Strictly Amenable Representations of Reduced Group C* -Algebras

We show that a locally compact group $G$ is amenable if and only if its reduced group $C^* $-algebra $C^*_r(G)$ is nuclear and has a tracial state, which is also equivalent to $C^*_r(G)$ having a special kind of $^* $-representation that we propose to call “strictly amenable representation”. Consequently, if $G$ is separable and connected, then $G$ is amenable if and only if $C^*_r(G)$ has a tracial state.