The Baum-Connes property for a quantum (semi-)direct product

The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete \emph{quantum} groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes a result of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The $K$-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena. The analogous strategy can be applied for the dual of a quantum direct product. In this case, we obtain, in addition, a connection with the \emph{K\"{u}nneth formula}, which is the quantum counterpart to a result of J. Chabert, S. Echterhoff and H. Oyono-Oyono. Again the $K$-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena.