  We show that for any C*-algebra $A$, a sufficiently large Hilbert space $H$
and a unit vector $\xi \in H$, the natural application $rep(A:H) \to Q(A)$,
$\pi \mapsto \langle \pi(-)\xi,\xi \rangle$ is a topological quotient, where
$rep(A:H)$ is the space of representations on $H$ and $Q(A)$ the set of
quasi-states, i.e. positive linear functionals with norm at most $1$. This
quotient might be a useful tool in the representation theory of C*-algebras. We
apply it to give an interesting proof of Takesaki-Bichteler duality for
C*-algebras which allows to drop a hypothesis.
