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

Abstract

The well known “associativity property” of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete 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 the result [5] of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. TheK-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 Künneth formula, which is the quantum counterpart to the result [7] 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.