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  "body": "\n## Summary\n\nA positive element of a C*-algebra is a self-adjoint element whose [[spectrum|EH0CZ19RKP]] lies in $[0, \\infty)$.\nEquivalently, it has the form $b^* b$.\nPositive elements generalize nonnegative functions and [[positive semidefinite matrices|01MMVW8CKC]], and they define the order $a \\le b$ on self-adjoint elements.\nThey are the basis of the notions of [[states|F7MB2WREBJ]] and [[completely positive maps|CGY299QMC3]].\n\n## Explanations\n\n### L1 human\n\nIn the algebra of continuous functions, a positive element is a function that never takes a negative value.\nIn the algebra of matrices, it is a Hermitian matrix with no negative eigenvalue, like $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$, whose eigenvalues are $1$ and $3$.\nEvery element of the form $b^* b$ is positive, just as $\\abs{z}^2 = \\overline{z} z \\ge 0$ for a complex number $z$.\n",
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