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

2013/04/15 by Yuhjtman, Sergio Andrés
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1304.4260

Abstract

We show that for any C*-algebra A, a sufficiently large Hilbert space H and a unit vector ξ∈ H, the natural application rep(A:H) → Q(A), π↦ ⟨ π(-)ξ,ξ⟩ 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.

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• Topological description of pure invariant states of the Weyl C^*-algebra
• Invariant means and finite representation theory of C*-algebras