Quantum direct products and the Künneth class

Rubén Martos

doi:10.5802/ambp.424

Quantum direct products and the Künneth class
[Produits directs quantiques et la classe de Künneth]

Rubén Martos ¹

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

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

Résumé
Abstract

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.

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.

Publié le : 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

Affiliations des auteurs :

Rubén Martos ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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, Tome 31 (2024) no. 1, pp. 11-45. doi : 10.5802/ambp.424. https://ambp.centre-mersenne.org/articles/10.5802/ambp.424/

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