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

Abstract

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.