The Quasi-state Space of a C∗-algebra Is a Topological Quotient of the Representation Space
نویسنده
چکیده
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) θξ −−→ Q(A), π 7→ 〈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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