Haagerup approximation property for quantum reflection groups
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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.References
- T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), no. 1, 3–37. MR 2779129, DOI 10.4153/CJM-2010-060-6
- Teodor Banica, Théorie des représentations du groupe quantique compact libre $\textrm {O}(n)$, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 3, 241–244 (French, with English and French summaries). MR 1378260
- Teodor Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), no. 4, 763–780. MR 1709109, DOI 10.1007/s002080050315
- Teodor Banica and Roland Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), no. 3, 327–359. MR 2511633, DOI 10.4171/JNCG/39
- Julien Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), no. 4, 343–362. MR 2096666, DOI 10.1023/B:ALGE.0000042148.97035.ca
- Michael Brannan, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math. 672 (2012), 223–251. MR 2995437, DOI 10.1515/crelle.2011.166
- Michael Brannan, Reduced operator algebras of trace-preserving quantum automorphism groups, preprint arXiv:1202.5020, 2012.
- Indira Chatterji, Cornelia Druţu, and Frédéric Haglund, Kazhdan and Haagerup properties from the median viewpoint, Adv. Math. 225 (2010), no. 2, 882–921. MR 2671183, DOI 10.1016/j.aim.2010.03.012
- Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR 1852148, DOI 10.1007/978-3-0348-8237-8
- Yves Cornulier, Yves Stalder, and Alain Valette, Proper actions of wreath products and generalizations, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3159–3184. MR 2888241, DOI 10.1090/S0002-9947-2012-05475-4
- Matthew Daws, Pierre Fima, Adam Skalski, and Stuart White, The Haagerup property for locally compact quantum groups, preprint, arXiv:1303.3261 (2013).
- Pierre Fima, Kazhdan’s property $T$ for discrete quantum groups, Internat. J. Math. 21 (2010), no. 1, 47–65. MR 2642986, DOI 10.1142/S0129167X1000591X
- Amaury Freslon, Examples of weakly amenable discrete quantum groups, J. Funct. Anal. 265 (2013), no. 9, 2164–2187. MR 3084500, DOI 10.1016/j.jfa.2013.05.037
- Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082
- Nigel Higson and Gennadi Kasparov, $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74. MR 1821144, DOI 10.1007/s002220000118
- Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934 (English, with English and French summaries). MR 1832993, DOI 10.1016/S0012-9593(00)01055-7
- Sorin Popa, On a class of type $\textrm {II}_1$ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), no. 3, 809–899. MR 2215135, DOI 10.4007/annals.2006.163.809
- Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
- Stefaan Vaes and Roland Vergnioux, The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J. 140 (2007), no. 1, 35–84. MR 2355067, DOI 10.1215/S0012-7094-07-14012-2
- Shuzhou Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671–692. MR 1316765, DOI 10.1007/BF02101540
- Shuzhou Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195–211. MR 1637425, DOI 10.1007/s002200050385
- Moritz Weber, On the classification of easy quantum groups, Adv. Math. 245 (2013), 500–533. MR 3084436, DOI 10.1016/j.aim.2013.06.019
- S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
- S. L. Woronowicz, Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted $\textrm {SU}(N)$ groups, Invent. Math. 93 (1988), no. 1, 35–76. MR 943923, DOI 10.1007/BF01393687
- S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 845–884. MR 1616348
Additional Information
- 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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3314111