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  "body": "\n## Summary\n\nThe Gelfand–Naimark–Segal (GNS) construction starts from a [[state|F7MB2WREBJ]] $\\rho$ on a C*-algebra $A$.\nIt builds a Hilbert space $H_\\rho$, a representation $\\pi_\\rho$ of $A$ on it and a cyclic vector $\\xi_\\rho$ with $\\rho(a) = \\ip{\\pi_\\rho(a)\\xi_\\rho}{\\xi_\\rho}$.\nIt shows that states and cyclic representations are two views of the same thing.\nApplied to enough states, it proves that every C\\*-algebra is isomorphic to an algebra of operators on a Hilbert space.\nThe same idea turns a positive-definite function on a group into a unitary representation.\n\n## Explanations\n\n### L1 human\n\nA state assigns a number to each element of an algebra, like the average value of an observable in physics.\nThe GNS construction uses the state to define an inner product on the algebra itself, by $\\ip{a}{b} = \\rho(b^* a)$.\nAfter removing elements of length zero and filling in limits, the algebra acts on this space by multiplication on the left.\nFor the algebra of continuous functions on $[0, 1]$ with the state \"integrate against $dx$\", the result is the space $L^2[0, 1]$, on which functions act by multiplication.\n",
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