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  "body": "\n## Definition\n\nLet $A$ be a unital C*-algebra and let $\\rho$ be a [[state|F7MB2WREBJ]] on $A$, a positive linear functional with $\\rho(1) = 1$.\nThe *GNS construction* for $\\rho$ is the triple $(H_\\rho, \\pi_\\rho, \\xi_\\rho)$ defined as follows.\n\n1. The set $N_\\rho = \\set{a \\in A : \\rho(a^*a) = 0}$ is a closed left ideal of $A$, and $\\ip{a + N_\\rho}{b + N_\\rho} = \\rho(b^*a)$ is an inner product on $A/N_\\rho$.\n2. $H_\\rho$ is the [[Hilbert space|HM2HHE3J51]] completion of $A/N_\\rho$ for this inner product.\n3. For $a \\in A$, $\\pi_\\rho(a)$ is the bounded operator on $H_\\rho$ that extends $b + N_\\rho \\mapsto ab + N_\\rho$.\n4. $\\xi_\\rho = 1 + N_\\rho$.\n\n## Remarks\n\nThen $\\pi_\\rho$ is a $*$-representation of $A$, the vector $\\xi_\\rho$ is cyclic, that is, $\\pi_\\rho(A)\\xi_\\rho$ is dense in $H_\\rho$, and $\\rho(a) = \\ip{\\pi_\\rho(a)\\xi_\\rho}{\\xi_\\rho}$ for all $a \\in A$.\nAny other cyclic representation with a cyclic vector $\\xi$ satisfying $\\rho(a) = \\ip{\\pi(a)\\xi}{\\xi}$ is unitarily equivalent to $\\pi_\\rho$ by a unitary sending $\\xi$ to $\\xi_\\rho$.\nThe boundedness in clause 3 comes from $\\rho(b^*a^*ab) \\le \\norm{a}^2 \\rho(b^*b)$.\n\nExample: $A = C[0, 1]$ and $\\rho(f) = \\int_0^1 f(x) \\, dx$.\nThen $N_\\rho = \\set{0}$, since $\\int_0^1 \\abs{f}^2 \\, dx = 0$ forces a continuous $f$ to vanish, and the inner product is $\\int_0^1 f \\overline{g} \\, dx$.\nSo $H_\\rho = L^2[0, 1]$, $\\pi_\\rho(f)$ is multiplication by $f$ and $\\xi_\\rho$ is the constant function $1$.\n\nNon-example: the functional $\\sigma(f) = -f(0)$ on $C[0, 1]$ is linear but not a state, and the construction fails at clause 1.\nThe form $\\ip{a}{b} = \\sigma(b^*a)$ gives $\\ip{1}{1} = -1 \u003c 0$, so it is not an inner product.\n\nFor a non-unital C\\*-algebra the construction works in the same way for positive functionals, with the cyclic vector obtained from an approximate unit.\n",
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  "name": "Gelfand–Naimark–Segal construction",
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