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Gelfand-Naimark-Segal construction

Given a C* algebra and a positive state over it, ρ, we can construct a unitary representation of it with a special vector in it, Ω such that <Ω|A|Ω>=ρ(A) for all elements A in .

The construction is done by taking the quotient algebra of over the left ideal given by the subalgebra of satisfying ρ(A*A)=0 and then taking its Cauchy completion. The element corresponding to the operator 1 (if is unital) is Ω