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  "body": "\n## Definition\n\nLet $A$ be a C*-algebra.\nFor $a \\in A$, let $\\sigma(a)$ be its [[spectrum|EH0CZ19RKP]], computed in the [[unitization|7SB03M850M]] of $A$ when $A$ has no unit.\nAn element $a \\in A$ is *positive*, written $a \\ge 0$, if\n\n1. $a$ is self-adjoint, that is, $a^* = a$, and\n2. $\\sigma(a) \\subseteq [0, \\infty)$.\n\n## Remarks\n\nThe following are equivalent for $a \\in A$: $a$ is positive; $a = b^* b$ for some $b \\in A$; $a = c^2$ for some self-adjoint $c \\in A$.\nThe positive elements form a closed convex cone $A_+$ with $A_+ \\cap (-A_+) = \\set{0}$.\nFor self-adjoint $a, b$ one writes $a \\le b$ when $b - a \\ge 0$.\n\nExample: in $C(X)$ with $X$ compact Hausdorff, $\\sigma(f) = f(X)$, so $f \\ge 0$ exactly when $f$ is real-valued with $f(x) \\ge 0$ for all $x$.\nIn $M_n(\\C)$ the positive elements are the [[positive semidefinite matrices|01MMVW8CKC]].\n\nNon-example: $a = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ in $M_2(\\C)$ has $\\sigma(a) = \\set{0} \\subseteq [0, \\infty)$, but $a^* \\ne a$, so clause 1 fails.\nAlso $\\operatorname{diag}(1, -1)$ is self-adjoint but has $-1$ in its spectrum, so clause 2 fails.\n",
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