Quantum direct products and the Künneth class

Rubén Martos

doi:10.5802/ambp.424

Rubén Martos ¹

¹ Department of Mathematical Sciences University of Copenhagen Universitetspark 5 2100 Copenhagen Ø Denmark

Annales mathématiques Blaise Pascal, Volume 31 (2024) no. 1, pp. 11-45.

Abstract
Résumé

We introduce a Künneth class in the quantum equivariant setting inspired by the pioneer work by J. Chabert, H. Oyono-Oyono and S. Echterhoff, which allows to relate the quantum Baum–Connes property with the Künneth formula by generalising some key results of Chabert–Oyono-Oyono–Echterhoff to discrete quantum groups. Finally, we make the observation that the $C^{*}$ -algebra defining a compact quantum group with dual satisfying the strong quantum Baum–Connes property belongs to the Künneth class. This allows to obtain some K-theory computations for quantum direct products based on earlier work by Voigt and Vergnioux–Voigt.

Nous introduisons une classe de Künneth dans le cadre quantique équivariant inspirés par les travaux pionniers de J. Chabert, H. Oyono-Oyono et S. Echterhoff ; qui permet de relier la propriété de Baum–Connes quantique à la formule de Künneth en généralisant certains résultats clés de Chabert–Oyono-Oyono–Echterhoff aux groupes quantiques discrets. Enfin, nous observons que la $C^{*}$ -algèbre définissant un groupe quantique compact dont le dual satisfait la propriété de Baum–Connes quantique forte appartient à la classe de Künneth. Ceci permet d’obtenir des calculs de K-théorie pour des produits directs quantiques basés sur des travaux antérieurs de Voigt et Vergnioux–Voigt.

Published online: 2024-10-23

DOI: 10.5802/ambp.424

Classification: 46L80, 19K35, 46L06, 22D25, 20G42, 18G15, 18E30
Keywords: Baum–Connes conjecture, compact/Discrete quantum groups, K-theory, Künneth formula, quantum direct product, tensor product of C$^*$-algebras, torsion, triangulated categories, universal coefficient theorem
Mot clés : Conjecture de Baum–Connes, groupes quantiques compacts/discrets, K-théorie, formule de Künneth, produits directs quantiques, produit tensoriel de C$^*$-algèbres, torsion, catégories triangulées, théorème universel des coefficients

Author's affiliations:

Rubén Martos ¹

¹ Department of Mathematical Sciences University of Copenhagen Universitetspark 5 2100 Copenhagen Ø Denmark

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Quantum direct products and the {K\"unneth} class},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {11--45},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Rubén Martos. Quantum direct products and the Künneth class. Annales mathématiques Blaise Pascal, Volume 31 (2024) no. 1, pp. 11-45. doi : 10.5802/ambp.424. https://ambp.centre-mersenne.org/articles/10.5802/ambp.424/

References
Cited by

[1] Yuki Arano; Kenny De Commer Torsion-freeness for fusion rings and tensor $C^{*}$ -categories, J. Noncommut. Geom., Volume 13 (2019) no. 1, pp. 35-58 | DOI | Zbl

[2] Yuki Arano; Adam Skalski On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture, Proc. Am. Math. Soc., Volume 149 (2021), pp. 5237-5254 | DOI | Zbl

[3] Saad Baaj; Georges Skandalis $C^{*}$ -algèbres de Hopf et théorie de Kasparov équivariante, $K$ -Theory, Volume 2 (1989) no. 6, pp. 683-721 | DOI | Zbl

[4] Bruce Blackadar $K$ -theory for operator algebras, Mathematical Sciences Research Institute Publications, 5, Cambridge University Press, 1998 | Zbl

[5] Jérôme Chabert; Hervé Oyono-Oyono; Siegfried Echterhoff Going-down functors, the Künneth formula, and the Baum–Connes conjecture, Geom. Funct. Anal., Volume 14 (2004) no. 3, pp. 491-528 | Zbl

[6] Matthew Daws; Paweł Kasprzak; Adam Skalski; Piotr M. Sołtan Closed quantum subgroups of locally compact quantum groups, Adv. Math., Volume 231 (2012) no. 6, pp. 3473-3501 | DOI | Zbl

[7] Kenny De Commer Actions of compact quantum groups, Topological quantum groups (Banach Center Publications), Volume 111, Polish Academy of Sciences, 2017, pp. 33-100 | Zbl

[8] Kenny De Commer; Rubén Martos; Ryszard Nest Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map (2021) (to appear in J. Noncommut. Geom.) | arXiv

[9] George A. Elliot On the classification of inductive limits of sequences of semisimple finitedimensional algebras, J. Algebra, Volume 38 (1976) no. 1, pp. 29-44 | DOI | Zbl

