Gelfand-Naimark-Segal construction
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.
Referenced By
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