Permanence of approximation properties for discrete quantum groups
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1437-1467.

We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.

Nous prouvons plusieurs résultats concernant la permanence de la moyennabilité faible et de la propriété de Haagerup pour les groupes quantiques discrets. En particulier, nous améliorons des résultats connus sur les produits libres en autorisant l’amalgamation sur un sous-groupe quantique fini. Nous définissons également une notion de moyennabilité relative pour les groupes quantiques discrets et nous la relions à l’équivalence moyennable d’algèbres de von Neumann, ce qui donne de nouvelles propriétés de permanence.

DOI: 10.5802/aif.2963
Classification: 20G42, 46L65
Keywords: Quantum groups, approximation properties, relative amenability
Mot clés : Groupes quantiques, propriétés d’approximation, moyennabilité relative
Freslon, Amaury 1

1 Univ. Paris Diderot, Sorbonne Paris Cité IMJ-PRG, UMR 7586 CNRS Sorbonne Universités UPMC Univ. Paris 06 F-75013, Paris (France)
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Freslon, Amaury. Permanence of approximation properties for discrete quantum groups. Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1437-1467. doi : 10.5802/aif.2963. http://www.numdam.org/articles/10.5802/aif.2963/

[1] Anantharaman-Delaroche, C. Action moyennable d’un groupe localement compact sur une algèbre de von Neumann, Math. Scand, Volume 45 (1979), pp. 289-304 | MR | Zbl

[2] Anantharaman-Delaroche, C. Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math., Volume 171 (1995) no. 2, pp. 309-341 | DOI | MR | Zbl

[3] Baaj, S.; Skandalis, G. Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres, Ann. Sci. École Norm. Sup., Volume 26 (1993) no. 4, pp. 425-488 | Numdam | MR | Zbl

[4] Banica, T. Le groupe quantique compact libre U(n), Comm. Math. Phys., Volume 190 (1997) no. 1, pp. 143-172 | DOI | MR | Zbl

[5] Banica, T. Symmetries of a generic coaction, Math. Ann., Volume 314 (1999) no. 4, pp. 763-780 | DOI | MR | Zbl

[6] Bannon, J.P.; Fang, J. Some remarks on Haagerup’s approximation property, J. Operator Theory, Volume 65 (2011) no. 2, pp. 403-417 | MR | Zbl

[7] Bédos, E.; Conti, R.; Tuset, L. On amenability and co-amenability of algebraic quantum groups and their corepresentations, Canad. J. Math., Volume 57 (2005) no. 1, pp. 17-60 | DOI | MR | Zbl

[8] Boca, F. On the method of constructing irreducible finite index subfactors of Popa, Pacific J. Math., Volume 161 (1993) no. 2, pp. 201-231 | DOI | MR | Zbl

[9] Bożejko, M.; Picardello, M.A. Weakly amenable groups and amalgamated products, Proc. Amer. Math. Soc., Volume 117 (1993) no. 4, pp. 1039-1046 | DOI | MR | Zbl

[10] Brannan, M. Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math., Volume 672 (2012), pp. 223-251 | MR | Zbl

[11] Brown, Nathanial P.; Ozawa, Narutaka C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, Providence, RI, 2008, pp. xvi+509 | DOI | MR | Zbl

[12] Connes, A.; Jones, V.F.R. Property T for von Neumann algebras, Bull. London Math. Soc., Volume 17 (1985) no. 1, pp. 57-62 | DOI | MR | Zbl

[13] Cowling, M.; Haagerup, U. Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math., Volume 96 (1989) no. 3, pp. 507-549 | DOI | MR | Zbl

[14] Daws, M.; Fima, P.; Skalski, A.; White, S. The Haagerup property for locally compact quantum groups (2014) (http://arxiv.org/abs/1303.3261, to appear in J. Reine Angew. Math.)

[15] Daws, Matthew Completely positive multipliers of quantum groups, Internat. J. Math., Volume 23 (2012) no. 12, pp. 1250132, 23 | DOI | MR | Zbl

[16] De Commer, K.; Freslon, A.; Yamashita, M. CCAP for universal discrete quantum groups, Comm. Math. Phys., Volume 331 (2014) no. 2, pp. 677-701 | DOI | MR

[17] Eymard, P. Moyennes invariantes et représentations unitaires, Lecture notes in mathematics, 300, Springer, 1972 | MR | Zbl

[18] Fima, P. Kazhdan’s property T for discrete quantum groups, Internat. J. Math., Volume 21 (2010) no. 1, pp. 47-65 | DOI | MR | Zbl

[19] Fima, P. K-amenability of HNN extensions of amenable discrete quantum groups, J. Funct. Anal., Volume 265 (2013) no. 4, pp. 507-519 | DOI | MR