[10] Pierre Fima; Arthur Troupel Operator Algebras of Free Wreath Products, Adv. Math., Volume 441 (2024), 109546, 53 pages | Zbl

[11] Amaury Freslon; Rubén Martos Torsion and $K$ -theory for some free wreath products, Int. Math. Res. Not., Volume 2020 (2018) no. 6, pp. 1639-1670 | DOI | Zbl

[12] Édouard Goursat Sur les substitutions orthogonales et les divisions régulières de l’espace, Ann. Sci. Éc. Norm. Supér., Volume 3 (1889) no. 6, pp. 9-102 | DOI | Numdam | Zbl

[13] Nigel Higson A characterization of $K K$ -theory, Pac. J. Math., Volume 126 (1987) no. 2, pp. 253-276 | DOI | Zbl

[14] Nigel Higson; Gennadi Kasparov E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math., Volume 144 (2001) no. 1, pp. 23-74 | DOI | Zbl

[15] Triangulated Categories (Thorsten Holm; Peter Jorgensen; Raphaël Rouquier, eds.), London Mathematical Society Lecture Note Series, 375, Cambridge University Press, 2010 | DOI | Zbl

[16] Vincent Lafforgue La conjecture de Baum–Connes à coefficients pour les groupes hyperboliques, J. Noncommut. Geom., Volume 6 (2012) no. 1, pp. 1-197 | DOI | Zbl

[17] François Lemeux; Pierre Tarrago Free wreath product quantum groups: the monoidal category, approximation properties and free probability, J. Funct. Anal., Volume 270 (2016) no. 10, pp. 3828-3883 | DOI | Zbl

[18] Rubén Martos The Baum–Connes conjecture for Quantum Groups. Stability properties and $K$ -theory computations, Ph. D. Thesis, Université Paris VII (2018)

[19] Rubén Martos The Baum–Connes property for a quantum (semi-)direct product, J. Noncommut. Geom., Volume 13 (2020) no. 4, pp. 1295-1357 | DOI | Zbl

[20] Rubén Martos Permanence of the torsion-freeness property for divisible discrete quantum subgroups, Math. Scand., Volume 130 (2024) no. 2, pp. 257-299 | Zbl

[21] Ralf Meyer Homological algebra in bivariant K-theory and other triangulated categories II, Tbil. Math. J., Volume 1 (2008), p. 165--210 | Zbl

[22] Ralf Meyer; Ryszard Nest The Baum–Connes conjecture via localisation of categories, Topology, Volume 45 (2006) no. 2, pp. 209-259 | DOI | Zbl

[23] Ralf Meyer; Ryszard Nest An analogue of the Baum-Connes isomorphism for coactions of compact groups, Math. Scand., Volume 100 (2007) no. 2, pp. 301-316 | DOI | Zbl

[24] Ryszard Nest; Christian Voigt Equivariant Poincaré duality for quantum group actions, J. Funct. Anal., Volume 258 (2010) no. 5, pp. 1466-1503 | DOI | Zbl

[25] Dmitri Nikshych; Leonid Vainerman Finite quantum groupoids and their applications, New directions in Hopf algebras (Mathematical Sciences Research Institute Publications), Volume 43, Cambridge University Press, 2002, pp. 211-262 | Zbl

[26] Jonathan Rosenberg; Claude Schochet The Künneth theorem and the universal coefficient theorem for equivariant $K$ -theory and $K K$ -theory, Memoirs of the American Mathematical Society, 348, American Mathematical Society, 1986, 95 pages

[27] Ken-ichi Tahara On the Second Cohomology Groups of Semidirect Products, Math. Z., Volume 129 (1972), pp. 365-379 | DOI | Zbl

[28] Klaus Thomsen The universal property of equivariant $K K$ -theory, J. Reine Angew. Math., Volume 504 (1998), pp. 55-71 | DOI | Zbl

[29] Roland Vergnioux; Christian Voigt The $K$ -theory of free quantum groups, Math. Ann., Volume 357 (2013) no. 1, pp. 355-400 | DOI | Zbl

[30] Christian Voigt The Baum–Connes conjecture for free orthogonal quantum groups, Adv. Math., Volume 227 (2011) no. 5, pp. 1873-1913 | DOI | Zbl

[31] Christian Voigt On the structure of quantum automorphism groups, J. Reine Angew. Math., Volume 732 (2017), pp. 255-273 | DOI | Zbl

[32] Shuzhou Wang Tensor products and crossed products of compact quantum groups, Proc. Lond. Math. Soc., Volume 71 (1995), pp. 695-720 | DOI | Zbl

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