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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] with identity $e$.\nA continuous function $\\varphi \\colon G \\to \\C$ is *positive-definite* if for every $n \\ge 1$, all $g_1, \\dots, g_n \\in G$ and all $c_1, \\dots, c_n \\in \\C$,\n$$\\sum_{i=1}^{n} \\sum_{j=1}^{n} \\overline{c_i} \\, c_j \\, \\varphi(g_i^{-1} g_j) \\ge 0.$$\n\n## Remarks\n\nEquivalently, each matrix $\\big(\\varphi(g_i^{-1}g_j)\\big)_{i,j=1}^{n}$ is Hermitian and [[positive semidefinite|0KX66CZMBB]].\nIt follows that $\\varphi(e) \\ge 0$, $\\varphi(g^{-1}) = \\overline{\\varphi(g)}$ and $\\abs{\\varphi(g)} \\le \\varphi(e)$ for all $g$.\nFor a discrete group, or when continuity is not part of the setting, the same condition defines positive-definite functions without the continuity requirement.\n\nExample: let $\\pi$ be a [[unitary representation|74BPYJNZTM]] of $G$ on $H$ and $v \\in H$, and put $\\varphi(g) = \\ip{\\pi(g)v}{v}$.\nThen $\\varphi$ is continuous, and\n$$\\sum_{i,j} \\overline{c_i} c_j \\ip{\\pi(g_i^{-1}g_j)v}{v} = \\sum_{i,j} \\overline{c_i} c_j \\ip{\\pi(g_j)v}{\\pi(g_i)v} = \\Big\\lVert \\sum_j c_j \\pi(g_j)v \\Big\\rVert^2 \\ge 0.$$\n\nNon-example: the constant function $\\varphi \\equiv -1$.\nWith $n = 1$ and $c_1 = 1$, the sum is $\\varphi(e) = -1 \u003c 0$.\n\nConversely, every continuous positive-definite function has the form $g \\mapsto \\ip{\\pi(g)v}{v}$ for some unitary representation $\\pi$ and a cyclic vector $v$, by a construction of Gelfand–Naimark–Segal type.\n",
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