Stable group algebras

Question

Let $G$ be a discrete group and let $C^*_r(G)$ be its reduced group C*-algebra. Is there any group $G$ for which we have $C^*_r(G)\cong M_2(C^*_r(G))$? Or more generally, $C^*_r(G) \cong C^*_r(G)\otimes \mathcal{K}(\ell_2)$?

Answer 1

Note that for a discrete group $C^*_r(G)$ has a faithful tracial state. The compact operators cannot have such a state thus $C^*_r(G)$ cannot have the compact operators as a subalgebra. In fact using a little von Neumann algebra theory you can get that $C^*_r(G)$ cannot contain any compact operators at all.