Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 2003-2027

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand–Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies * –regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C * –norm.

Banica et Vergnioux ont montré que le groupe quantique discret dual d’un groupe de Lie compact et simplement connexe a croissance polynomiale de degré égal à la dimension réelle de la variété. On étend ce résultat aux groupes compactes quelconques et à leur dimension topologique, en la reliant à la dimension de Gelfand–Kirillov d’une algèbre. De plus, on prouve que la croissance polynomiale, pour un groupe quantique compact de Kac G, implique la * -régularité de l’algèbre de Fourier A(G), c’est-à-dire que tout idéal fermé de C(G) a intersection dense avec A(G). En particulier, A(G) admet une unique norme C * .

Received : 2016-05-23
Revised : 2016-10-01
Accepted : 2016-12-08
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3127
Classification:  58B32,  46L65,  43A20,  16P90
Keywords: quantum group, topological dimension, polynomial growth, Fourier algebra
@article{AIF_2017__67_5_2003_0,
     author = {D'Andrea, Alessandro and Pinzari, Claudia and Rossi, Stefano},
     title = {Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2003-2027},
     doi = {10.5802/aif.3127},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_5_2003_0}
}
D’Andrea, Alessandro; Pinzari, Claudia; Rossi, Stefano. Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2003-2027. doi : 10.5802/aif.3127. http://www.numdam.org/item/AIF_2017__67_5_2003_0/

[1] Baaj, Saad; Skandalis, Georges Unitaires multiplicatifs et dualité pour les produits croisés de C * –algèbres, Ann. Sci. Éc. Norm. Supér., Tome 26 (1993) no. 4, pp. 425-488 | Article | Zbl 0804.46078

[2] Banica, Teodor Representations of compact quantum groups and subfactors, J. Reine Angew. Math., Tome 509 (1999), pp. 167-198 | Article | Zbl 0957.46038

[3] Banica, Teodor; Vergnioux, Roland Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Tome 12 (2009) no. 2, pp. 321-340 | Article | Zbl 1189.46059

[4] Barnes, Bruce A. Ideal and representation theory of the L 1 –algebra of a group with polynomial growth, Colloq. Math., Tome 45 (1981), pp. 301-315 | Article | Zbl 0497.43001

[5] Barnes, Bruce A. The properties * –regularity and uniqueness of * –norm in a general * –algebra, Trans. Am. Math. Soc., Tome 279 (1983), pp. 841-859 | Zbl 0532.46031

[6] Bichon, Julien; De Rijdt, An; Vaes, Stefaan Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Commun. Math. Phys., Tome 262 (2006) no. 3, pp. 703-728 | Article | Zbl 1122.46046

[7] Boidol, Joachim Group algebras with a unique C * –norm, J. Funct. Anal., Tome 55 (1984), pp. 220-232 | Article | Zbl 0529.22004

[8] Boidol, Joachim; Leptin, Horst; Schürman, Jürgen; Vahle, D. Räume primitiver Ideale in Gruppenalgebren, Math. Ann., Tome 236 (1978), pp. 1-13 | Article | Zbl 0363.46057

[9] Bröcker, Theodor; Tom Dieck, Tammo Representations of compact Lie groups, Springer, Graduate Texts in Mathematics, Tome 98 (1985), x+313 pages | Zbl 0581.22009

[10] Caspers, Martijn The L p –Fourier transform on locally compact quantum groups, J. Oper. Theory, Tome 69 (2013) no. 1, pp. 161-193 | Article | Zbl 1313.43003

[11] Daws, Matthew Multipliers of locally compact quantum groups via Hilbert C * –modules, J. Lond. Math. Soc., Tome 84 (2011) no. 2, pp. 385-407 | Article | Zbl 1235.43004

[12] Dixmier, Jacques Opérateurs de rang fini dans les représentation unitaires, Publ. Math., Inst. Hautes Étud. Sci., Tome 6 (1960), pp. 13-25 | Article | Zbl 0100.32303

[13] Eymard, Pierre L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. Fr., Tome 92 (1964), pp. 181-236 | Article | Zbl 0169.46403

[14] Gelfand, Israel Moiseevich; Kirillov, Alexandre Aleksandrovich Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math., Inst. Hautes Étud. Sci., Tome 31 (1966), pp. 506-523 | Zbl 0144.02104

[15] Grigorchuk, Rostislav; Musat, Magdalena; Rørdam, Mikael Just-infinite C * –algebras (2016) (https://arxiv.org/abs/1604.08774v1 )

