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  "body": "\n## Definition\n\nLet $A$ be a C*-algebra.\nA linear functional $\\varphi \\colon A \\to \\C$ is *positive* if $\\varphi(a) \\ge 0$ for every [[positive element|NA8DF5X3M4]] $a \\in A$.\nA *state* on $A$ is a positive linear functional $\\varphi$ with $\\norm{\\varphi} = 1$.\n\n## Remarks\n\nA positive linear functional is automatically bounded, so it lies in the [[dual space|4A82S5FVRT]] $A^*$.\nIf $A$ has a unit, then $\\norm{\\varphi} = \\varphi(1)$ for positive $\\varphi$, so a state is a positive linear functional with $\\varphi(1) = 1$.\nThe set $S(A)$ of states of a unital $A$ is convex and weak-* compact; its [[extreme points|3MSGFJM2NX]] are the *pure states*.\n\nExample: let $H$ be a Hilbert space, $A$ the bounded operators on $H$, and $\\xi \\in H$ a unit vector.\nThe vector state $\\varphi(T) = \\ip{T \\xi}{\\xi}$ is linear, satisfies $\\varphi(T^* T) = \\norm{T \\xi}^2 \\ge 0$, and $\\varphi(I) = \\norm{\\xi}^2 = 1$; so it is a state.\nOn $M_n(\\C)$, the normalized trace $\\tfrac{1}{n} \\tr$ is a state.\n\nNon-example: the trace $\\tr$ on $M_2(\\C)$ is positive, but $\\tr(I) = 2$, so its norm is $2$ and it is not a state.\n",
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