Topological boundaries of covariant representations

We associate a boundary $\mathcal B_{\pi ,u}$ to each covariant representation $(\pi ,u,H)$ of a $C^*$-dynamical system $(G,A,\alpha )$ and study the action of $G$ on $\mathcal B_{\pi ,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product $C^*$-algebra to faithfulness of the action of the group on this boundary.