[16] Hewitt, Edwin; Ross, Kenneth A. Abstract harmonic analysis: Volume II: Structure and Analysis for Compact Groups Analysis on Locally Compact Abelian Groups, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 152 (1994), viii+771 pages | Zbl 0830.43001

[17] Hu, Zhiguo; Neufang, Matthias; Ruan, Zhong-Jin On topological centre problems and SIN quantum groups, J. Funct. Anal., Tome 257 (2009) no. 2, pp. 610-640 | Article | Zbl 1184.46047

[18] Hu, Zhiguo; Neufang, Matthias; Ruan, Zhong-Jin Multipliers on a new class of Banach algebras, locally compact quantum groups and topological centres, Proc. Lond. Math. Soc., Tome 100 (2010) no. 2, pp. 429-458 | Article | Zbl 1192.43002

[19] Kahng, Byung-Jay Fourier transform on locally compact quantum groups, J. Oper. Theory, Tome 64 (2010) no. 1, pp. 69-87 | Zbl 1224.46129

[20] Krause, Günter R.; Lenagan, Thomas H. Growth of algebras and Gelfand-Kirillov dimension, American Mathematical Society, Graduate studies in Mathematics, Tome 22 (2000), x+212 pages | Zbl 0957.16001

[21] Kustermans, Johan Locally compact quantum groups in the universal setting, Int. J. Math., Tome 12 (2001) no. 3, pp. 289-338 | Article | Zbl 1111.46311

[22] Leung, Chi-Wang; Ng, Chi-Keung Some permanence properties of C * –unique groups, J. Funct. Anal., Tome 210 (2004) no. 2, pp. 376-390 | Article | Zbl 1048.43002

[23] Leung, Chi-Wang; Ng, Chi-Keung Functional calculus and *–regularity of a class of Banach algebras, Proc. Am. Math. Soc., Tome 134 (2005), pp. 755-763 | Article

[24] Maes, Ann; Van Daele, Alfons Notes on compact quantum groups, Nieuw Arch. Wiskd., Tome 16 (1998) no. 1-2, pp. 73-112 | Zbl 0962.46054

[25] Neshveyev, Sergey; Tuset, Lars Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Commun. Math. Phys., Tome 312 (2012) no. 1, pp. 223-250 | Article | Zbl 1250.22017

[26] Neshveyev, Sergey; Tuset, Lars Compact quantum groups and their representation categories, Société Mathématique de France, Cours Spécialisés, Tome 20 (2014), iv+169 pages | Zbl 1316.46003

[27] Neshveyev, Sergey; Yamashita, Makoto Classification of non-Kac compact quantum groups of SU(n) type, Int. Math. Res. Not., Tome 2016 (2016) no. 11, pp. 3356-3391 | Article

[28] Pinzari, Claudia; Roberts, John E. A duality theorem for ergodic actions of compact quantum groups on C * –algebras, Commun. Math. Phys., Tome 277 (2008) no. 2, pp. 385-421 | Article | Zbl 1160.46045

[29] Podleś, Piotr Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU (2) and SO (3) groups, Commun. Math. Phys., Tome 170 (1995) no. 1, pp. 1-20 | Article | Zbl 0853.46074

[30] Takahashi, Shuichi Dimension of compact groups and their representations, Tôhoku Math. J., Tome 5 (1953), pp. 178-184 | Article | Zbl 0053.21104

[31] Vergnioux, Roland The property of rapid decay for discrete quantum groups, J. Oper. Theory, Tome 57 (2007) no. 2, pp. 303-324 | Zbl 1120.58004

[32] Wang, Shuzhou Free products of compact quantum groups, Commun. Math. Phys., Tome 167 (1995) no. 3, pp. 671-692 | Article | Zbl 0838.46057

[33] Wang, Simeng Lacunary Fourier series for compact quantum groups (2015) (https://arxiv.org/abs/1507.01219v2 )

[34] Wang, Simeng L p –Improving convolution operators on finite quantum groups (2015) (https://arxiv.org/abs/1412.2085v2 )

[35] Woronowicz, Stanisław Lech Compact matrix pseudogroups, Commun. Math. Phys., Tome 111 (1987), pp. 613-665 | Article | Zbl 0627.58034

[36] Woronowicz, Stanisław Lech Compact quantum groups, Symétries quantiques, North-Holland (1998), pp. 845-884 | Zbl 0997.46045

[37] Zhuang, Guangbin Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. Lond. Math. Soc., Tome 87 (2013) no. 3, pp. 877-898 | Article | Zbl 1365.16018