[20] Fima, P.; Freslon, A. Graphs of quantum groups and K-amenability, Adv. Math., Volume 260 (2014), pp. 233-280 | DOI | MR | Zbl

[21] Freslon, A. A note on weak amenability for free products of discrete quantum groups, C. R. Acad. Sci. Paris Sér. I Math., Volume 350 (2012), pp. 403-406 | DOI | MR | Zbl

[22] Freslon, A. Examples of weakly amenable discrete quantum groups, J. Funct. Anal., Volume 265 (2013) no. 9, pp. 2164-2187 | DOI | MR

[23] Freslon, A. Propriétés d’approximation pour les groupes quantiques discrets, Université Paris VII (France) (2013) (Ph. D. Thesis)

[24] Freslon, A. Fusion (semi)rings arising from quantum groups, J. Algebra, Volume 417 (2014), pp. 161-197 | DOI | MR

[25] Joita, M.; Petrescu, S. Amenable actions of Katz algebras on von Neumann algebras, Rev. Roumaine Math. Pures Appl., Volume 35 (1990) no. 2, pp. 151-160 | MR | Zbl

[26] Joita, M.; Petrescu, S. Property (T) for Kac algebras, Rev. Roumaine Math. Pures Appl., Volume 37 (1992) no. 2, pp. 163-178 | MR | Zbl

[27] Kraus, J.; Ruan, Z-J. Approximation properties for Kac algebras, Indiana Univ. Math. J., Volume 48 (1999) no. 2, pp. 469-535 | DOI | MR | Zbl

[28] Lemeux, F. Fusion rules for some free wreath product quantum groups and applications, J. Funct. Anal., Volume 267 (2014) no. 7, pp. 2507-2550 | DOI | MR

[29] Monod, N.; Popa, S. On co-amenability for groups and von Neumann algebras, C. R. Math. Acad. Sci. Soc. R. Can., Volume 25 (2003) no. 3, pp. 82-87 | MR | Zbl

[30] Pestov, V. On some questions of Eymard and Bekka concerning amenability of homogeneous spaces and induced representations, C. R. Math. Acad. Sci. Soc. R. Can., Volume 25 (2003) no. 3, pp. 76-81 | MR | Zbl

[31] Ricard, E.; Xu, Q. Khintchine type inequalities for reduced free products and applications, J. Reine Angew. Math., Volume 599 (2006), pp. 27-59 | MR | Zbl

[32] Rieffel, M.A. Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra, Volume 5 (1974) no. 1, pp. 51-96 | DOI | MR | Zbl

[33] Timmermann, T. An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond, EMS, 2008 | MR | Zbl

[34] Tomatsu, R. Amenable discrete quantum groups, J. Math. Soc. Japan, Volume 58 (2006) no. 4, pp. 949-964 | DOI | MR | Zbl

[35] Tomiyama, J. On tensor products of von Neumann algebras, Pacific J. Math., Volume 30 (1969) no. 1, pp. 263-270 | DOI | MR | Zbl

[36] Vaes, S. The unitary implementation of a locally compact quantum group action, J. Funct. Anal., Volume 180 (2001) no. 2, pp. 426-480 | DOI | MR | Zbl

[37] Vaes, S. A new approach to induction and imprimitivity results, J. Funct. Anal., Volume 229 (2005) no. 2, pp. 317-374 | DOI | MR | Zbl

[38] Vaes, S.; Vergnioux, R. The boundary of universal discrete quantum groups, exactness and factoriality, Duke Math. J., Volume 140 (2007) no. 1, pp. 35-84 | DOI | MR | Zbl

[39] Vergnioux, R. K-amenability for amalgamated free products of amenable discrete quantum groups, J. Funct. Anal., Volume 212 (2004) no. 1, pp. 206-221 | DOI | MR | Zbl

[40] Vergnioux, R.; Voigt, C. The K-theory of free quantum groups, Math. Ann., Volume 357 (2013) no. 1, pp. 355-400 | DOI | MR | Zbl

[41] Wang, S. Free products of compact quantum groups, Comm. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | DOI | MR | Zbl

[42] Wang, S. Tensor products and crossed-products of compact quantum groups, Proc. London Math. Soc., Volume 71 (1995) no. 3, pp. 695-720 | DOI | MR | Zbl

[43] Woronowicz, S.L. Compact quantum groups, Symétries quantiques (Les Houches, 1995) (1998), pp. 845-884 | MR | Zbl

[44] Wu, JinSong Co-amenability and Connes’s embedding problem, Sci. China Math., Volume 55 (2012) no. 5, pp. 977-984 | DOI | MR | Zbl

[45] Zimmer, R.J. Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Funct. Anal., Volume 27 (1978) no. 3, pp. 350-372 | DOI | MR | Zbl

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