Gelfand-Naimark-Segal construction

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In functional analysis, given a C* algebra and a state ρ on , we can construct a Hilbert space and a *-representation of on with a distinguished vector x in , with the property that

for every A in .

The construction is done by taking the quotient algebra of over the left ideal of consisting of elements A satisfying ρ(A*A)=0. is then taken to be the Cauchy completion of this quotient space, where we complete in the norm induced by the seminorm on . The element of corresponding to the identity operator 1 (if is unital) is x.

This construction is at the heart of the proof of the Gelfand-Naimark theorem.