Gelfand-Naimark-Segal construction

GivenC* algebra andpositive state over it, ρ, we can constructunitary representationit withspecial vectorit, Ω such that <Ω|A|Ω>=ρ(A)all elements A.

The constructiondone by takingquotient algebra overleft ideal given bysubalgebra satisfying ρ(A*A)=0then taking its Cauchy completion. The element corresponding tooperator 1 (if unital)Ω