Unitary representation acting on the K-theory of the reduced group $C^*$-algebra

Question

Let $G$ be a group (usually infinite), $R$ a ring and $\rho: G \rightarrow Gl_n(\mathbb{Z})$ a finite-dimensional representation of $G$. Then we can define a functor from the category of projective $RG$-modules to itself by sending a projective module $P$ to $\mathbb{Z}^n \otimes_{\mathbb{Z}} P$ with the diagonal action of $G$, where $G$ acts on $\mathbb{Z}^n$ via $\rho$. Hence we get an induced map on the algebraic $K$-theory of $RG$.

Now let $\rho: G \rightarrow U_n(\mathbb{C})$ be a unitary representation of $G$. Does $\rho$ induce a map on the topological $K$-theory of the reduced group $C^*$-algebra $C_r^* G$?

Answer 1

Even more is true: denote by $R(G)$ the ring of homotopy classes of finite-dimensional representations of $G$ (not necessarily unitary ones); then there is a module action of $R(G)$ on $K_*(C_r^*G)$. See my memoir: ``Les fibr\'es en th\'eorie de Kasparov'', Acad. Royale de Belgique, M\'emoire Classe des Sciences, 2eme s\'erie, T. XLV, Fasc.6, 1988.