  We show that the Baum-Connes morphism twisted by a non-unitary
representation, defined in [GA08], is an isomorphism for a large class of
groups satisfying the Baum-Connes conjecture. Such class contains all the real
semi-simple Lie groups, all hyperbolic groups and many infinite discret groups
having Kazhdan's property (T). We define a tensorisation by a non-unitary
finite dimensional representation on the left handside of the Baum-Connes
morphism and we show that its analogue in K-theory must be defined on the
K-theory of the twisted group algebras introduced in [GA07b].
