Baum-Connes conjecture for some semi-direct products

Let G be a locally compact topological group, and B be a G-algebra. P. Baum and A. Connes conjectured that the K-theory of the reduced crossed product of B by G is isomorphic to the C-equivariant K-homology with compact suppo rt and with coefficients in B of the classifying space for proper actions o f G. We consider the case when G = N x H is a semi-direct product group. We prov ide a way to link this conjecture for G to the similar statement for N by b uilding an H-equivariant assembly map for N. We then use this construction and the Dirac/dual-Dirac method to prove that the conjecture is true for G if it is true for N, under the following assumptions: G has a gamma-element in the sense of Kasparov, N is an amenable group and H is almost connected or has a compact-open subgroup.