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 of Kac type implies –regularity of the Fourier algebra , that is every closed ideal of has a dense intersection with . In particular, has a unique –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 , implique la -régularité de l’algèbre de Fourier , c’est-à-dire que tout idéal fermé de a intersection dense avec . En particulier, admet une unique norme .
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Keywords: quantum group, topological dimension, polynomial growth, Fourier algebra
Mot clés : Groupe quantique, dimension topologique, croissance polynomiale, algèbre de Fourier
D’Andrea, Alessandro 1; Pinzari, Claudia 1; Rossi, Stefano 2
@article{AIF_2017__67_5_2003_0, author = {D{\textquoteright}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}, pages = {2003--2027}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {5}, year = {2017}, doi = {10.5802/aif.3127}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3127/} }
TY - JOUR AU - D’Andrea, Alessandro AU - Pinzari, Claudia AU - Rossi, Stefano TI - Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra JO - Annales de l'Institut Fourier PY - 2017 SP - 2003 EP - 2027 VL - 67 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3127/ DO - 10.5802/aif.3127 LA - en ID - AIF_2017__67_5_2003_0 ER -
%0 Journal Article %A D’Andrea, Alessandro %A Pinzari, Claudia %A Rossi, Stefano %T Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra %J Annales de l'Institut Fourier %D 2017 %P 2003-2027 %V 67 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3127/ %R 10.5802/aif.3127 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.3127/
[1] Unitaires multiplicatifs et dualité pour les produits croisés de –algèbres, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 4, pp. 425-488 | DOI | Zbl
[2] Representations of compact quantum groups and subfactors, J. Reine Angew. Math., Volume 509 (1999), pp. 167-198 | DOI | Zbl
[3] Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 12 (2009) no. 2, pp. 321-340 | DOI | Zbl
[4] Ideal and representation theory of the –algebra of a group with polynomial growth, Colloq. Math., Volume 45 (1981), pp. 301-315 | DOI | Zbl
[5] The properties –regularity and uniqueness of –norm in a general –algebra, Trans. Am. Math. Soc., Volume 279 (1983), pp. 841-859 | Zbl
[6] Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Commun. Math. Phys., Volume 262 (2006) no. 3, pp. 703-728 | DOI | Zbl
[7] Group algebras with a unique –norm, J. Funct. Anal., Volume 55 (1984), pp. 220-232 | DOI | Zbl
[8] Räume primitiver Ideale in Gruppenalgebren, Math. Ann., Volume 236 (1978), pp. 1-13 | DOI | Zbl
[9] Representations of compact Lie groups, Graduate Texts in Mathematics, 98, Springer, 1985, x+313 pages | Zbl
[10] The –Fourier transform on locally compact quantum groups, J. Oper. Theory, Volume 69 (2013) no. 1, pp. 161-193 | DOI | Zbl
[11] Multipliers of locally compact quantum groups via Hilbert –modules, J. Lond. Math. Soc., Volume 84 (2011) no. 2, pp. 385-407 | DOI | Zbl
[12] Opérateurs de rang fini dans les représentation unitaires, Publ. Math., Inst. Hautes Étud. Sci., Volume 6 (1960), pp. 13-25 | DOI | Zbl
[13] L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. Fr., Volume 92 (1964), pp. 181-236 | DOI | Zbl
[14] Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math., Inst. Hautes Étud. Sci., Volume 31 (1966), pp. 506-523 | Zbl
[15] Just-infinite –algebras (2016) (https://arxiv.org/abs/1604.08774v1)
[16] Abstract harmonic analysis: Volume II: Structure and Analysis for Compact Groups Analysis on Locally Compact Abelian Groups, Grundlehren der Mathematischen Wissenschaften, 152, Springer, 1994, viii+771 pages | Zbl
[17] On topological centre problems and SIN quantum groups, J. Funct. Anal., Volume 257 (2009) no. 2, pp. 610-640 | DOI | Zbl
[18] Multipliers on a new class of Banach algebras, locally compact quantum groups and topological centres, Proc. Lond. Math. Soc., Volume 100 (2010) no. 2, pp. 429-458 | DOI | Zbl
[19] Fourier transform on locally compact quantum groups, J. Oper. Theory, Volume 64 (2010) no. 1, pp. 69-87 | Zbl
[20] Growth of algebras and Gelfand-Kirillov dimension, Graduate studies in Mathematics, 22, American Mathematical Society, 2000, x+212 pages | Zbl
[21] Locally compact quantum groups in the universal setting, Int. J. Math., Volume 12 (2001) no. 3, pp. 289-338 | DOI | Zbl
[22] Some permanence properties of –unique groups, J. Funct. Anal., Volume 210 (2004) no. 2, pp. 376-390 | DOI | Zbl
[23] Functional calculus and –regularity of a class of Banach algebras, Proc. Am. Math. Soc., Volume 134 (2005), pp. 755-763 | DOI
[24] Notes on compact quantum groups, Nieuw Arch. Wiskd., Volume 16 (1998) no. 1-2, pp. 73-112 | Zbl
[25] Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Commun. Math. Phys., Volume 312 (2012) no. 1, pp. 223-250 | DOI | Zbl
[26] Compact quantum groups and their representation categories, Cours Spécialisés, 20, Société Mathématique de France, 2014, iv+169 pages | Zbl
[27] Classification of non-Kac compact quantum groups of type, Int. Math. Res. Not., Volume 2016 (2016) no. 11, pp. 3356-3391 | DOI
[28] A duality theorem for ergodic actions of compact quantum groups on –algebras, Commun. Math. Phys., Volume 277 (2008) no. 2, pp. 385-421 | DOI | Zbl
[29] Symmetries of quantum spaces. Subgroups and quotient spaces of quantum and groups, Commun. Math. Phys., Volume 170 (1995) no. 1, pp. 1-20 | DOI | Zbl
[30] Dimension of compact groups and their representations, Tôhoku Math. J., Volume 5 (1953), pp. 178-184 | DOI | Zbl
[31] The property of rapid decay for discrete quantum groups, J. Oper. Theory, Volume 57 (2007) no. 2, pp. 303-324 | Zbl
[32] Free products of compact quantum groups, Commun. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | DOI | Zbl
[33] Lacunary Fourier series for compact quantum groups (2015) (https://arxiv.org/abs/1507.01219v2)
[34] –Improving convolution operators on finite quantum groups (2015) (https://arxiv.org/abs/1412.2085v2)
[35] Compact matrix pseudogroups, Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | DOI | Zbl
[36] Compact quantum groups, Symétries quantiques (1998), pp. 845-884 | Zbl
[37] Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. Lond. Math. Soc., Volume 87 (2013) no. 3, pp. 877-898 | DOI | Zbl
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