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  "body": "\n## Summary\n\nA state on a C*-algebra is a positive linear functional of norm $1$.\nOn a unital C\\*-algebra this means a linear functional that takes nonnegative values on [[positive elements|NA8DF5X3M4]] and the value $1$ at the unit.\nStates play the role of probability measures.\nFor a compact Hausdorff space $X$, the states on $C(X)$ are the integrals against regular Borel probability measures on $X$.\nIn quantum mechanics, the states of a system with observable algebra $A$ are the states on $A$.\n\n## Explanations\n\n### L1 human\n\nA state assigns to each observable an expected value.\nFor a quantum system described by a unit vector $\\xi$, the state assigns to an operator $T$ the number $\\ip{T \\xi}{\\xi}$.\nFor a classical system on a space $X$, a state assigns to a function its average with respect to a probability distribution on $X$.\nThe states that cannot be written as mixtures of two other states are called pure states.\n",
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