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

TL;DR

The paper demonstrates that the quasi-state space of a C*-algebra can be obtained as a topological quotient of its representation space, providing new insights into the algebra's structure and duality theory.

Contribution

It establishes that the map from the representation space to the quasi-state space is a topological quotient, simplifying the understanding of their relationship.

Findings

The quotient map is continuous and surjective.

This approach offers a new proof of Takesaki-Bichteler duality.

The method potentially simplifies the study of C*-algebra representations.

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.