Gelfand–Naimark–Segal construction

Summary

The Gelfand–Naimark–Segal (GNS) construction starts from a state $\rho$ on a C*-algebra $A$. It builds a Hilbert space $H_\rho$, a representation $\pi_\rho$ of $A$ on it and a cyclic vector $\xi_\rho$ with $\rho(a) = \ip{\pi_\rho(a)\xi_\rho}{\xi_\rho}$. It shows that states and cyclic representations are two views of the same thing. Applied to enough states, it proves that every C*-algebra is isomorphic to an algebra of operators on a Hilbert space. The same idea turns a positive-definite function on a group into a unitary representation.