove that the duals of the quantum reflection groups $ H_N^{s+}$ have the Haagerup property for all $ N\ge 4$ and $ s\in [1,\infty )$. We use the canonical arrow $ \pi : C(H_N^{s+})\to C(S_N^+)$ onto the quantum permutation groups, and we describe how the characters of $ C(H_{N}^{s+})$ behave with respect to this morphism $ \pi $ thanks to the description of the fusion rules binding irreducible corepresentations of $ C(H_N^{s+})$ as in Banica and Vergnioux, 2009. This allows us to construct states on the central $ C^*$-algebra $ C(H_N^{s+})_0$ generated by the characters of $ C(H_{N}^{s+})$ and to use a fundamental theorem proved by M. Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group $ \mathbb{G}$ of Kac type.


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François Lemeux
Affiliation: Laboratoire de mathématiques de Besançon, UFR Sciences et Techniques, Université de Franche-Comté, 16 route de Gray, 25000 Besançon, France
Email: francois.lemeux@univ-fcomte.fr

DOI: https://doi.org/10.1090/S0002-9939-2015-12402-1
Received by editor(s): March 8, 2013
Received by editor(s) in revised form: September 5, 2013
Published electronically: January 21, 2015
Communicated by: Marius Junge
Article copyright: © Copyright 2015 American Mathematical Society