Torsion for quantum direct products and the K\"unneth class

Rub\'en Martos

Mar 2023

摘要

We classify torsion actions of quantum direct products. On the one hand, thisallows to improve several results appearing already in a previous work of theauthor around permanence properties of the (resp. strong) quantum Baum-Connesproperty with respect to quantum direct products. On the other hand, weintroduce a K\"unneth class in the quantum equivariant setting inspired by thepioneer work by J. Chabert, H. Oyono-Oyono and S. Echterhoff, which allows torelate the quantum Baum-Connes property with the K\"unneth formula bygeneralising some key results of Chabert-Oyono-Oyono-Echterhoff to discretequantum groups. Finally, we make the observation that the C$^*$-algebradefining a compact quantum group with dual satisfying the strong quantumBaum-Connes property belongs to the K\"unneth class. This allows to obtain someK-theory computations for quantum direct products based on earlier work byVoigt and Vergnioux-Voigt.

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