In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account of the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C*-algebras over $S^1$ admits faithful representations, and when the net is covariant under Diff$(S^1)$, it admits representations covariant under any amenable subgroup of Diff$(S^1)$.