Topological boundaries of covariant representations

We associate a boundary ${\mathcal{B}}_{\pi ,u}$ to each covariant representation $(\pi ,u,H)$ of $C^{\ast }$-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 action of the group on this boundary.