Is there a locally compact group $G$ such that the canonical map from $C^{\*}(G)$ to $C^{\*}\_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{\*}$ algebras are isomorphic $C^{\*}$ algebras via another morphism? 



Can the full and reduced group $C^*$-algebras be "noncanonically" isomorphic?
If $C^\*(G)$ is isomorphic to $C^\*\_r(G)$, then $C^\*\_r(G)$ has a 1-dimensional representation, i.e. $G$ has a 1-dimensional representation $\chi$ weakly contained in the regular representation $\lambda\_G$. Then $1\_G=\chi\otimes\overline{\chi}$ is weakly contained in $\lambda\_G\otimes\overline{\lambda\_G}\simeq \infty.\lambda\_G$, hence $G$ is amenable.


