Quantum direct products and the Künneth class

Martos, Rubén

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

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 ${\mathrm{C}}^{*}$-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 ${\mathrm{C}}^{*}$-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.

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
Mots-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

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