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.