Quantum direct products and the Künneth class

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi mathvariant="normal">C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:math> -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.