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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 ofquasi-states, i.e. positive linear functionals with norm at most $1$. Thisquotient might be a useful tool in the representation theory of C*-algebras. Weapply it to give an interesting proof of Takesaki-Bichteler duality forC*-algebras which allows to drop a hypothesis.

The quasi-state space of a C*-algebra is a topological quotient of the representation space